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Australian and New Zealand Industrial and Applied Mathematics ANZIAM 2015 The 51st ANZIAM Conference Outrigger Surfers Paradise, 22 View Avenue Surfers Paradise QLD 1–5 February 2015 The abstracts of the talks in this volume were set individually by the authors. Only minor typographical changes have been made by the editors. The opinions, findings, conclusions and recommendations in this book are those of the individual authors. We are grateful to the organisers of the ANZIAM 2012 conference for providing the template files that they used for their book. Editors: Scott W. McCue and Matthew J. Simpson Email: [email protected] Web: http://anziam15.com// ISBN: 978-0-9942562-0-1 (softcover) ISBN: 978-0-9942562-1-8 (portable document format) 2 3 The Queensland Cyber Infrastructure Foundation (QCIF) QCIF operates QRIScloud, a Queensland-based large-scale cloud computing and data storage service. Part of a federally-funded, national network of research computing infrastructure, QRIScloud is designed to provide researchers with access to high-speed, high-capacity computing services. Using QRIScloud, researchers can: • Share datasets with collaborators worldwide • Manage and control access to their data collections • Leverage data collections stored in state and national nodes • Integrate access to Queensland-based HPC facilities and specialised cloud services • Access virtual labs with national communities • Launch on-demand computation • Host web services • Access and use a wide range of existing eResearch services, tools and applications QRIScloud is managed by QCIF and jointly operated through The University of Queensland and James Cook University. www.qriscloud.edu.au 4 5 1 Conference Details and History 1.1 Organising Committee • Matthew Simpson (Queensland University of Technology) – Co-Director • Scott McCue (Queensland University of Technology) – Co-Director • Peter van Heijster (Queensland University of Technology) – SIAM representative • Owen Jepps (Griffith University) • Peter Johnston (Griffith University) • Barbara Johnston (Griffith University) • Zoltan Neufeld (University of Queensland) • Graeme Pettet (University of Queensland) • Tony Roberts (University of Queensland) 1.2 Plenary Speakers Committee • Stan Miklavcic (University of South Australia) – Chair • Matthew Simpson (Queensland University of Technology) • Scott McCue (Queensland University of Technology) • Kate Miles-Smith (Monash University) • Georg Gottwald (University of Sydney) • Michael Plank (University of Canterbury) • Nigel Bean (University of Adelaide) • Yvonne Stokes (University of Adelaide) • Antoinette Tordesillas (University of Melbourne) 1.3 Plenary Speakers • Hugh Possingham (University of Queensland) • Thomas Witelski (Duke University) • Leah Edelstein-Keshet (University of British Columbia) • Anne Juel (University of Manchester) • Mary Myerscough (University of Sydney) • Michael Small (University of Western Australia) • Gary Froyland (University of New South Wales) • Kerry Landman (University of Melbourne) – 2014 ANZIAM Medallist • Ngamta Thamwattana (University of Wollongong) – 2014 J.H. Michell Medallist 6 1.4 Past Conference Locations ,2015 Rot or ua2014 1966 Kangaroo Island 1967 Adelaide 1968 Hall’s Gap 1969 Victor Harbor 1970 Lorne 1971 Smiggin’s Hole 1972 Wollongong 1973 Surfers Paradise 1974 Lorne 1975 Tanunda 1976 Jindabyne 1977 Terrigal 1978 Broadbeach 1979 Leura 1980 Cowes 1981 Victor Harbor 1982 Bundanoon (August) 1966 Coorong (December) 1999 Mollymook 1983 Perth 2000 Waitangi 1984 Merimbula 2001 Barossa Valley 1985 Launceston 2002 Canberra 1986 Wirrina 2003 Sydney 1987 Wairakei 2004 Hobart 1988 Leura 2005 Napier 1989 Ballarat 2006 Mansfield 1990 Coolangatta 2007 Fremantle 1991 Hanmer Springs 2008 Katoomba 1992 Bateman’s Bay 2009 Caloundra 1993 Hahndorf 2010 Queenstown 1994 Pokolbin 2011 Glenelg 1995 Busselton 2012 Warrnambool 1996 Masterton 2013 Newcastle 1997 Lorne 2014 Rotorua 1998 Coolangatta 7 1.5 The T.M. Cherry Student Prize A student prize was introduced in 1969 at Victor Harbor and is awarded annually for the best student talk presented at the conference. In May 1976, ANZIAM (then the Division of Applied Mathematics) adopted the title “T.M. Cherry Student Prize” in honour of one of Australia’s leading scientists, Professor Sir Thomas MacFarland Cherry. Past recipients are listed below. 1969 1970 1971 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 R. Jones J. Rickard J. Jones R. P. Oertel R. E. Robinson J. P. Abbott J. Finnigan S. Bhaskaran B. Hughes P. Robinson J. R. Coleby B. Hughes M. Lukas A. Plank G. Fulford J. Gear P. Kovesi A. Kucera S. Wright G. Fulford F. Murrell A. Becker K. Thalassoudis M. Rumsewicz W. Henry M. Myerscough J. Roberts University of Adelaide University College London Mount Stromlo University of Adelaide University of Sydney Australian National University CSIRO University of Adelaide Australian National University University of Queensland University of Adelaide Australian National University Australian National University University of New South Wales University of Wollongong University of Melbourne University of Western Australia University of Wollongong University of Queensland University of Wollongong University of Melbourne Monash University University of Adelaide University of Adelaide Australian National University University of Oxford University of Melbourne 8 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 J. Best S. K. Lucas A. Tordesillas S. F. Brown D. Standingford B. Barnes A. Buryak A. Gore D. Scullen S. Cummins J. Clark T. Gourlay E. Ostrovskaya C. Reid M. Haese V. Gubernov W. Megill K. Mustapha J. Looker C. Fricke S. Harper E. Button M. Haythorpe S. Cohen L. Mitchell S. Butler J. Caffrey J. Nassios D. Khoury T. Vo M. Chan D. Khoury University of Wollongong University of Sydney University of Wollongong University of Sydney University of Adelaide Monash University Australian National University University of Newcastle University of Adelaide Monash University University of Sydney University of Adelaide Australian National University Massey University University of Adelaide Australian Defence Force Academy University of British Columbia/University of Wollongong University of New South Wales University of Melbourne University of Melbourne Massey University University of Melbourne University of South Australia University of Adelaide University of Sydney University of Sydney University of Melbourne University of Melbourne University of New South Wales University of Sydney University of Sydney University of New South Wales 9 1.6 The Cherry Ripe Prize Since 1995 the students have run an alternative competition for the best non-student talk. The past recipients are listed below. 1995 1996 1997 1998 1999 2000 2001 2002 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Natashia Boland Andrew Pullan Neville de Mestre David Stump Mark McGuinness Joseph Monaghan Andy Philpott Phil Broadbridge Ernie Tuck Larry Forbes Stephen Lucas Kerry Landman Vicky Mak James Sneyd Geoffry Mercer Neville de Mestre Philip Maini Larry Forbes Larry Forbes Darren Crowdy Martin Wechselberger Scott McCue Sheehan Olver Peter Kim University of Melbourne University of Auckland Bond University University of Queensland Victoria University of Wellington Monash University University of Auckland University of Wollongong University of Adelaide University of Tasmania University of South Australia University of Melbourne Deakin University University of Auckland University of New South Wales Bond University University of Oxford University of Tasmania University of Tasmania Imperial College, London University of Sydney Queensland University of Technology University of Sydney University of Sydney 10 1.7 The J.H. Michell Medal The J.H. Michell Medal is awarded to outstanding new researchers who have carried out distinguished research in applied or industrial mathematics, where a significant proportion of the research work has been carried out in Australia or New Zealand. The past recipients are listed below. 1999 2000 2001 2002 2004 2006 2007 2008 2009 2011 2012 2013 2014 1.8 Harvinder Sidhu Antoinette Tordesillas Nigel Bean Stephen Lucas Mark Nelson Sanjeeva Balasuriya Yvonne Stokes Carlo Laing Scott McCue Frances Kuo Matthew Simpson Terence O’Kane Ngamta Thamwattana University of New South Wales University of Melbourne University of Adelaide University of South Australia University of Wollongong University of Sydney University of Adelaide Massey University Queensland University of Technology University of New South Wales Queensland University of Technology CSIRO, Marine and Atmospheric Research University of Wollongong The E.O. Tuck Medal In honour of the late Ernest Oliver Tuck, FAustMS, FTSE and FAA, ANZIAM has instituted a midcareer award for outstanding research and distinguished service to the field of Applied Mathematics. The inaugural EO Tuck Medals were presented at ANZIAM 2013. 2013 Shaun Hendy Geoffry Mercer Victoria University of Wellington and Callaghan Innovation Australian National University 11 1.9 The ANZIAM Medal The ANZIAM Medal is awarded on the basis of research achievements or activities enhancing applied or industrial mathematics and contributions to ANZIAM. The first award was made in 1995. The past recipients are listed below. 1995 1997 1999 2001 2004 2006 2008 2010 2012 2014 1.10 Renfrey Potts Ian Sloan Ernie Tuck Charles Pearce Roger Grimshaw Graeme Wake James Hill Bob Anderssen Robert McKibbin Kerry Landman University of Adelaide University of New South Wales University of Adelaide University of Adelaide Loughborough University Massey University University of Wollongong CSIRO Massey University University of Melbourne The AF Pillow Applied Mathematics Top-up Scholarship The AF Pillow Applied Mathematics Trust offers an annual ‘top-up’ scholarship to a student holding either an Australian Postgraduate Award (APA) or equivalent award for full-time research in Applied Mathematics leading to the award of a PhD. The aim of the AF Pillow Applied Mathematics Top-up Scholarship is to increase the quality of postgraduate students in the field of applied mathematics in Australia. The past recipients are listed below. 2009 2010 2011 2012 2013 2014 Christopher Lustri Alex Badran Michael Dallaston Hayden Tronnolone Lisa Mayo Audrey Markowskei Queensland University of Technology University of Wollongong Queensland University of Technology University of Adelaide Queensland University of Technology Macquarie University 12 1.11 Acknowledgements The Organising Committee gratefully acknowledges the financial support of the Mathematical Science School at Queensland University of Technology (QUT), Hearne Scientific Software, the Queensland Cyber Infrastructure Foundation (QCIF), Pearson, and Taylor & Francis. The Organising Committee is especially thankful to CSIRO for sponsoring the following students to attend the ANZIAM 2015 conference: David Arnold Rachelle Binny Jesse Collis Saber Dini Adam Ellery Ashish Goyal David Harman Andrew Holder Wang Jin Daniel Ladiges Michael McCullough Nicholas Read Konstantinos Sakellariou David Skene Minh Tran Ada Wing Chi Yan Andrea Babylon Chen Chen Eamon Conway Carson Drummond Soorena Ezzati Adrian Grantham Xinjiang He Hamidul Islam Stuart Johnston Guiyan Ma Ellen Muir Nicolas Rebuli Kate Saunders Mingmei Teo Hayden Tronnolone Lucas Yiew 13 Peter Ballard Luigi Cirocco Heather Davidson Tom Dyer Megan Farquhar Rachael Griffiths Alexandra Hogan Michael Jackson Laura Karantgis Karen McCulloch Ravindra Pethiayagoda James Reoch Shrupa Shah Jakub Tomczyk James Walker Ayham Zaitouny 14 Conference venue Conference Events, Venues and Facilities Level 2 The Outrigger floor plan for Level 2 and 4. Level 4 The conference is being held at Outrigger Surfers Paradise, 22 View Avenue Surfers Paradise QLD. 2.1 2 2.2 Conference Reception A Welcome reception will be held from 6-8pm on Sunday 1 February at the Level 4 Palm Prefunction area. All registered delegates and their accompanying guests are invited. 2.3 One Day Workshop A one-day workshop on Discrete Mathematical Models in the Life Sciences will held in the new Science and Engineering Centre at Queensland University of Technology, Brisbane, on Friday 6 February 2015. The Plenary Speakers for the one-day workshop are Leah Edelstein-Keshet, Kerry Landman and Michael Small, who are also the invited speakers for ANZIAM 2015. 2.4 Conference Banquet The banquet dinner will be held at SkyPoint, with pre-dinner drinks from 6:30pm on 4 February 2015 at the venue. SkyPoint is situated on level 77 of the iconic Q1 Building, Surfers Paradise Boulevard, Surfers Paradise. It is just a short 6 minute walk from central Surfers Paradise and FREE undercover parking is available for visitors via Hamilton Avenue, disabled parking is available. Public parking is available nearby at Centro Surfers Paradise in Hanlan St or Bruce Bishop car park on Beach Rd. 2.5 Refreshment Breaks and Lunches Morning and afternoon tea and light refreshments will be available in the Pre-Function area outside Boulevard 2. Lunches are included in the registration fee for delegates and their registered guests. They will be available after the last presentations of the morning sessions at Deja View Restaurant. 15 2.6 Women in Mathematics Lunch The Women in Mathematics Special Interest Group is running this special lunch, which will be held at the Level 4 Palm pre-function area at lunchtime on Tuesday 3 February. The lunch will be hosted by Dr Joanne Hall and supported by Prof. Nalini Joshi’s Georgina Sweet Australian Laureate Fellowship. Delegates will receive an email message with a request to RSVP before the conference. 2.7 Internet Access Delegates will be provided internet access from Monday–Thursday. Instructions will be provided at the registration desk. 2.8 Plenary Lectures and Contributed Talks All invited plenary lectures will take place in Boulevard 1–2. Contributed talks will be held in five parallel sessions in Boulevard 1, 2, 3 and Palm 1, 2. The duration of each contributed talk will be fifteen minutes with an additional five minutes for questions and room changes. 16 17 10:40–11:00 10:20–10:40 10:00–10:20 9:40–10:00 8:00–8:15 8:15–8:30 8:30–9:30 Boulevard 2 * denotes student talk. Boulevard 1 Palm 2 Registration at the ANZIAM desk, Pre-function area Conference Opening, Boulevard 1–2 Plenary: Prof. Hugh Possingham, University of Queensland Title: Formulating and solving biodiversity conservation problems Chair: Scott McCue Chair: Ed Green Chair: Rosyln Hickson Chair: Graeme Hocking Chair: Shev MacNamara Graeme Wake Mod- Michael Plank What’s Larry Forbes What is Jerome Droniou A hyelling Growth Variability the catch? Fluid Turbulence? brid higher-order numeriin Cell Populations cal scheme for convectiondiffusion problems Ali A. Zaidi* Solutions Peter Johnston Aggres- Michael Page Singular- Dylan Lusmore* Using to an advanced functional sion Model for Wolbachia ities in diffusion-driven a biharmonic equation to partial differential equa- Flies flows extend velocity fields in tion of the pantographlevel set methods, with type applications to melanoma tumour growth Bruce van Brunt A Cell Rebecca Turner* De- Jim Denier The un- Megan Farquhar* Growth Model Adapted veloping a Model of Bird steady flow due to a spin- GPU accelerated algofor Minimum Cell Size Di- Navigation ning toroidal mass rithms for computing vision matrix function vector products student Morning tea Boulevard 3 Monday morning Ellen Muir* A mechanism design approach to efficient dynamic market clearing Chair: Andrew Eberhard Ian Sloan The ANOVA decomposition of a nonsmooth function of an infinite number of variables Carson Drummond* Making Waves: High Frequency Volatility Estimation and the Hilbert-Huang Transform Palm 1 18 12:55–1:50 12:10–12:55 11:40–12:00 11:20–11:40 11:00–11:20 Chair: Josh Ross John Hearne Mobile kangaroo to sedentary Abalone - what scale to manage? Meksianis Ndii* The effects of Wolbachia on dengue transmission dynamics Chair: Kerry Landman Rachelle Binny* Defining Moments: Spatial Structure in a Model of Collective Cell Movement Catherine Penington Dying in order: how crowding affects particle lifetimes * denotes student talk. Sergey Suslov Nonlinear thermomagnetic instabilities in a vertical layer of a ferromagnetic fluid Md. Habibur Rahman* Effects of oblique magnetic field on mixed ferrofluid convection Chair: Yvonne Stokes Jesse Collis* The Physics of Suspended Microchannel Resonators Boulevard 1 Chair: Tim Moroney Frank de Hoog Applications of Compressive Sensing Palm 2 Minh Tran* A New Approach For Solving A Sparse Linear System With Periodic Boundary Conditions Stuart Johnston* How Andrea Babylon* Josh Chopin* The inmuch information can be Modelling Leptospirosis fluence of object shape on obtained from tracking in Livestock and Wildlife the convergence of active the position of the leading contour models for image edge in a scratch assay? segmentation student Plenary: A/Prof. Natalie Thamwattana (2014 J.H. Michell Medallist), University of Wollongong Title: Mathematical modelling in nanotechnology Chair: Mary Myerscough Lunch at Deja View Restaurant Boulevard 2 Boulevard 3 Monday morning continued Chair: Jerzy Filar Jin Liang A Free Boundary Problem for Corporate Bond with Credit Rating Migration Xin-Jiang He* A new closed-form formula for pricing European options under a skew Brownian motion Guiyuan Ma* Pricing European options written on a hard to borrow stock Palm 1 19 4:00–4:20 David Harman* Applying Polynomial Chaos to Epidemic Models Pascal Buenzli Curvature effects in the evolution of bone tissues during bone remodelling 3:40–4:00 3:20–3:40 3:00–3:20 Plenary: Prof. Tom Witelski, Duke University Title: Multiscale dynamics of dewetting fluid films Chair: Kerry Landman Chair: Zoltan Neufeld Chair: Vivien Kirk Bruce Gardiner Ayham Zaitouny* Achilles tendon turnover Tracking and Predicting and adaptive remodelling Multiple Object Dynamics in a Complex Environment (Animal’s Behaviour) Edward Green Mathe- Matthew Chan* Mathmatical models for cell- ematical Modelling of extracellular matrix inter- Spatial Sorting and Evoactions in tissue develop- lution in a Host-Parasite ment System Boulevard 2 1:50–2:50 Boulevard 3 Afternoon tea Andrew Cramer* Microstructure Interpolation for Macroscopic Design Adam Tunney* A new mode of instability in compressible boundarylayer flows Chair: John Knight Jason Cosgrove* Polar vortices on celestial bodies Boulevard 1 * denotes student talk. Monday afternoon Tim Moroney Preconditioned finite volume methods on non-uniform grids for one-dimensional fractional diffusion equations Harish Sankaranarayanan* Fractionalin-space partial differential equations on finite intervals, boundary conditions, and associated stochastic processes Chair: Barbara Johnston Christopher Angstmann From stochastic processes to numerical schemes for fractional DEs, and PDEs Palm 2 Rahela Abdul Rahim The Evaluation of Faculty Employments Policies Using Markov Chain Model Louis Bhim* A stochastic analysis approach to placing upper bounds on solutions to free boundary problems Chair: Ian Sloan Hongmei Zhang The numerical simulation of a fractional Black-Scholes model for European call Palm 1 20 5:50–7:20 7:40–8:40 5:20–5:40 5:00–5:20 4:40–5:00 4:20–4:40 Chair: Mick Roberts John Murray Agentbased modelling of hepatitis B virus infection and clearance Chair: Mike Plank Adam Ellery* Characterising transport through a crowded environment with different obstacle sizes Saber Dini* Quantifying spatial distributions using a pair correlation function based on generalized measures of separation Ali Eshragh The Complexity of Optimal Experimental Design: A Tour from Applied Probability to Experimental Mathematics James Nichols Modelling reaction-diffusion systems with anomalous diffusion using a discrete time random walk, with examples in modelling of HIV * denotes student talk. Mike Chen Drawing of microstuctured optical fibres with pressurisation of the internal channels Yvonne Stokes The (un)importance of the temperature gradient in fibre drawing Hayden Tronnolone* Extruding Complicated Fluid Structures Chair: Larry Forbes Bronwyn Hajek Stretching viscous threads Boulevard 1 Barbara Johnston A monomial transformation for evaluating two-dimensional nearly singular boundary element integrals Hao Wang* Analytical and numerical solutions of the multi-term timespace fractionaldiffusion equations with a fractional Laplacian operator Shev MacNamara The wave equation is Toeplitz plus Hankel Chair: Peter Johnston Silvestru Dragomir Accurate Approximations of the Riemann-Stieltjes Integral Palm 2 Mathematics-in-Industry Special Interest Meeting, Boulevard 1 Mathematics-in-Biology Special Interest Meeting, Boulevard 1 Shrupa Shah* An Individual-based model approach to analyse the spatio-temporal dynamics of Influenza in Melbourne Ada Yan* Modelling the role of innate and adaptive immune responses in controlling influenza infection Ashish Goyal* Impact of delta hepatitis on hepatitis B epidemiology and optimal intervention policies Boulevard 2 Boulevard 3 Monday afternoon continued Azam Asanjarani* Relations Between the Markovian Transition Counting Process and the Markov Modulated Poisson Process Jerzy Filar Patient Flows and Markov Decision Processes Jeffrey Hunter A comparison of computational techniques of the key properties of Markov Chains Chair: Dion O’Neale Peter Taylor How old is this bird? Palm 1 21 10:40–11:00 Steve Taylor On solutions of a functional PDE for cell growth and division Andrew Black Modelling the evolution of unito multi-cellular life 10:00–10:20 10:20–10:40 Matthew Simpson Do pioneer cells exist? 9:40–10:00 8:30–9:30 Alexandra Hogan* Exploring bifurcations and seasonality in a mathematical model of childhood disease Mingmei Teo* Optimal vaccine allocation for structured populations James McCaw Exploring long-term drivers of pertussis resurgence and improved vaccine control strategies Morning tea David Arnold* Thinfilm flow in helical channels Lisa Mayo* The effect of surface wettability on droplet dynamics Sue Ann Chen Osmotically driven deformation of a stable water film Boulevard 3 Boulevard 2 Boulevard 1 Plenary: Prof. Leah Edelstein-Keshet, University of British Columbia Title: Models of cell polarization and motility Chair: Ed Green Chair: Bruce Gardiner Chair: Owen Jepps Chair: John Sader * denotes student talk. Tuesday morning Adrian Grantham* Bootstrapping methods for prediction intervals for solar radiation forecasts Rachael Griffiths* Nonparametric comparison of regression surfaces to assess the impacts of vegetation re-growth on wind fields Chair: Winston Sweatman John Boland Spatialtemporal forecasting of solar radiation Palm 2 Michelle Dunbar Improving Public Transport Accessibility via the Optimisation and Synchronisation of Schedules for Key Transport Modes Soorena Ezzati* An Investigation into Probabilistic Constraint in Optimal Power Flow Model Philipp Braun* A Distributed Optimization Algorithm with an Application to a Smart Grid Chair: Mark Fackrell Palm 1 22 1:50–7:30 7:30–9:30 9:30–11:30 12:50–1:50 Michael Jackson* The Saffman-Taylor instability with a finite amount of viscous fluid Scott McCue Selecting the appropriate SaffmanTaylor finger Boulevard 1 Chair: Michael Page Palm 2 Chair: Silvestru Sever Dragomir Dougal McQueen* Wind power simulation using Correlated Innovation Matrix and Wavelet Multi-resolution Analysis approaches Luigi Cirocco* Optimising Revenue for Concentrating Solar Thermal Power Plants with Limited Thermal Energy Storage Matthew Tam* Reflection methods for Euclidean distance matrix reconstruction Vivien Challis Can we optimise the strength of porous materials? Palm 1 Chair: Julia Piantadosi Women in Mathematics Lunch at the Level 4 Palm pre-function area Chair: Joanne Hall Lesley Ward and Nalini Joshi Free time ANZIAM AGM, Boulevard 1 ANZIAM Executive Meeting, Boulevard 1 Nicolas Rebuli* Hybrid Markov chain models for disease dynamics James Walker* Inference Methods for First Few Hundred Studies Boulevard 2 Chair: Jeffrey Hunter Plenary: Prof. Anne Juel, University of Manchester Title: Interfacial instabilities on the pore scale Chair: Larry Forbes Normal lunch at Deja View Restaurant for delegates Andrew Holder* The interaction of acidic tumours and chemotherapy 11:20–11:40 11:50–12:50 Wang Jin* Incorporating the effects of chemotherapeutic drugs into a multiphase model of cancer spheroid growth 11:00–11:20 Boulevard 3 Chair: James Osborne Tuesday morning continued 23 10:40–11:00 10:20–10:40 10:00–10:20 9:40–10:00 8:30–9:30 (* denotes student talk) Boulevard 3 Boulevard 2 Boulevard 1 Palm 2 Plenary: A/Prof. Mary Myerscough, University of Sydney Title: Modelling atherosclerotic plaque formation: Boundaries, balances and bifurcations Chair: Nalini Joshi Chair: Harvi Sidhu Chair: Carlo Laing Chair: Noel Barton Chair: Troy Farrell Jason Sharples Mod- Peter van Heijster A Duncan Farrow Peri- Mark Nelson Biogas elling the intrinsic dy- geometric approach to odically forced circulation production in anaerobic namics of bushfire propa- stationary defect solu- near the shore of a lake bioreactors gation using plane curva- tions in one space dimenture flow sion Claire Miller Spark - a Chen Chen* Macroscale Heather Davidson* Eamon Conway* new research tool for in- model and boundary con- Geothermal Spring Mathematical modelling vestigating novel bushfire ditions for spring mass Temperature Analysis and numerical simulation spread concepts system with fine structure of nanopores Nicholas Read* The Lotte Sewalt* A ge- Emma Greenbank* Tony Miller Efficient Probability of Bushfire ometric construction of Modelling Surtseyan and robust iterative soIgnition shock waves in a tumour Ejecta lutions of the potential growth model, incorpoequation applied to modrating the Allee effect elling of electrochemical electrodes Morning tea Wednesday morning Dale Ward DES simulation for modelling patient congestion within a SA metropolitan hospital Alexander Gilbert* A modification to the component-bycomponent construction that simultaneously chooses the weights Chair: Peter Taylor Mark Fackrell Modeling the care pathway for stroke patients Palm 1 24 Adrian Noppe* Modeling wound closure in an epethelial cell sheet using the Cellular Potts Model 12:00–12:20 11:40–12:00 Yan Ding Mathematical modelling of atherosclerosis - atheoma plaque formation, development and rupture Md Hamidul Islam* Numerical Continuation of Equilibria of an Atherosclerosis Model Christopher Kellett Activating Lyapunovbased Feedback Design: Nonsmoothness and State Constraint Kerry-Lyn Roberts* Folded Saddle-Node Bifurcations Vivien Kirk Dynamics of systems with three timescales Chair: Peter van Heijster Cris Hasan* MixedMode Oscillations and Canard orbits in Chemical Oscillators Chair: Mat Simpson Francis Woodhouse Long-term stochastic modelling to predict and prevent osteoarthritis 11:20–11:40 11:00–11:20 Boulevard 2 Boulevard 3 David Skene* Modelling Overwash on Ice Floes by Water Waves Jeong Ryeol Choi Analysis of novel oscillations of quantized mechanical energy in mass-accreting nanooscillator systems Winston Sweatman Symmetric 4-body motions Chair: Vivien Challis Tom Dyer* Modelling of graphene oxide and carbon nanotubes in a nematic liquid crystal using continuum mechanics Steve Walters* The flux paradox in gravitational lensing Palm 2 (* denotes student talk) Chair: Tim Marchant Tony Roberts Highorder evolution PDEs model nonlinear dispersive waves over large scales Ravindra Pethiyagoda* The apparent wake angle of a ship travelling in a fluid of finite depth Lucas Yiew* Modelling the Motions of a Sea Ice Floe in Waves Boulevard 1 Wednesday morning continued Pouya Baniasadi* Solving DNA Sequencing Problems by Efficient Traveling Salesman Problem Heuristics Konstantinos Sakellariou* Complex Network Transformations of Time Series: the Ordinal Partitions Method Chair: John Hearne Lewis Mitchell How much does your social network reveal about you? Predictability and social information flow Dion O’Neale (Re)wiring network models to understand the economics of innovation Palm 1 25 4:00–4:20 3:40–4:00 3:20–3:40 3:00–3:20 12:40–1:40 1:40–2:50 12:20–12:40 Boulevard 2 John Mitry* I’ve Got Fauxs in Different Area Codes Boulevard 1 Luke Bennetts A transfer matrix method for multiple wave scattering in 2d (* denotes student talk) Palm 2 Nick Fewster-Young Existence Results for the Thomas–Fermi model of the atom with Bohr boundary conditions Lunch at Deja View Restaurant Plenary: Prof. Michael Small, University of Western Australia Title: What is a random graph, and why should we care? Chair: Mick Roberts Chair: Francis Wood- Chair: John Murray Chair: Jim Denier Chair: Luke Bennetts house James Reoch* Multi- Peter Ballard* Calcu- Ying Wan Yap Rarefied Audrey Markowskei* phase modelling of biolog- lating the probability of gas flow generated by an Scattering of acoustic ical gel mechanics an epidemic dying out af- oscillating sphere plane waves by obstacles ter the initial outbreak with corners: the effect of rounding Andras Czirok Elasto- Anna McGann* Daniel Ladiges* Aimin Chen A twoplastic tissue deforma- Derivation of Fractional Frequency-domain Monte dimensional finite volume tions in multicellular mor- SIR Model Carlo method for linear unstructured trianphogenesis oscillatory gas flows gular mesh method forfractional-in-space Allen-Cahn equations in irregular domains James Osborne Multi- Karen McCulloch* Laura Karantgis* Shanlin Qin* Numeriscale modelling of mul- Analytical expressions for Steady Saturated- cal and analytical soluticellular biological sys- infection path probabili- Unsaturated Water tions of confined subdiffutems: mechanics, devel- ties of an SIR model on Flow in a Sloping Do- sion in three dimensions opment and disease small networks main and its Application to Landslides Afternoon tea Boulevard 3 Catheryn Gray* It’s not just what you do, it’s where you do it: Signalling through Akt Wednesday afternoon Pieter Roffelsen* Solutions of the discrete Painlev´e equation q-P (A∗1 ) which are meromorphic at the origin or infinity Kate Saunders* The probability of extreme rain on your parade given the El Nio Southern Oscillation Andrew Eberhard Symmetry, Invariance and Criticality Chair: Chris Lustri Palm 1 Maryam AlaviShoshtari Limited resource maintenance of sensor networks 26 Lynne McArthur Depicting the outbreak and spread of algal blooms in New South Wales Lagoons using NOVA Rowena Ball In the beginning, there was hydrogen peroxide: periodicity, gradients, and chirality at the dawn of life Marianito Rodrigo A nonlinear least squares approach to time of death estimation via body cooling 4:40–5:00 6:30pm 5:20–5:40 5:00–5:20 William Holmes Asymmetries in the distribution of gene expression noise direct spatial organization in the developing mammalian embryo Elliot Carr Two-scale computational modelling of unsaturated water flow in soils exhibiting smallscale heterogeneity Andrey Pototsky Stability of liquid films covered by a carpet of selfpropelled surfactant particles Noel Barton How hot is hot (with reference to a solar collector)? Chair: Tony Roberts (UQ) John Knight Lewis Fry Richardson: pioneer of finite difference methods for partial differential equations Boulevard 1 Luke Fullard ResidenceTime Distribution of Heaped and Sloped Powder Layers in a Conical Mass-Flow Hopper Chris Lustri Nanopterons in a granular chain Mark McGuinness Erupting Dusts Graeme Hocking Washing sugar pulp with dirty maths Chair: Jim Hill Palm 2 (* denotes student talk) Pre-dinner drinks and Conference Banquet at SkyPoint, level 77 of Q1 Building Owen Jepps Influence of homeostasis on the longtime-limit behaviour of an autoimmune disease Mick Roberts Exponential growth and the final size of an epidemic Robert Moss Epidemic detection and forecasting from surveillance data via Bayesian estimation Joshua Ross Computation of epidemic final size distributions Chair: Jen Flegg Chair: Steve Taylor 4:20–4:40 Boulevard 2 Boulevard 3 Wednesday afternoon continued Jin Hyup Hong On the Euclidean Dimension of Graphs Phil Broadbridge Exact non-classical symmetry solutions of reaction-diffusion equations: Arrhenius combustion and logistic population growth Nobutaka Nakazono The Lax pairs of discrete Painlev´e equations arising from the integer lattice: (A2 + A1 )(1) case Yang Shi Symmetry and combinatorics of Coxeter groups and discrete integrable systems Chair: Nalini Joshi Palm 1 27 Sarthok Sircar Chemotactic adhesion in bacterial flocs in shear flow: a multi-scale model 10:20–10:40 Chair: Catherine Penington Adelle Coster Vesicle Queues: Insulin Regulation in Glucose Transport Carlo Laing Exact derivation of a neural field model from a network of theta neurons 10:40–11:00 Boulevard 2 Boulevard 1 Palm 2 Jennifer Flegg Spatiotemporal mathematical modelling of mutations of the dhps gene in African Plasmodium falciparum malaria Roslyn Hickson A model of Ebola for evaluating control Chair: James McCaw Tony Roberts Elena Vynogradova Scalar diffraction by a rotationally symmetric ensemble of arbitrarily shaped screens: rigorous solution Morning tea Chair: (Adel) Awad Al-Mohy Numerical Algorithms to Compute the Sine and the Cosine of a Matrix Dwi Lestari Solving capacitated vehicle routing problems with time windows by goal programming approach Sheehan Olver Fast and stable spectral methods for PDEs Chair: Bob Anderssen Plenary: Prof. Kerry Landman (2014 ANZIAM Medallist), University of Melbourne Title: Tracing genealogy within Fisher’s travelling wave Chair: Mat Simpson 10:00–10:20 9:40–10:00 8:30–9:30 Boulevard 3 Thursday morning Laleh Tafakori Random coefficient autoregressive model and Maximum quasi likelihood estimation Robert Scheichl Tractable Quadrature in Infinite Dimensions and Applications in Uncertainty Quantification Chair: Lewis Mitchell Palm 1 28 12:50–1:50 11:50–12:50 11:20–11:40 11:00–11:20 Boulevard 2 Chair: Mark Nelson Edward Waters The vital role of animals in the transmission of waterborne disease in rural Australia John Shepherd Modelling Tumour Treatment using the Single Species Gompertz Population Model Debadi Chakraborty Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures Boulevard 1 Chair: Frank de Hoog Bob Anderssen Solving the Interconversion Equation of Rheology for Sums of Exponentials Palm 2 Chair: Mark McGuinness Melanie Roberts Understanding risk through virtual sensing: an application to the agricultural industry Shinya Miyajima Enclosing solutions of the delay eigenvalue problem Lunch at Deja View Restaurant Plenary: Prof. Gary Froyland, University of New South Wales Title: Dynamics, Probability, and Predictability Chair: Peter Taylor Boulevard 3 Chair: Peter van Heijster Zoltan Neufeld Mathematical modelling of the self-renewal of the epidermis Thursday morning continued Boris Baeumer Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise Palm 1 Chair: Jerome Droniou Vera Roshchina Condition numbers in conic feasibility problems Plenary Sessions Models of cell polarization and motility Leah Edelstein-Keshet University of British Columbia, Canada [email protected] In this lecture, I will survey work done over the past few years with group members on modeling the polarization of motile cells (such as white blood cells), their internal signaling, their temporal dynamics, and their evolving shapes. I will describe how a combination of biological facts and mathematical models (e.g, reaction diffusion equations) helped us to gain a better understanding of how internal signaling (of proteins such as Rho GTPases, PI3K), and the response of the structural proteins (cytoskeleton elements such as actin) can orchestrate some of the complex and interesting cell motility behaviour. I will describe some recent mathematical analysis that helped us probe these models, as well as experimental collaborations with Andre Levchenko and William Bement. Contributers to this research include former students and postdoctoral fellows: AFM Maree, AT Dawes, A Jilkine, Y Mori, WR Holmes, V Grieneisen, and others. Dynamics, Probability, and Predictability Gary Froyland University of New South Wales [email protected] Many interesting natural phenomena are hard to predict beyond the immediate future. The medium to longterm behaviours of dynamical systems models of these phenomena are often best studied using probabilistic approaches. Following a gentle introduction to some fundamental results in ergodic theory, I will illustrate how functional analytic tools can be used to extract robust structures from chaotic flows in Earth’s oceans and atmosphere and in Jupiter’s atmosphere. 29 Interfacial instabilities on the pore scale Anne Juel University of Manchester, UK [email protected] What links a baby’s first breath to adhesive debonding, enhanced oil recovery, or even drop-on-demand devices? All these processes involve moving or expanding bubbles displacing fluid in a confined space, bounded by either rigid or elastic walls. In this talk, we show how spatial confinement may either induce or suppress interfacial instabilities and pattern formation in such flows. We demonstrate that a simple change in the bounding geometry can radically alter the behaviour of a fluiddisplacing air finger both in rigid and elastic vessels. A rich array of propagation modes, including symmetric, asymmetric and localised fingers, is uncovered when air displaces oil from axially uniform tubes that have local variations in flow resistance within their cross-sections. An unexpected and novel propagation mode exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed relative distance behind the moving finger-tip. The presence of multiple steady and unsteady modes is in contrast to the single, symmetric mode observed in tubes of regular cross-section, e.g. circular, elliptical, rectangular and polygonal. Moreover, we show that the experimentally observed states are all captured by a two-dimensional depth-averaged model for bubble propagation through wide channels with a smooth occlusion, which is similar to a model describing viscous fingering, but with a spatially varying channel height. Viscous fingering in Hele-Shaw cells is a classical and widely studied fluid-mechanical instability: when air is injected into the narrow, liquid-filled gap between parallel rigid plates, the axisymmetrically expanding air-liquid interface tends to be unstable to non-axisymmetric disturbances. We show how the introduction of wall elasticity (via the replacement of the upper bounding plate by an elastic membrane) can weaken or even suppress the fingering instability by allowing changes in cell confinement through the flow-induced deflection of the boundary. The presence of a deformable boundary also makes the system to additional solid-mechanical instabilities, so that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin and elastic sheets, the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the F¨ oppl-von K´ arm´ an equations, which describe the deformation of the thin elastic sheet. 30 Tracing genealogy within Fisher’s travelling wave Kerry Landman University of Melbourne [email protected] Cell invasion, whereby cells move and undergo cell division into previously unoccupied substrates or tissues, occurs in tumour growth and wound healing. Continuum models of cell invasion typically employ the wellknown partial differential equation called the Fisher equation (1937). The equation supports travelling wave solutions, making the population-level behaviour highly predictable. Discrete agent-based models, governed by agent probabilities, reproduce the population-level behaviour of the Fisher equation. However, individual agent contributions to the total population, measured by agent lineage, are highly variable. Both behaviours have been verified in a developmental invasion system. In order to understand such seemingly paradoxical findings, we examine an intermediate level by tracking of the number of divisions (generation number) that cells undergo within an invasion wave. The spatial and temporal dynamics of cell generation number is determined two ways, using agent lineage tracings and a multispecies Fisher equation. An interesting inverse problem arises. Can the lineage tracings of all agents at any given time be determined through knowledge of the generation distributions? We answer this question by constructing a generation-dependent Galton-Watson process. The method provides a potentially useful technique for deducing cell lineage data when imaging every cell is not feasible. Modelling atherosclerotic plaque formation: Boundaries, balances and bifurcations Mary Myerscough University of Sydney [email protected] Why do some people develop cardiovascular disease while others with similar risk profiles do not? What causes plaques to regress? Why does raising HDL (“good cholesterol”) reduce cardiovascular disease in some but not all cases? Why does atherosclerosis sometimes progress in fits and starts? Can atherosclerotic plaques ever disappear once they have formed? At first sight, none of these appears to be a question that can be answered mathematically. But the formation and progress of atherosclerotic plaques are outcomes of many interlinked biochemical, physiological and cellular processes. Most of these processes are nonlinear and many are influenced by slow changes in physiological conditions both in the arteries where the plaques form, and in the body as a whole. I will present models for the formation of plaques in the artery wall based on ordinary and partial differential equations. The solutions show a variety of bifurcation behaviour and nonlinear dynamical effects that explain published experimental outcomes and are relevant to drug therapies that are currently under clinical trial. 31 Formulating and solving biodiversity conservation problems Hugh Possingham The University of Queensland [email protected] Conservation science is booming and as it matures many of its components are becoming more quantitative. Traditionally most conservation scientists have had quantitative training in statistics and sometimes basic applied mathematics. They rarely, if ever, are trained in operations research. This leaves a huge gap in the field — particular when one recognizes that conservation is an applied science that is all about achieving conservation outcomes within a constrained budget. In this talk I will discuss how we have been formulating and solving nature conservation problems — most of which have never been formulated before. In particular I will discuss: 1) how to allocate resources across the globe to minimize species loss 2) how monitoring is first and foremost an optimization problem, not a statistical problem; and 3) how our software, Marxan http://www.uq.edu.au/marxan/, is being used to create marine and terrestrial reserve systems in over 110 countries. The tools we have used to formulate and solve these problems are fairly standard— differential equations, optimal control theory, integer linear programming and simulated annealing. For each example I will define the problem verbally, then mathematically, with a brief discussion of results. The focus of the examples will be on problem formulation because I have found that the most significant challenge is formulating the correct problem and communicating the approach to end users. These, and other issues, are discussed in an informal way in our centre’s monthly magazine — “Decision Point” — http://www.decision-point.com.au/. What is a random graph, and why should we care? Michael Small University of Western Australia [email protected] Complex networks, and in particular scale-free networks (graphs with a power-law degree distribution), have been observed in a wide range of natural and man-made systems: the Internet and telecommunication networks, power grids, neuronal networks, social networks and disease transmission networks. However, how to generate random members of a class of networks (graphs) with a given degree distribution (vertex valency sequence) has not been properly addressed. While algorithms to generate random networks exist, they all introduce biases in the specific realisations that result. We introduce an unbiased algorithm to randomly select networks based on degree distribution (and possibly other properties) and show that typical properties of such graphs differ from many of the properties claimed of (for example) scale-free networks. This allows us to probe experimentally obtain network data and seek out atypical features — we can determine, in a statistically quantifiable manner, exactly what properties of a given network make it special. 32 Mathematical modelling in nanotechnology Natalie Thamwattana University of Wollongong [email protected] In this talk, we discuss the mechanical behaviour for non-bonded interactions between various nanostructures by using applied mathematical modelling techniques and continuum mechanical approach. Particularly, we look at modelling nanostructures, such as nanotubes, aromatics rings and nanopores, which have potential applications in nanomedicine and clean energy storage. The talk also touches on modelling polymer chains and proteins using classical calculus of variations. Multiscale dynamics of dewetting fluid films Tom Witelski National Centre for Epidemiology and Population Health, Australian National University. [email protected] Instabilities of thin liquid films spreading on solid surfaces are of great concern for many applications involving coating flows. Generally called “dewetting instabilities”, several stages of dynamics yield rupture, growth of dry spots, and ultimately break-up of the film into sets of droplets. These instabilities can be captured by a lubrication model consisting of a fourth-order nonlinear parabolic PDE for the film height. The long-time behavior can be reduced to a finite-dimensional system for the dynamics of the remaining droplets as interacting quasi-steady localized structures. The final stage, “coarsening”, is the successive re-arrangement and merging of smaller drops into fewer larger drops. Mean field models can be constructed to describe the evolution of the number of droplets and the distribution of drop sizes yielding macro-scale system properties from the underlying small-scale nonlinear dynamics. 33 Normal Sessions Numerical Algorithms to Compute the Sine and the Cosine of a Matrix Awad Al-Mohy King Khalid University, Abha, Saudi Arabia email: [email protected] Coauthors: Nicholas Higham and Samuel Relton The importance of the matrix sine and cosine stems from their role in the solution of second order differential equations y (t) + Ay(t) = g(t), y(0) = y0 , y (o) = y0 , where A is a square matrix. This equation arises in finite element semidiscretization of the wave equation. We derive new algorithms to evaluate sin(A) and cos(A) separately and together employing both Pad´e approximants of the sine function and new rational approximants to the cosine and sine functions obtained from Pad´e approximants to the exponential function. By rigorous analysis we prove that the algorithms are backward stable in exact arithmetic; and our numerical experiments show that they behave in a forward stable manner in floating point arithmetic and outperform existing algorithms. Limited resource maintenance of sensor networks Maryam Alavi-Shoshtari University of Auckland, Auckland, New Zealand email: [email protected] Coauthors: David E. Williams, Jennifer A. Salmond and Jarip P. Kaipio Due to the increasing availability of sensors of moderate cost, large scale sensor networks are currently considered for different tasks. One of these is the monitoring of air quality with several, possibly hundreds of sensors that may cover an extensive area, or may be difficult to access. Maintenance tasks such as periodic external calibration of the sensors can be tedious and expensive, or render long periods of data ambiguous. When such maintenance tasks over the whole network are planned and assessed, the related costs are usually fixed. In this talk, we consider a maintenance strategy that targets at controlling the calibration schedule, with a fixed average cost. The strategy is based on modelling the measurement process, focuses on the conditional stationarity of the model, and takes a decision theoretic approach to detect local disturbances, which may be due either to the actual change in the environment or malfunction of the sensor. The approach also allows for controlling the trade-off between the probability of detecting a potential sensor malfunction and the related degree of errors. As a case study, we consider ozone data from the Metro Vancouver air quality sensor network. Solving the Interconversion Equation of Rheology for Sums of Exponentials Bob Anderssen DPAS CSIRO, Canberra, ACT, Australia email: [email protected] Coauthors: Frank de Hoog and Rick Loy For linear viscoelasticity, the interconversion equation is fundamental. It characterizes, in terms of a linear Volterra convolution relationship, how, for a linear viscoelastic material, its relaxation modulus G and creep modulus J are related. Consequently, only a single instrument is required to measure experimentally either G or J. The interconversion relationship can then be solved to obtain the other. A recent stability analysis has established that recovering J from G is always stable, whereas that of G from J can be unstable. When G is modelled as a sum of exponentials, then the interconversion algorithm must be solved for J which will also be a sum of exponentials. A new algorithm for solving the interconversion equation for this situation will be discussed. 34 From stochastic processes to numerical schemes for fractional DEs, and PDEs Christopher Angstmann UNSW, Sydney, Australia email: [email protected] Coauthors: Bruce Henry In this talk I will outline a method of obtaining numerical schemes for certain classes of DEs and PDEs as well as generalisations to fractional DEs and PDEs. The fundamental idea is to use a discrete time and space random walk and show that in the diffusive limit, that is the limit as the time and space grid spacings go to zero, the governing equations of the random walk become the equation of interest. In such a manner the governing equations of the discrete random walk can then be taken as approximating the diffusion limit equations. As a the numerical scheme corresponds to the governing equation of a stochastic process the resulting solution must obey certain regularity conditions. For simple PDEs, such as the diffusion equation, this random walk scheme corresponds to well known numerical schemes. But in slightly more complicated cases, such as Fokker-Planck equations, the discrete random walk gives different schemes. In the case of fractional DEs and PDEs a discrete time random walk is chosen such that the waiting time probability is dependent on the time since rival at the site. With the appropriate choice of probabilities, fractional derivatives appear in the diffusive limit of the governing equations. Examples of where we have used this include solving fractional Fokker-Planck equations, fractional reactiondiffusion equations, as well as a fractional SIR compartment model. Thin-film flow in helical channels David Arnold School of Mathematical Sciences, The University of Adelaide, Adelaide, South Australia, Australia email: [email protected] Flows in helical channels have applications to static spiral separators used in mineral processing, and microfluidic lab-on-a-chip devices used to separate different types of cells in blood test samples. In this talk I will describe solutions for thin-film flows in helical channels with rectangular cross-section, and arbitrary centreline torsion and curvature. We characterise the effects of changing the radius and pitch of the channel centreline by considering the balance of centrifugal and gravitational effects. In a region of the parameter space we see the emergence of two rotating cells of fluid, a result that may have implications for spiral separators. Relations Between the Markovian Transition Counting Process and the Markov Modulated Poisson Process Azam Asanjarani The University of Queensland, Brisbane, QLD, Australia email: [email protected] Coauthors: Sophie Hautphene and Yoni Nazarathy Comparison of different stochastic processes to find a versatile model for describing observed data in an accurate manner is a fundamental objective in stochastic modelling. In modelling a variety of phenomena such as queueing processes, traffic in telecommunication networks, requests for Web pages, the frequency of bank transactions, rainfall, and optical communications a special Markovian arrival process known as the Markov Modulated Poisson Process (MMPP) is often applied. In this talk we revisit the MMPP and introduce an alternative that we call Markovian Transition Counting Process (MTCP). The latter is simply a point process counting the number of transitions of a finite continuoustime Markov chain. Our motivation for studying the MTCP is due to the fact that this process can serve as a useful substitute for MMPPs whose arrival rate in any phase is greater than the total rate of leaving that phase. We focus on mathematical peculiarities of this comparison. Specifically, we show that in the stationary case, both processes can be set to have the same first and second moments at any time, but different third 35 moments. In the non-stationary case, we show that these processes have the same first moments but different second moments. Modelling Leptospirosis in Livestock and Wildlife Andrea Babylon Massey University, Auckland, New Zealand email: [email protected] Leptospirosis is a disease resulting from a bacterial infection. It occurs when contaminated material, such as water polluted with the urine of an infected animal, comes into contact with broken skin, mucus membranes or is ingested internally. It causes abortions and decreased weight gain in livestock. In humans, symptoms are usually flu-like and result in an average of six weeks off work. It is the highest occurring occupational disease in New Zealand with between 80 and 180 cases per year, 60% of which result in hospitalisation. Two mathematical models are proposed here. The first is a simple model with age structure, of the spreading of infection in wildlife, originally motivated by the infection in rats in Tanzania. The dynamics of the system, including fixed points, stability criteria, bifurcation diagrams, the next-generation matrix and basic reproduction number (R0 ), will be addressed. The second is a cyclical model, showing the dynamics of the infection in farmed livestock in New Zealand. The system is reset to the initial condition (for livestock) at the beginning of each year. The limit cycle, bifurcation diagram and quasi-R0 value of the system will be determined. Both models are used to predict conditions under which the infection will persist in the population, and will be used to derive protocols for minimising the incidence of disease in humans. Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise Boris Baeumer University of Otago, Dunedin, New Zealand email: [email protected] Coauthors: Matthias Geissert, Mih´ aly Kov´ acs We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-p H older continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space- time (Sobolev-H older) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family. 36 In the beginning, there was hydrogen peroxide: periodicity, gradients, and chirality at the dawn of life. Rowena Ball The Australian National University, Canberra, ACT, Australia email: [email protected] Coauthors: John Brindley The story of the relationship between hydrogen peroxide and life is complex and dynamic, and fraught with certain natural tensions which have led to human misapprehensions. Modern cells make and break hydrogen peroxide and, after a long period of misunderstanding when it was reviled as a toxic cell-vandal and saboteur of gene transcription fidelity, evidence is mounting that its relationship with living organisms is intimate and vital. Yet non-biologically produced hydrogen peroxide existed in the environment before the first photosynthetic organisms appeared, and primitive anaerobes must have come to some arrangement with it. From various lines of evidence we have been led to the hypothesis that the dependence of life on hydrogen peroxide is even far more ancient,and that this small, energetic, chiral molecule was the agent that enabled the very first non-cellular, self-replicating and evolving systems of the RNA world. In this presentation I will report some key tests that support this hypothesis. Numerical simulations show that the thiosulfate-hydrogen peroxide (THP) thermopH oscillator can drive RNA replication and RNA enzyme activity, effectively driving evolution of the RNA world. In a spatially extended system the pH oscillations manifest as travelling waves, so the THP oscillator may have initiated the ubiquitous dependence of all life on pH gradients. I will discuss how consideration of the unique physical properties of hydrogen peroxide can explain outstanding mysteries of the origin of life: 1) its chirality makes the evolution of homochiral biology inevitable, and 2) its high surface tension provides a favourable environment for vesicles or proto-cells to form. An interesting and more subtle point is that in each of its actions on the RNA world - direct and indirect, physical and chemical - hydrogen peroxide is effectively preparing the RNA world for continued existence, evolution, and reliance on an alternative energy source in its absence. Calculating the probability of an epidemic dying out after the initial outbreak Peter Ballard University of Adelaide, Adelaide, SA, Australia email: [email protected] After an initial outbreak, an epidemic may go extinct (“epidemic fade-out”), or become endemic due to sufficient replenishment of susceptible individuals. We consider the probability of epidemic fade-out in the Markovian SIRS (susceptible-infectious-recovered-susceptible) model. An exact calculation is computationally intensive, while previously published approximations are not always accurate. We propose a simplified computation method. This gives only a small error, and is fast enough to be practical even for large population sizes. Solving DNA Sequencing Problems by Efficient Traveling Salesman Problem Heuristics Pouya Baniasadi Flinders University, SA, Australia email: [email protected] The ground-breaking discovery of DNA structure in the 1950s opened up an unparalleled opportunity for multidisciplinary efforts, such as the multibillion dollar Human Genome Project, to come together in a quest for understanding “life”. Mathematics has proved to be vital in many such efforts, specifically the DNA Sequencing Problem; aligning and merging fragments of DNA to construct the original sequence. The importance and mathematical beauty in the DNA-Sequencing Problem stem from its close ties to fundamental problems in Combinatorial Optimization and Complexity Theory. In particular, the basic idealized DNA-sequencing Problem can be easily embedded in a Traveling Salesman Problem (TSP) which, arguably, is the most widely studied problem in combinatorial optimization thanks to its theoretical importance and its wide range of applications. While the close relationship between the two problems is underexploited due 37 to the computational difficulty of TSP, recent advances in the quality of TSP heuristic algorithms provide a compelling opportunity for a new approach to DNA-Sequencing Problem. Our project is aimed at exploring this opportunity for developing TSP-based models and algorithms to advance our mathematical understanding of the DNA-Sequencing Problem as well as offering practical solutions to the DNA-sequencing Problem. How hot is hot (with reference to a solar collector)? Noel Barton Sunoba Pty Ltd, Macquarie Park, NSW, Australia email: [email protected] A heat transfer analysis is presented for a gas-cooled solar collector for domestic space heating, process heat or power generation. The heat transfer gas flows along a duct enclosed within an insulated box. Solar radiation enters the collector assembly through a double-glazed window with a low-emissivity coating. The geometry of the collector is invariant in the flow direction. The model includes convective heat transfer to the gas flow in the duct, molecular diffusion of heat in the insulation and in the air-gap of the double-glazed window, molecular diffusion of heat in the outer glass sheet of the window, and convective heat transfer from the window to the surrounding atmosphere. Effects of infrared radiation are included, as are absorption of solar and infrared radiation in the outer glass sheet. The model equations are solved by a combined analytical-numerical approach compact enough to be coded in a spreadsheet. Checks and examples will be given. A transfer matrix method for multiple wave scattering in 2d Luke Bennetts Uni Adelaide, Adelaide, SA, Australia email: [email protected] Coauthors: Fabien Montiel A new method will be outlined to solve time-harmonic multiple wave scattering problems in 2d. The method uses a decomposition of the 2d domain into slabs. Transfer matrices, which map wave fields from one side of a slab to the other, are derived using integral transforms. The method is designed for problems involving large numbers of scatterers and disordered arrays of scatterers, for which direct methods are not effective. A stochastic analysis approach to placing upper bounds on solutions to free boundary problems. Louis Bhim The University of Sydney, Sydney, NSW, Australia email: [email protected] Coauthors: Reiichiro Kawai We approach the problem of placing tight bounds on solutions to obstacle style free boundary problems by bounding the stochastic representation for the solution using results from stochastic analysis. After establishing this bound we then formulate the problem as a semidefinite programming problem and tighten the bounds using numerical optimization. This approach makes use of the Dynkin formula from stochastic analysis to obtain the bound and sums of squares relaxations to arrive at a computationally tractable problem. We also discuss applications of this approach to the problem of pricing American style options and the potential for this approach to be extended to bound solutions to more general free boundary problems. 38 Defining Moments: Spatial Structure in a Model of Collective Cell Movement Rachelle Binny University of Canterbury, Christchurch, New Zealand email: [email protected] The ability of cells to migrate plays a fundamental role in tissue repair, development and the immune response. Pathologies such as cancer can arise when the regulatory mechanisms controlling this movement are disrupted. Interactions occurring at the level of individuals may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that cells interact with one another in proportion to their average density and ignores the presence of small-scale spatial structure. In this talk, we will describe an individualbased model (IBM) that uses random walk theory to model stochastic interactions occurring at the scale of individual migrating cells in continuous space. The IBM incorporates a mechanism for local directional bias such that an individual’s direction of movement is dependent on the degree of cell crowding in its neighbourhood. We will then discuss an alternative to the mean-field approach which employs spatial moment theory in order to account for spatial structure and predict how individual-level interactions propagate to the scale of the whole population. Modelling the evolution of uni- to multi-cellular life Andrew Black University of Adelaide, Adelaide, Australia email: [email protected] The transition from uni-cellular life to multi-cellular is one of the best known examples of the emergence of a new level of biological organisation. To understand how this transition proceeded we need to know how early groups of cells came to have the properties needed for Darwinian evolution, i.e. groups must possess some form of reproduction with heritable variations in fitness. In this talk I will present some recent work towards modelling the very early stages of this transition and the emergence of crude forms of group level reproduction in a system in which groups can form when individual cells remain attached to their parents after reproduction. Spatial-temporal forecasting of solar radiation John Boland University of South Australia, Mawson Lakes, Australia email: [email protected] Coauthors: Ted Soubhdan We apply the Combined Autoregressive Dynamical System (CARDS) solar forecasting tool, developed at the University of South Australia, to forecasting of solar radiation series at three sites in Guadeloupe in the Caribbean. After performing the model estimates at each individual site, forecast errors were tested for cross correlation. It was found that on an hourly time scale, there was small but significant correlation between sites, and this was taken into account in refining the forecast. Cross correlation was found to be insignificant at the ten minute time scale so this effect was not included in the forecasting. Also, the final error series in each case was tested for an Autoregressive Conditional Heteroscedastic (ARCH) effect, finding that to construct prediction intervals for the forecast a conditional forecasting model had to be constructed for the variance. Note that cross correlation between sites has to be included for this procedure as well as in the forecasting of the radiation. 39 A Distributed Optimization Algorithm with an Application to a Smart Grid Philipp Braun University of Bayreuth, 95440 Bayreuth, Germany email: [email protected] Coauthors: Lars Gr¨ une, Christopher Kellett, Steven Weller and Karl Worthmann We present a novel hierarchical scheme for optimal control of large scale systems. The scheme is based on distributed controllers connected to a central entity. The distributed optimization ensures flexibility while the central entity coordinates the exchange of information. The algorithm is compared with a fully centralized and a fully decentralized optimal control scheme. We prove that the distributed control scheme generates the same results as the centralized control scheme. We illustrate the results of the control schemes applied to a model of a residential level electricity network with distributed power generation and distributed storage devices. The algorithms are used in a model predictive control framework to compute optimal charging/discharging rates of the storage devices to reduce the fluctuations in the power demand over the entire network based on predicted load and predicted generation. Exact non-classical symmetry solutions of reaction-diffusion equations: Arrhenius combustion and logistic population growth. Phil Broadbridge La Trobe University, Melbourne, VIC, Australia email: [email protected] Only a special class of reaction-diffusion equations has full non-classical reductions to solutions that are not invariant under Lie’sclassical symmetries. However some of those equations have important applications. For 1+1 dimensional linear diffusion with a nonlinear reaction, only equations such as the Fitzhugh-Nagumo equation, and the Huxley equation, with cubic source terms, have strictly non-classical invariant solutions. Under the assumptions set down by Fisher in 1930, the advance of a new advantageous gene through a diploid population is governed not by Fisher’s equation but by Huxley’s equation. For nonlinear reaction-diffusion equationsin n spatial dimensions, there is a single restriction relating nonlinear diffusivity to nonlinear reaction that always allows non-classical reduction to the linear Helmholtz equation. This allows us to construct unsteady solutions to a reaction-diffusion equation with any differentiable reaction term. We demonstrate the radial solutions with logistic source term of population dynamics, and with Arrhenius reaction term of combustion, that follows from the Gibbs non-analytic temperature-dependent canonical probability distribution. The extinguishing solution is stable to small perturbations. Curvature effects in the evolution of bone tissues during bone remodelling Pascal Buenzli School of Mathematical Sciences, Monash University, Clayton, VIC, Australia email: [email protected] Coauthors: Almie Alias Bone tissues undergo continual remodelling by cells resorbing old, damaged bone and other cells re-forming new bone. The evolution of the bone interface during resorption and formation is determined in particular by the local surface cell density. A significant influence on surface cell density is the contraction or expansion of the local surface area (depending on curvature) when the surface undergoes resorption or formation. In this contribution, we propose a continuous mathematical model of bone formation to simulate the co-evolution of the bone interface and of the surface cell density. The concentration or dilution of surface cell density by the contracting or expanding bone surface requires a regularisation of the equations, specified in the model as a diffusive redistribution of the cells. Depending on the strength of cell diffusion, initial undulations of the bone interface either (i) oscillate such that concave regions become convex and convex regions become concave, or (ii) 40 become smoothed out at different rates. The model is compared with in-vitro experiments of bone deposition on substrates of different geometries. Finally, a general mathematical framework based on the level-set method is developed to describe the systematic effects of substrate geometry on tissues or materials that change shape through formation or resorption by cells from the surface. The advantage of the level-set based framework is to avoid parameterisation of the surface, which enables the consideration of topological changes of the surface as may occur for example in osteoporosis. Two-scale computational modelling of unsaturated water flow in soils exhibiting small-scale heterogeneity Elliot Carr Queensland University of Technology, Brisbane, QLD, Australia email: [email protected] Coauthors: Ian Turner and Patrick Perr´e Unsaturated water flow refers to flow in a partially saturated porous medium where the pores are filled with air in addition to water. A classical continuum description of the process is given by Richards’ equation, a simplified two-phase model based on Darcy’s Law, taking the form of a single PDE for the water saturation (or equivalent). For the case when the domain exhibits small-scale heterogeneities in hydraulic properties, numerical solution of the problem is prohibitively expensive due to the excessive mesh resolution required to capture the fine detail. This talk presents a two-scale computational model for overcoming this issue for a specific subset of such problems. The resulting two-scale numerical simulations are shown to produce good qualitative agreement with the full fine-scale model with a significant reduction in computation time. Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures Debadi Chakraborty Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria-3010, Australia email: [email protected] Coauthors: Matthew Pelton, Edward Malachosky, Philippe Guyot-Sionnest and John E. Sader Newtonian fluid mechanics, in which the shear stress is proportional to the strain rate, is synonymous with the flow of simple liquids like water. While this paradigm holds widely, the fluid-structure interaction of mechanical devices at nanometre scales can probe the intrinsic molecular relaxation processes in a surrounding liquid. In this talk, I will report our recent theoretical and experimental work on the non-Newtonian, viscoelastic flow phenomena produced by the high-frequency (¿20 GHz) vibration of gold nanoparticles immersed in simple liquids. The observed viscoelasticity is not due to molecular confinement, but is a bulk continuum effect arising from the short time scale of vibration. This represents the first direct mechanical measurement of the intrinsic viscoelastic properties of simple bulk liquids, and opens a new paradigm for understanding extremely high frequency fluid mechanics, nanoscale sensing technologies, and biophysical processes. 41 Can we optimise the strength of porous materials? Vivien Challis The University of Queensland, Brisbane, QLD, Australia email: [email protected] The computational solution of the equations of linear elasticity leads to excellent predictions of the elastic properties of porous and composite materials. In contrast, the failure strength of porous and composite materials is poorly understood and is therefore difficult to predict with computational methods. I’ll overview some of our recent work in this area. Our goal is to computationally optimise the strength of porous scaffolds for bone replacement applications. Mathematical Modelling of Spatial Sorting and Evolution in a Host-Parasite System Matthew Chan University of Sydney, NSW, Australia email: [email protected] Coauthors: Richard Shine, Gregory Brown and Peter Kim There have been numerous empirical and agent-based modelling studies on the spatial self-structuring of traits, particularly in regard to dispersal ability (termed spatial sorting) of cane toads in northern Australia and the resultant accelerating invasion, but few mathematical modelling studies. In this study, we formulate a reaction-diffusion based partial-integro-differential equation model based on an earlier model by Bouin et al. (2012, Comptes Rendus Mathematique) to examine this spatial self-structuring of traits in both a cane toad population and lungworm parasite population, which evolves with the cane toad population. In particular, the traits we focus on are dispersal ability for the cane toad population and both prepatent period and larval size for the lungworm parasite population. Apart from the spatial self-structuring of these traits, our results confirm a number of observations made in empirical and agent-based studies; particularly, that there is a noticeable lag between the host and parasite population which is critically dependent on the parasite functional response to host densities, that older populations regress back to lower dispersal speeds and that spatial sorting can still occur with a disadvantage in reproductivity and/or survival in more motile individuals. Moreover, we find that such a disadvantage in reproductivity and/or survival is unlikely to be large if spatial sorting is to have a noticeable effect on the rate of range expansion, as it has been observed to have over the last 60 years in northern Australia. A two-dimensional finite volume unstructured triangular mesh method for fractional-in-space Allen-Cahn equations in irregular domains Aimin Chen Henan University, Kaifeng, China email: [email protected] Coauthors: Fawang Liu, Ian Turner, Kevin Burrage and Qianqian Yang Fractional-in-space Allen-Cahn equations(FISAC) containing strong nonlinear source term and small perturbation shows metastability and a quartic double well potential. Using finite volume unstructured triangular mesh method, the present paper solve the two-dimensional FISAC equations with homogeneous Neumann boundary conditions in different irregular domains. By using linear interpolation shape function method and matrix transfer technigues and the Lanczos interation method, the two-dimensional FISAC equations are computed numerically on different irregular domains. 42 Macroscale model and boundary conditions for spring mass system with fine structure. Chen Chen The University of Adelaide, Adelaide, Australia email: [email protected] Multiscale modelling methodologies build macroscale models of materials with complicated fine structure. My innovation is to derive correct boundary conditions to use with the macroscale model. We derive macroscopic boundary conditions for a microscale discrete spring mass system with microscale structure. The spring mass system has two strands and the strands are linked by springs. The derived macroscale boundary conditions improved the accuracy of macroscale model. We verify the new boundary conditions by numerical methods. The techniques developed here can be adapted to a range of multiscale modelling situations to provide boundary conditions for accurate predictions. Drawing of microstuctured optical fibres with pressurisation of the internal channels Mike Chen University of Adelaide, Adelaide, SA, Australia email: [email protected] Coauthors: Yvonne Stokes, Peter Buchak, Darren Crowdy and Heike Ebendorff-Heidepriem Microstructured optical fibres are distinguished from solid optical fibres by the large number of internal air channels running along their length. These fibres are manufactured by heating and stretching a preform, which has some cross-sectional pattern of holes. In stretching the preform with a diameter of 1-3cm to a fibre with a diameter of the order of 100 micrometers, the cross-sectional hole pattern changes in scale but is also deformed due to surface tension. A practical way of countering this deformation is to introduce pressurisation in the internal channels. This pressure acts against surface tension and potentially provides an extra degree of control over the shape of the internal channel geometry. We generalise an existing model of fibre drawing to include channel pressurisation and present examples of pressurised fibre drawing for several cross-sectional geometries of practical importance. Osmotically driven deformation of a stable water film Sue Ann Chen IBM Research, Melbourne, Australia email: [email protected] Coauthors: Lucy Y. Clasohm, Roger G. Horn and Steven L. Carnie Recent experiments on a mercury drop near a mica surface show that a dimple forms on the mercury/water interface when there is a sudden change in the electric potential of the mercury drop coated with a self-assembled monolayer (SAM) of 11-mercapto-1-undecanoic acid thiol molecules. It is suggested that the dimple formation is due to the desorption of a fraction of the SAM from the mercury drop surface when the surface potential is changed. The osmotic pressure in the thin film region increases as a result of the presence of the thiol molecules in the region, giving rise to the observed dimple. The solute concentration is introduced as a new dependent variable in the system and the transport of the solute is described by a convection-diffusion equation. The thin film and convection-diffusion equations form a system of coupled partial differential equations. The effects of disjoining pressure, hydrodynamic pressure and total pressure are discussed. As the simplest version of the model, in which desorption is assumed to be uniform and instantaneous, could not explain some features of the experimental observations, this suggests the presence of a more refined mechanism. We explore some of the more complicated models here. 43 Analysis of novel oscillations of quantized mechanical energy in mass-accreting nano-oscillator systems Jeong Ryeol Choi Department of Radiologic Technology, Daegu Health College, Daegu, Republic of Korea email: Coauthors: Ji Nny Song Quantum characteristics of mass-accreting oscillator which can be applied to analyzing quantum features of nanomechanical mass sensing are investigated by making use of the invariant operator theory. The invariant operator theory is one of rigorous quantum theories for dynamical systems that have time-varying parameters. In particular, quantum energy of the system is analyzed in detail and compared it to the classical one. We focused on two particular cases, which are linearly mass-accreting oscillator and exponentially mass-accreting one. It is confirmed that quantum energy agrees well with the classical one in the limit h → 0 where h is the Planck’s constant. We showed that not only the classical but also the quantum energy oscillates with time. A reasonable explanation for this energy oscillation is given. The influence of object shape on the convergence of active contour models for image segmentation Josh Chopin University of South Australia, Adelaide, South Australia, Australia email: [email protected] Coauthors: Hamid Laga and Stanley Miklavcic Modern genomics experiments are often conducted in controlled environments on hundreds of plants at a time. As such, manual and destructive phenotyping techniques are no longer suitable. Recently there has been a trend toward automated high-throughput phenotyping facilities, in an attempt to relieve the phenotyping bottleneck. A majority of these facilities focus on the use of cameras for non-destructive recording of plant data. Hence, the new challenge lies in accurately segmenting the plants from 2D images in an automated manner and then analysing structural and statistical information about them. One such technique for image segmentation is known as the active contour model. Active contour models have seen widespread success throughout a number of applied fields due to their versatility and semi-automated nature. However, a high majority of these models rely on arbitrary parameters that are required to be selected manually. Furthermore, small variations in these parameters can produce substantial variations in the method’s overall accuracy. This makes them unsuitable for use by non-experts and also for the analysis series of images that can change drastically over time, such as the growth of a plant. In this talk we attempt to establish relationships between the parameter values of active contour models and the geometry of the objects/shapes that they are segmenting. Our goal is for users to be able to utilise some basic a-priori knowledge about the geometry of the object in order to automatically select a range of suitable parameter values. We analyse the accuracy of active contour models over multiple series of shapes that exhibit some pattern, such as decreasing number of sides or increasing concavity. We present a novel normalization technique so that the parameter values are of a similar scale. We also carefully design an experimental setup that ensures no bias between different shapes or parameter values. We show that over a series of shapes the range of parameters that provide convergence do follow a trend. We also show that not all contours that converge to the objects boundary do so in a stable manner, with a substantial amount oscillating continuously. However more information, such as more shapes and more parameter values, is required to draw meaningful and quantitative conclusions from such an analysis. Future work includes incorporating more of this information along with the application to more active contour models. Another exciting future direction is the use of 2D shape diagrams to quantify relationships between shapes, parameter values and levels of accuracy. 44 Optimising Revenue for Concentrating Solar Thermal Power Plants with Limited Thermal Energy Storage Luigi Cirocco University of South Australia, Mawson Lakes, SA, Australia email: [email protected] Coauthors: John Boland, Frank Bruno, Peter Pudney and Martin Belusko The optimal control strategy for maximising of revenue for a Concentrating Solar Thermal (CST) power plant with unlimited Thermal Energy Storage (TES) operating in an electrical energy market has been established in our previous work where the storage state space problem was resolved using the necessary conditions of Pontryagins Maximum Principle. This presentation focuses on further refinements the problem where we consider the constrained storage state using a direct adjoining technique to investigate the properties of the optimal control strategy for an extended set of necessary conditions. We also present the development of a solution algorithm for the constrained problem and discuss how the necessary conditions are enacted within the algorithm. The discussion concludes with the identification of the possible refinements to the modelling and the development of demand side models for the investigation of optimal control for electricity customers. The Physics of Suspended Microchannel Resonators Jesse F. Collis The University of Melbourne, Melbourne, Victoria, Australia email: [email protected] Coauthors: John E. Sader Suspended Microchannel Resonators enable mass sensing in liquid with exquisite resolution, with measurements at the attogram level being demonstrated recently. In this talk, I will discuss our recent work on analysing the performance of these devices using both asymptotic and numerical analyses. This work is in conjunction with our experimental collaborators at the Massachusetts Institute of Technology (USA). Mathematical modelling and numerical simulation of nanopores Eamon Conway QUT, Queensland, Australia email: [email protected] Coauthors: Steven Psaltis and Troy Farrell The transport of ions through nanometre sized channels (nanopores) leads to complicated two-dimensional behaviour due to the development of double layers at the electrode and nanopore/electrolyte interfaces. Notably, the double layer forma- tion inside the nanopore causes selective motion, inhibiting the transport of one of the ionic species. This behaviour, known as ion current rectification, produces an asymmetric current-voltage (I-V) curve. We propose a mathematical model of ion transport through nanopores based on the transient PoissonNernst-Planck (PNP) equations with diffuse Butler-Volmer kinetics applied at the electrolyte/electrode interface. Such kinetics model ionic current at the working electrode. This framework allows for novel extensions to the model, such as the inclusion of steric effects and the production of simulated I-V curves. In this talk, we present the initial results from our proposed model. 45 Polar vortices on celestial bodies Jason Cosgrove University of Tasmania, Hobart, Tasmania, Australia email: [email protected] Atmospheric vortices over the polar regions of a celestial body are considered, with a particular focus on modelling the famed North Polar Hexagon (NPH) on Saturn. The atmosphere is weakly compressible, and the fluid motion is subject to the Coriolis pseudo-force, due to celestial bodies being in a non-inertial reference frame. The NPH rotates in an anti-clockwise direction when viewed from above and so will be initially modelled by a low pressure system in the form an exponential function. The standard f-plane and beta-plane approximations are invalid over the poles and thus non-linear solutions are presented using a planar approximation where the Coriolis parameter varies quadratically away from the pole. Vesicle Queues: Insulin Regulation in Glucose Transport Adelle Coster UNSW Australia, Sydney, Australia email: [email protected] Mammalian cells regulate glucose levels by translocating membrane embedded glucose transporter proteins to and from their outer cell membranes. The predominant transporter in fat and muscle cells is Glucose Transporter 4 (GLUT4) which is insulin-responsive. GLUT4 is embedded in vesicles (small spheres of membrane) for transport. The vesicles then fuse with the destination membrane, and the GLUT4 is released. It has been suggested that the insulin regulation of vesicle fusion is not only limiting the appearance of GLUT4 at the cell surface but also regulating the drain on internal stores, as well as the extent to which vesicles are associated with the cell cytoskeleton. There is also evidence that the vesicles transit along microtubules suggesting that there might be a queueing protocol for fusion. Here, closed Markovian queueing networks with finite capacity queues are explored to determine whether the experimentally observed features of GLUT4 translocation can be described by such systems. Microstructure Interpolation for Macroscopic Design Andrew Cramer The University of Queensland, Brisbane, Australia email: [email protected] Coauthors: Vivien Challis and Anthony Roberts Structural optimisation seeks to design the best possible structure for a given load case. Typically the problem is to determine where to place a finite amount of material in a domain to minimise compliance or to maximise stiffness under the assumption of linear elasticity. Multi-scale optimisation methods have been developed to expand the space of allowed solutions and make use of developments in additive manufacturing. We propose a new method of multi-scale optimisation which interpolates optimised microstructures for a material distribution method. The technique is benchmarked against existing structural optimisation methods for some test problems. We also discuss the benefits over existing multi-scale optimisation methods. 46 Elasto-plastic tissue deformations in multicellular morphogenesis Andras Czirok University of Kansas Medical Center, Kansas City, KS, USA email: [email protected] Coauthors: Dona Isai Multicellular pattern formation is an important aspect of both embryonic development and certain pathologies like tumor formation and invasion. Large-scale cell movements often involve cell-exerted mechanical forces and suitably controlled changes in cell adjacency. Based on empirical data, we propose a three dimensional mechanical model of multicellular assemblies. Mechanically coupled adherent cells are represented as particles interconnected with elastic beams which can exert non-central forces and torques. Tissue plasticity is modeled by a stochastic process consisting of a connectivity change (addition or removal of a single link) followed by a complete relaxation to mechanical equilibrium. In particular, we assume that (i) two non-connected, but adjacent particles can form a new link; and (ii) the lifetime of links is reduced by tensile forces. We demonstrate that the proposed model yields a realistic macroscopic elasto-plastic behavior and we establish how microscopic model parameters determine material properties at the macroscopic scale. In addition to their mechanical role, model particles can also act as simulation agents and actively modulate their connectivity according to specific rules. As an example, anisotropic link insertion and removal probabilities can give rise to local cell intercalation and large scale convergent extension movements. The proposed stochastic simulation of cell activities yields fluctuating tissue movements which exhibit the same autocorrelation properties as empirical data from avian embryos. Geothermal Spring Temperature Analysis Heather Davidson Massey University, Albany, New Zealand email: [email protected] The aim of this study was to examine temperature time-series data recorded from several geothermal features in the Taupo Volcanic Zone, New Zealand. Pool temperatures were recorded at 17 features at various times between 1996 and 2011. The monitored features ranged from geysers with regular eruption cycles to hot springs with erratic temperature cycles. The length and number of records differs for each site. The effects of rainfall, air pressure and air temperature were analysed to ascertain whether there is a relationship between the pool temperatures and any of these factors. Water level data from Lake Ohakuri was also examined to determine whether it had any influence on the features at the nearby Orakei Korako geothermal area. Earlier research on a small subset of data from the Waiotapu Geyser had found that variations in air pressure could trigger or halt eruptions as well as affecting eruption frequency (Nikrou et al., 2013). Analysis of all available datasets from the Waiotapu Geyser confirmed the existence of such a relationship. No other monitored features appear to be influenced significantly by air pressure. No correlation between pool temperatures and rainfall or air temperature were detected for any geothermal features in this study. Lake Ohakuri is part of the Waikato River hydro scheme causing the water level to fluctuate daily. This does not appear to affect the recharge or temperature cycles of nearby geothermal features. Applications of Compressive Sensing Frank de Hoog CSIRO, Canberra, ACT, Australia email: [email protected] In applications it is often the case that we wish to that we wish to find an approximation to a set of linear equations for which there are fewer equations than unknowns. In general of course this is an insoluble problem but, if it is known that the solution can be well approximated by a sparse solution, then progress can be often be made. 47 In this presentation, we briefly describe conditions under which error bounds can be established and describe applications to which compressive sensing techniques have been applied. The unsteady flow due to a spinning toroidal mass Jim Denier University of Auckland, Auckland, New Zealand email: [email protected] Coauthors: Sophie Calabretto and Trent Mattner The flow due to a rotating torus provides a paradigm for the study of many phenomena that arise in unsteady fluid flows. It also provides a model for the flow induced by the colelctive motion of fish contained within an aquaculture pen. This talk will present some new results demonstrating that this flow evolves through a series of well defined stages, starting from a collision of viscous boundary layers which result in the development of a radial jet. This radial jet is preceded by a starting vortex which subsequently detaches forming an isolated toroidal vortex. The radial jet then develops an absolute instability which leads to a turbulent flow in the vicinity of the sphere’s equator. In the context of the collective motion of fish contained in aquaculture pens, the fluid flow they induced would serve to provide an induced force which would impact upon the overall pen structure. Such interactions are observed as a draw-up of the aquaculture pen. Mathematical modelling of atherosclerosis - atheoma plaque formation, development and rupture Yan Ding RMIT University, Melbourne, Victoria, Australia email: [email protected] Coauthors: John A. Gear Atherosclerosis is a medical terminology meaning artery hardening. It is resulted by the presence of fatty deposits, such as low density lipoprotein (LDL), monocytes and macrophages, etc. accumulated in the walls of the arteries, which results thickening the artery wall, narrowing the passageway of the blood flow and leading to a reduction in the blood flow through the blood vessels. This process is called plaque formation. The initial stage of a plaque build-up in an artery wall is asymptomatic. However, as a plaque grows, the blockage of blood flow becomes severe. In some serious cases, a plaque is ruptured, which releases thrombogenic agents into the blood stream resulting fatal damage. Atherosclerosis occurring in different areas of the body has different effects. In the brain, atherosclerosis would cause thrombus, meaning oxygen is being cut off from the brain, leading to brain damage and stroke. In the aorta, plaque building-up would cause artery wall rupture due to the ballooning of the artery wall. Atherosclerosis in legs would decrease blood circulation, and in serious cases, amputations may be the only option to save the lives. Thrombosis would occur in coronary arteries, causing heart attack. Atherosclerosis is, in fact, the main contributor to myocardial and cerebral infarctions, which had been linked to 50% of all fatality across USA, Europe and Japan in 1990’s. According to the fact sheet No317 of the World Health Organization (WHO), released in March 2013, cardiovascular disease (CVD), mainly from heart disease and stroke, are the first leading cause of death globally; and the number of death will reach 23.3 million by year 2030. The initiation and progression of atherosclerosis involve many biomedical and biochemical aspects. In this presentation, we report our current research activities and achievements in the mathematical modelling of the complex phenomenon. We also discuss the main objectives of the research and present our frame work for achieving these objectives. 48 Quantifying spatial distributions using a pair correlation function based on generalized measures of separation Saber Dini The University of Adelaide, Adelaide, SA, Australia email: [email protected] Pair correlation functions are statistical tools for analyzing spatial distributions. They describe the frequency of distances between pairs of data points. Pair correlation functions have been used to analyze the distributions of objects in various fields, from the stars in the galaxy to cells in a dish. In this talk, we derive a pair correlation function based on a generalized distance obtained by projecting the positions of points from a higher dimensional space into a one dimension (this includes the Euclidean distance but also e.g. angular separation). By using the features of order statistics on the distribution of generalized distances, we are able to estimate the expected value of the frequency of pairs for objects distributed with a given probability density function. We can thus normalize an observed distribution of interest relative to any reference distribution (which need not be uniform), which allows us to analyze the deviation of the data from this reference distribution. We present preliminary results of applying our method to analyze experimental data concerning the distribution of cells within spheroids. Accurate Approximations of the Riemann-Stieltjes Integral Silvestru Sever Dragomir Victoria University, Melbourne, Victoria, Australia email: [email protected] Riemann-Stieltjes integral for complex or real-valued functions plays a key role in various fields of Mathematics, including Probability Theory & Statistics, Complex Functions Theory, representation of selfadjoint operators on complex Hilbert spaces, in Numerical Analysis and for Quadrature rules etc. In this presentation we survey some recent inequalities of Ostrowski, Trapezoid and Gruss type for RiemannStieltjes integral obtained by the author and show how these can be used to provide accurate quadrature rules in approximating the Riemann-Stieltjes integral. A hybrid higher-order numerical scheme for convection-diffusion problems Jerome Droniou Monash University, Clayton, VIC, Australia email: [email protected] Coauthors: Daniele Di Pietro and Alexandre Ern Convection-diffusion equations permeate a variety of fluid flows models, including in particular flows in porous media. In such models, the natural diffusion can be in some places much smaller than the convection driven by the Darcy velocity, and it is therefore essential to dispose of numerical methods that can automatically and locally adapt to the flow regime (diffusion-dominated or convection-dominated). Some practical constraints must also be taken into account, such as e.g. the capacity for the method to be efficiently implemented in a parallel environment. In this talk, we will present a numerical scheme of arbitrary order to deal with convection-diffusion equations. This scheme uses separate degrees of freedom on cells and faces, and has a local connectivity (each cell is only connected to its faces) which makes it a good candidate for parallel implementations. The discretisation of the convective terms uses a stabilisation which automatically adjusts to all regimes (including vanishing viscosity). I will also present error estimates which are optimal in all regimes, thanks to the usage of local P´eclet numbers. 49 Making Waves: High Frequency Volatility Estimation and the Hilbert-Huang Transform Carson Drummond University of Wollongong, Wollongong, NSW, Australia email: [email protected] In this presentation a new way to estimate the spot volatility of high frequency foreign exchange data using the Hilbert-Huang transform is introduced. The proposed volatility estimate was designed to overcome the difficulties encountered when microstructure noise is present. The problem of assessing the validity of latent variable estimates is overcome by setting up a virtual options trading market in which competing volatility forecasts buy and sell straddle options to one another using real high frequency foreign exchange data. Improving Public Transport Accessibility via the Optimisation and Synchronisation of Schedules for Key Transport Modes. Michelle Dunbar University of Wollongong, Wollongong, NSW, Australia email: [email protected] As the population within modern metropolitan cities continues to grow, greater population dispersion means that daily commuters are increasingly faced with longer commute times and journeys consisting of multiple legs; often involving more than one mode of transport. In a bid to discourage the use of the private motor-car and facilitate the uptake of public transport, there is a developing trend towards the construction of centrallylocated Transport Hubs, allowing passengers to connect with multiple modes of transport. To assist passengers in connecting with their outbound mode more efficiently, it is desirable to synchronise connecting modal services within the Transport Hub. In this presentation we consider the problem of designing shuttle-bus routes for passengers connecting with one of four different modes of transport at a Transportation Hub. We seek to minimise the average waiting time for passengers, the cost of missed connections at the Hub and the total travel time. Furthermore, we incorporate time-of-day effects and passenger heterogeneity with respect to value-oftime. In addition to commuters, the framework developed is amenable and directly extensible to the perishable good delivery problem for which items possess heterogeneity in delivery priority. Our model is posed as an extension of the vehicle routing problem with time windows, and solved using column generation. We provide a brief outline of our optimisation formulation and preliminary results for a number of datasets. Modelling of graphene oxide and carbon nanotubes in a nematic liquid crystal using continuum mechanics Tom Dyer University of Wollongong, NSW, Australia email: [email protected] Coauthors: Ngamta Thamwattana In this study, we construct a continuum model for graphene oxide based upon the Lerf-Klinowski structure to investigate the idea of liquid crystal dispersions. We use the Lennard-Jones potential and Coulombs law to determine the potential energy between sheets of graphene oxide. Our model is then modified to investigate different levels of hydration and oxidation within the system. We find that the results discovered using the continuum approach match experimental results found in literature. This model is then used to explore the idea of liquid crystal constructions of graphene oxide and nanotubes, where the tubes and sheets prevent each other from aggregation. We investigate the orientation and position of a single-walled carbon nanotube between our sheets of graphene oxide at equilibrium. We find that sufficiently short nanotubes prefer to orient perpendicular between the sheets. Additionally, the nanotubes do not form bundles and instead form some alignment. Our investigations are reconstructed using the LAMMPS molecular dynamics simulations and we compare them with our mathematical modelling results. 50 Symmetry, Invariance and Criticality Andrew Eberhard RMIT University, Melbourne, Australia email: [email protected] Coauthors: Vera Roshchina The aim of this talk is to summarise, relate, explain and generalise a range of results in nonsmooth, and predominantly nonconvex analysis, that exploit the symmetry of underlying problems. Results of this kind date back to the work of Palais on the principle of symmetric criticality but there more recent results that can be placed in a similar framework. We will discuss some old and new results. Characterising transport through a crowded environment with different obstacle sizes. Adam Ellery Queensland University of Technology, Brisbane, QLD, Australia email: [email protected] Coauthors: Matthew Simpson, Scott McCue and Ruth Baker Transport through crowded environments is often classified as anomalous, rather than classical, Fickian diffusion. Several studies have sought to describe such transport processes using either a continuous time random walk or fractional order differential equation. For both these models the transport is characterized by a parameter α, where α = 1 is associated with Fickian diffusion and α < 1 is associated with anomalous subdiffusion. Here, we simulate a single agent migrating through a crowded environment populated by impenetrable, immobile obstacles and estimate α from mean squared displacement data. We also simulate the transport of a population of such agents through a similar crowded environment and match averaged agent density profiles to the solution of a related fractional order differential equation to obtain an alternative estimate of α. We examine the relationship between our estimate of α and the properties of the obstacle field for both a single agent and a population of agents; we show that in both cases, α decreases as the obstacle density increases, and that the rate of decrease is greater for smaller obstacles. Our work suggests that it may be inappropriate to model transport through a crowded environment using widely reported approaches including power laws to describe the mean squared displacement and fractional order differential equations to represent the averaged agent density profiles. The Complexity of Optimal Experimental Design: A Tour from Applied Probability to Experimental Mathematics Ali Eshragh The University of Newcastle, Newcastle, NSW, Australia email: [email protected] In this presentation, we deliver our theoretical and numerical results on the Fisher Information for the birth rate of a partially-observable simple birth process involving n observations. Our goal is to estimate the rate of growth, λ, of a population governed by a simple birth process. We may choose n time points at which to count the number of individuals present, but due to detection difficulties, or constraints on resources, we are able only to observe each individual independently with fixed probability p. We discuss the optimal times at which to make our n observations in order to maximize the Fisher Information for the birth rate λ. Finding an analytical form of the Fisher Information in general appears intractable. Nonetheless, we find a very good approximation for the Fisher Information by exploiting the probabilistic properties of the underlying stochastic process. Both numerical and theoretical results strongly support the latter approximation and confirm its high level of accuracy. However, this approximation is limited to the number of observations. Eventually, we utilized techniques from Experimental Mathematics to calculate the Fisher Information efficiently. 51 An Investigation into Probabilistic Constraint in Optimal Power Flow Model Soorena Ezzati Federation University, Ballarat, Victoria, Australia email: [email protected] Coauthors: Musa Mammadov and Sid Kulkarni There are many accepted optimisation models available for electricity distribution networks and one of these models is optimal power flow (OPF). The main objective of an OPF model is to minimise total cost (including manufacturing, instalment and maintenance costs) of an electricity network. Safety is one of the most challenging aspects of designing electricity networks. Although several reliability indices are introduced in the literature to take into account uncertainties, there is no probabilistic constraint/model for these networks. In this paper, we introduce a safety condition for an electricity network based on the existing constraints in the related OPF model. This condition will then be used to formulate a performance function for the electricity network. A probabilistic constraint is introduced using the defined performance function to keep failure probability of the network below than a predetermined accepted level. It’s anticipated to experience an extra cost while the probabilistic constraint is introduced. A non-deterministic optimisation problem must be formulated to minimise the extra cost which is experienced by considering network reliability using probabilistic constraint. Modeling the care pathway for stroke patients Mark Fackrell University of Melbourne, Melbourne, Victoria, Australia email: [email protected] Coauthors: Ria Szeredi and Peter Taylor When patients who have suffered a stroke present at a hospital emergency department, depending on the severity, it is vital that they are seen and treated quickly so that their health outcomes are optimal. In this talk we will analyse and discuss the care pathway for stroke patients at a major metropolitan hospital. We establish that while the number of stroke arrivals each day is distributed according to a Poisson distribution, the arrival process is not Poisson. Also, we fit phase-type distributions to the length of stay in hospital of stroke patients, which yields some interesting results. GPU accelerated algorithms for computing matrix function vector products Megan Farquhar Queensland University of Technology, Brisbane, Queensland, Australia email: [email protected] Coauthors: Timothy Moroney, Qianqian Yang and Ian Turner Recently, there has been increased interest in the use of fractional diffusion models in many applications. Computing the numerical solution of these models often requires the computation of a matrix function vector product involving a sparse matrix raised to a fractional power. The fractional power arises as a result of the nonlocal nature of the operator, and motivates numerical methods that avoid the direct computation and storage of the resulting dense matrix . In this talk we introduce methods for computing matrix function vector products such as these that take advantage of GPU acceleration. The introduction of GPU-accelerated techniques results in improved computational times compared to CPU-only algorithms. We use a contour integral approach that transforms the computation of matrix function vector products into the sum of weighted shifted linear system solves. We improve the convergence of our methods and thereby significantly reduce device memory overheads by introducing two levels of preconditioning. We demonstrate the effectiveness of our approach by presenting results that demonstrate an order of magnitude speed up. We also show how the method can be used to efficiently compute numerical solutions to a fractional diffusion equation. 52 Periodically forced circulation near the shore of a lake Duncan Farrow Murdoch University, Murdoch WA, Australia email: Approximate analytical and numerical solutions for a model of the near-shore circulation in a lake subject to two diurnal forcing mechanisms are presented. The first mechanism is a heating/cooling term in the heat equation representing the daytime heating and nighttime cooling of the diurnal cycle. The second is a periodic surface stress modelling a sea-breeze/gully wind system typical of some coastal regions. The two forcing mechanisms can either act together or against each other depending on their relative phase. When the forcing mechanisms are opposed and of sufficient strength unstable density profiles lead to secondary circulation. Existence Results for the Thomas–Fermi model of the atom with Bohr boundary conditions. Nick Fewster-Young UNSW Australia, Sydney, NSW, Australia email: [email protected] In 1927, L. H. Thomas and E. Fermi derived a nonlinear differential equation that models the electrical potential in an atom under varying conditions. This talk investigates the instance when the atom is neutral; presenting novel existence results concerning the theory and a numerical approximation to a solution. The results complement and extend on the work of R. Agarwal and D. O’Regan in Singular Differential Equations. Also, the work aligns with the computation methods derived by C. Chan and Y. Hon for a numerical solution in 1988. The methods used are differential inequalities to yield a priori bounds on possible solutions and toplogical methods to prove the existence results. Patient Flows and Markov Decision Processes Jerzy Filar Flinders University, Adelaide, Australia email: [email protected] Coauthors: Anthony Clissold While the flow of patients in a hospital is a complex stochastic process, bed occupancy data exhibit characteristic regularities that lend themselves to quantitative modelling. In this study midnight census data from a three year period at Flinders Medical Centre (FMC) were used to develop a coarse Markov Decision Process (MDP) model whose steady state behaviours were in general concordance with the observed data under the “business as usual” scenario. Of course, the model also included multiple, possible, actions to alleviate congestion, from which alternative operating scenarios could be derived. The model and some of its findings will be briefly described in this presentation. 53 Spatiotemporal mathematical modelling of mutations of the dhps gene in African Plasmodium falciparum malaria Jennifer Flegg Monash University, Melbourne, Australia email: [email protected] Plasmodium falciparum malaria has repeatedly evolved resistance to antimalarial drugs, thwarting efforts to control and eliminate the disease and contributing to an increase in mortality. In this talk I will discuss a mathematical model developed to map the spatiotemporal trends in the distribution of mutations in the dihydropteroate synthetase (dhps) gene that are highly correlated with resistance to sulphadoxine-pyrimethamine (SP). The aim of this work was to present a proof of concept for spatiotemporal modelling of trends in antimalarial drug resistance that can be applied to monitor trends in resistance of other antimalarials, as they emerge or spread. Prevalence data of single nucleotide polymorphisms in three codon positions of the dhps gene from published studies across Africa were used within a geostatistical model to create predictive surfaces of the dhps540E mutation over the spatial domain of sub-Saharan Africa from 1990-2010. The statistical model was implemented within a Bayesian framework and quantified the associated uncertainty of the prediction of the prevalence of the dhps540E mutation in sub-Saharan Africa. The maps presented visualize the changing prevalence of the dhps540E mutation in sub-Saharan Africa. These allow prediction of space-time trends in the parasite resistance to SP, and provide probability distributions of resistance prevalence in places where no data are available as well as insight on the spread of resistance in a way that the data alone do not allow. The results of this work will be extended to design optimal sampling strategies for the future molecular surveillance of resistance, providing a proof of concept for similar techniques to design optimal strategies to monitor resistance to ACT. What is Fluid Turbulence? Larry Forbes University of Tasmania, Hobart, Tasmania, Australia email: [email protected] Modelling turbulent fluid flow remains one of the difficult unsolved problems of classical mechanics. An enormous amount of work has been done on this over the past century, and much is known about the phenomenon experimentally. There are also many computer codes that account for turbulence, usually based on heuristic models involving somewhat arbitrary closure assumptions. Although it is usually stated that turbulence is governed by the Navier-Stokes equations, these equations do not predict the transition to turbulent flow correctly, nor are they used exactly in computer codes for turbulence. This talk examines the alternative hypothesis that turbulence is, in fact, a manifestation of weakly non-Newtonian behaviour. The difference from Navier-Stokes theory is discussed, and the transition to turbulence is investigated. The aim is to obtain a more coherent account of the underlying physics. Residence-Time Distribution of Heaped and Sloped Powder Layers in a Conical Mass-Flow Hopper Luke Fullard Massey University, Palmerston North, New Zealand email: [email protected] Coauthors: Clive Davies and Sam Irvine The flow of a hypothetical Coulomb material flowing under gravity from a conical mass-flow hopper is modelled using stress field theory. The assumptions inherent for a Coulomb material can be combined with the assumption of radial flow within the hopper to determine the velocity profile within the hopper. From the velocity profile, 54 ejection times and residence time distributions may be calculated. Since, in a real granular system, the powder layer interface is generally not flat, but sloped at some angle, (nominally the angle of repose), the residence time distribution and ejection times will be dependent on the initial geometry of the powder layers. Residence time distributions and ejection times are calculated for a given granular material in a conical mass-flow hopper firstly for the case of flat layers, secondly for the case where the powder forms a conical heap at the angle of repose, and thirdly for the case when the powder is sloping against a wall. It is found that the shape of the powder layers greatly changes the residence time distribution and ejection times in the system, and needs to be considered when performing residence time measurements in the industrial setting. Achilles tendon turnover and adaptive remodelling Bruce Gardiner The University of Western Australia, Crawley, WA, Australia email: [email protected] Coauthors: Stuart Young, Arash Mehdizadeh, Jonas Rubenson and David Smith A mathematical model of Achilles tendon damage, repair and adaption in normal daily activity is proposed. Key aspects of the model include (1) reproducing the non-linear stress-strain behaviour of tendon based on a distribution of collagen fibre lengths, (2) a stochastic model of individual fibre mechanical and proteolytic damage and cell-mediated repair based on known biology, and (3) incorporating the tendon model into a multitimescale model of the muscle-tendon unit. Tendon efficiency is assessed using metabolic energy costs. With this model, the predicted tendon remodelling, based on individual fibre damage and repair, is found to minimise metabolic costs of the tendon. That is, physical activity causing mechanical damage and repair to tendon fibres leads to shifts in the fibre length distribution and overall efficiency of the muscle-tendon unit. Conversely a lack of physical activity leads to reduction of tendon metabolic efficiency due to a reduced mechanical damage, but increased proteolytic damage. The proposed model therefore provides a framework for understanding the role of physical activity in tendon health and adaption. A modification to the component-by-component construction that simultaneously chooses the weights Alexander Gilbert University of New South Wales, NSW, Australia email: [email protected] Coauthors: Frances Kuo and Ian Sloan We present a novel modification of the component-by-component (CBC) algorithm for choosing the generating vector of a rank-1 lattice rule that simultaneously chooses the optimal weights. Assuming that a bound on the norm of the integrand is known we use a simple minimisation to obtain a formula for the weight in each dimension. Numerical results are also provided. 55 Impact of delta hepatitis on hepatitis B epidemiology and optimal intervention policies Ashish Goyal The University of NSW, Sydney, Australia email: [email protected] Coauthors: John M. Murray The major cause of liver cancer around the globe is hepatitis B virus (HBV) which also contributes to a large number of deaths due to liver failure. Hepatitis delta virus (HDV) is as potentially alarming as HBV since life threatening cases are 10 times more likely with HBV-HDV dual infection as compared to HBV mono-infection. Quantitative modelling can lead to a better understanding of HDV epidemiology and health policies to reduce its impact. Numerous studies have captured the transmission dynamics of HBV in a population, including determining optimal controls to curb HBV. However the impact of HDV has not been considered. Therefore, we construct a mathematical model to represent the transmission of HBV and HDV, and compare both the health benefit and cost outcomes of four interventions: testing with HBV adult vaccination (diagnosis), diagnosis with antiviral treatment for HBV mono-infected individuals, diagnosis with antiviral treatment for dually infected individuals and awareness programs. We find that the presence of HDV makes little difference to the structure of optimal control policies. However, HBV prevalence, HDV prevalence, the cost per capita at 50 years and the death toll all increase significantly in moderate and high HDV endemic regions compared to HBV mono-infected regions. Modelling also showed that in highly HDV endemic countries with poor infrastructure, high efficacy awareness programs can be used as a substitute for high cost antiviral treatment. These results can assist policymakers. Bootstrapping methods for prediction intervals for solar radiation forecasts Adrian Grantham University of South Australia, Adelaide, South Australia, Australia email: [email protected] Coauthors: Yulia Gel and John Boland We first develop a forecasting method to forecast one step-ahead hourly solar global horizontal irradiance. Fourier analysis is used to identify any periodicity in the series and then an autoregressive model is used to identify any serial correlation in the time series. The residuals are then placed into a 2-dimensional matrix according to the sun elevation and solar hour angle. The rows correspond to sun elevations in increments of 10 degrees and the columns correspond to solar hour angles in increments of 15 degrees. This takes care of the systematic variation in variance in the time series. We then generate N synthetic forecasts by using a bootstrapping technique. For each hour, a residual is randomly sampled (with replacement) from the 2-dimensional matrix, according to the sun’s elevation and hour angle, and is added to the forecast to generate a synthetic value. Then for each hour, the 2.5 and 97.5 percentile from the N synthetic forecasts are calculated. This provides the lower and upper prediction intervals respectively in which 95% of the forecasts will lie. We then compare this bootstrapping method to another method where the prediction intervals are generated using the percentiles from the residuals in the 2-dimensional matrix. We chose the simpler and more efficient percentile method for generating prediction intervals. 56 It’s not just what you do, it’s where you do it: Signalling through Akt Catheryn Gray University of New South Wales, NSW, Australia email: [email protected] Coauthors: Adelle Coster Akt/PKB (protein kinase B) is a key biochemical regulator within mammalian cells. It is a switch-point for numerous signalling pathways that display distinct signalling modalities. One of these key pathways is the regulation of glucose transport by insulin. The phosphorylation (activation) of only a small percentage of the Akt pool in insulin-sensitive cells results in maximal activation of downstream components: it is a very low threshold switch. Akt is phosphorylated at the plasma membrane (PM) but is found in the phosphorylated state both at the PM and in the cytosol. Recent experimental evidence suggests that the physical location of Akt, and not just its phosphorylation state, is an important determinant of downstream regulation. Here we present an experimentally validated four compartment model of Akt activation and its effect on some of its downstream components in the insulin signalling pathway. Mathematical models for cell-extracellular matrix interactions in tissue development Edward Green University of Adelaide, SA, Australia email: [email protected] Tissue engineers hope in future to be able to grow functional tissues in vitro to replace those that are damaged by injury, disease, or simple wear and tear. Many cell culture methods involve seeding cells within gels such as collagen, designed to mimic the cells’ environment in vivo. Amongst other factors, it is clear that mechanical interactions between cells and the extracellular matrix (ECM) in which they reside play an important role in tissue development. This talk presents a mathematical model which explores the role played by the anisotropic mechanics of the ECM in shaping the form of tissues grown in vitro. Modelling Surtseyan Ejecta Emma Greenbank Victoria University, Wellington, New Zealand email: [email protected] Coauthors: Mark McGuinness Surtseyan Ejecta are formed in shallow sub-aqueous eruptions. They occur when water containing sediments, or sediments saturated with water, sink, as a mud, into the magma during an eruption and are ejected from the volcano in a ball of magma. After ejection there is a large temperature gradient between the magma at approximately 1000 ◦ C and the mud at 20 ◦ C. As the temperature of the mud increases the water, in the mud, evaporates causing the pressure to increase until either the pressure exceeds the tensile strength of the magma, causing an explosion, or the water source is depleted due to the steam escaping through the magma. The volcanological question is whether the ball of magma ruptures. There is evidence of intact ejecta so we can conclude it does not always occur. I am developing a set of equations that accurately and transiently model the changes in temperature and pressure in surtseyan ejecta. These equations are then used to predict the conditions needed for an explosion to occur. In my presentation I will share some of the progress I have made. 57 Nonparametric comparison of regression surfaces to assess the impacts of vegetation re-growth on wind fields Rachael Griffiths University of New South Wales, Canberra, Australia email: [email protected] Bivariate distributions representing the wind direction response to changing prevailing wind conditions can be empirically estimated from discrete data using a variety of methods, including thin-plate-smoothing splines. These distributions are presented here as regression surfaces over the torus and a statistical test for equality between two surfaces is constructed. The Wild bootstrap algorithm is implemented to construct the distributions of two test statistics based on (1) the difference between the two estimated surfaces and (2) the difference between each surface and the surface constructed from the combined dataset. A comparison of wind response surfaces is used to investigate whether vegetation re-growth has a significant impact on wind behaviour across complex terrain. Stretching viscous threads Bronwyn Hajek University of South Australia, SA, Australia email: [email protected] Coauthors: Yvonne Stokes and Jonathan Wylie The drawing of optical fibres is an important example of a viscous extensional flow where an axisymmetric viscous thread is pulled at its ends. Other examples include the drawing of glass and polymer fibres for optical microscopy and for glass microelectrodes. Using a one dimensional model, we examine the evolution of axisymmetric viscous threads of various initial shapes that are pulled from both ends with a prescribed velocity which may vary with time. We find asymptotic expressions for the evolving thread shape, both when inertia is small, and when inertia becomes important at large times. Both the small inertia solutions and large time asymptotic expressions compare well with numerical solutions. Applying Polynomial Chaos to Epidemic Models David Harman Griffith University, Brisbane, Australia email: [email protected] Coauthors: Peter Johnston Epidemic Models constructed using compartment models consisting of systems of ordinary differential equations are widely used and studied. However, the parameters within these models, as well as their initial conditions, are rarely known with accuracy. It is important to include the uncertainty in these parameters and initial conditions into our model or our model may give misleading results. Generalised Polynomial Chaos (gPC) is a new method that incorporates the probability distribution of these parameters and initial conditions directly into the model. gPC involves two different methods: stochastic Galerkin and stochastic collocation. The stochastic Galerkin method expands the solution as the sum of deterministic solutions multiplied by orthogonal polynomials from the Askey Scheme, where the weight function of the orthogonal polynomials matches the probability density function of the parameters. The stochastic collocation method evaluates the model at specific nodes (which are the roots of the orthogonal polynomials used in the stochastic Galerkin method) and multiplies these node evaluations by predetermined weights. From both the stochastic Galerkin and stochastic collocation methods, the mean and variance can easily be determined. gPC has many advantages over existing methods, such as Monte Carlo sampling. gPC is much less computationally expensive. Additionally, once gPC has been applied to the model, the parameter values and initial conditions can easily be adjusted without having to rederive the gPC equations. 58 During my talk, I will look briefly at the gPC method and its applications to epidemic modeling. I will then compare the solutions found with gPC with Monte Carlo sampling as well as the time taken to determine each solution. Mixed-Mode Oscillations and Canard orbits in Chemical Oscillators Cris Hasan University of Auckland, Auckland, New Zealand email: [email protected] Coauthors: Hinke Osinga and Bernd Krauskopf A mixed-mode oscillation (MMO) is a complex waveform with a pattern of alternating small- and largeamplitude oscillations. MMOs have been observed experimentally in many physical and biological applications, but most notably in chemical reactions. We are interested in MMOs that appear in chemical systems with one fast and two slow variables. The presence of slow and fast epochs in such systems provides a mechanism for generating small and large oscillations. The mathematical analysis of MMOs is very geometric in nature and based on singular limits of the time-scale ratios. Near the singular limit one finds so-called slow manifolds that guide the dynamics on the slow time scale. In systems with one fast and two slow variables, slow manifolds are surfaces that can be either attracting or repelling. Transversal intersections between attracting and repelling slow manifolds are called canard orbits. Our aim is to analyse how slow manifolds and canard orbits organise the patterns of MMOs in a model of an autocatalytic chemical reaction. A new closed-form formula for pricing European options under a skew Brownian motion Xin-Jiang He University of Wollongong, Wollongong, NSW, Australia email: [email protected] Coauthors: Song-Ping Zhu In this paper, we present a new pricing formula based on a modified Black-Scholes model with the standard Brownian motion being replaced by a special type of skew Brownian motions. In particular, the adopted stochastic process for the dynamics of the underlying asset is the sum of a standard Brownian motion and a reflected Brownian motion that are independent of each other. The motivation for such a modification originates from observations of the non-normal distribution of asset log-price in the financial markets (see Peiro(1999), Rachev(2005), Kim(1999)), which are at odds with one of the fundamental assumptions in the Black-Scholes pricing theory. Although Corns & Satchell(2007) have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Corns & Satchell(2007) are, but also present a new closed-form pricing formula based on a new equivalent martingale measure. The newly derived option pricing formula takes the Black-Scholes formula as a special case and it does not add any significantly extra burden in terms of numerical computations involved in calculating option values, compared with those involved in invoking the Black-Scholes formula. Amazingly, the simple analytic form of the BlackScholes formula is preserved by the new formula and an elegant financial interpretation can also be given under the new martingale measure. 59 Mobile kangaroo to sedentary Abalone - what scale to manage? John Hearne RMIT University, Melbourne, Vic, Australia email: [email protected] We consider two species of contrasting mobility: sedentary abalone and free-roaming kangaroo. Abalone are managed at a zone level while commercial divers are profit-driven and make decisions regarding where to harvest at a reef level. What are the implications of this? It is claimed that kangaroo cannot be commercially ‘farmed’ as they are not constrained by standard fencing practices. They roam freely amongst different properties according to the availability of forage. Can a property owner manipulate matters to benefit from harvesting animals that have been raised largely on a neighbour’s property? Differential equation models of these systems will be formulated and analysed. Some results will be presented that yield some insights into these issues and the appropriate scale of management. A model of Ebola for evaluating control Roslyn Hickson IBM Research, Australia, Australia email: [email protected] Coauthors: Emma McBryde, Rob Moss, Nicholas Geard and Matthew Davis The recent outbreak of Ebola in Western Africa has infected an estimated twenty thousand people so far, with approximately eight thousand deaths. This outbreak has highlighted the importance of a strong healthcare system, with beds in isolation units, home isolation, and the important role of healthcare workers in stopping an epidemic. We have formulated a model of Ebola Virus Disease to consider these effects, with spatial components incorporated through a metapopulation model. We demonstrate possible results of control strategies in Western Africa, using the Spatio-Temporal Epidemiological Modeler (STEM). Washing sugar pulp with dirty maths. Graeme Hocking Maths & Stats, Murdoch University, Western Australia, Australia email: [email protected] Sugar pulp (Megasse) is flushed of the sugar it contains as it moves along a conveyor by cycling water through a system of sprinklers and collectors. The unit is completely enclosed and is therefore difficult to monitor for water level, overflow or drying. A patchwork of mathematical techniques is used to set up a very simple model to monitor the situation in real time. This was a problem at the 2014 Mathematics-in-Industry Conference in Johannesburg, South Africa. Some general management advice can be given from the simple model. 60 Exploring bifurcations and seasonality in a mathematical model of childhood disease Alexandra Hogan The Australian National University, Canberra, ACT, Australia email: [email protected] Coauthors: Kathryn Glass, Hannah Moore and Bob Anderssen Respiratory syncytial virus (RSV) is the main cause of acute lower respiratory tract infections in infants and young children, with almost all children infected by the age of two. Because of the significant health care and economic burden of RSV, an improved understand of its transmission dynamics will benefit health care and vaccination planning. In temperate climates, RSV dynamics are highly seasonal, with mid-winter peaks and low incidence during the summer months. While the dynamics of RSV infection are quite complex, we show that they can be captured by an ordinary differential equation model with sinusoidal forcing. Using parameter values derived from fitting the model to population-linked laboratory data for Western Australia, numerical bifurcation and phase plane analyses are presented that demonstrate the range of different seasonal patterns that the model can reproduce. Based on this analysis, we suggest possible drivers of the different seasonal patterns observed globally, and discuss the next steps for this research. The interaction of acidic tumours and chemotherapy Andrew Holder University of Wollongong, Wollongong, NSW, Australia email: [email protected] Coauthors: Marianito Rodrigo Tumour development is a complex process with many well observed yet poorly understood processes. As such there are many opportunities for mathematics to help contribute to this field of research. One such process with this potential is aerobic glycolysis. Aerobic glycolysis or the “Warburg Effect” is a type of metabolism that the vast majority of tumours possess. While the physical process of aerobic glycolysis is well understood, the reasons that tumours acquire this mechanism are still debated. A consequence of aerobic glycolysis is the production of excess protons that result in the acidification of the tumour microenvironment. This acidic environment is inhospitable to normal cells and as such results in normal cell death. It is therefore proposed that this acidification provides a mechanism for invasion known as the acid-mediation hypothesis. This hypothesis has been examined so far as a relatively closed system, in that very few external influences, such as a treatment, have been considered. As such, I have considered a model for acid-mediated tumour invasion with chemotherapy intervention. In this talk I will provide a brief explanation of the acid-mediation hypothesis and will present the results obtained from the analysis of the model that considers the interaction between an acidic tumour and chemotherapy drug. Asymmetries in the distribution of gene expression noise direct spatial organization in the developing mammalian embryo William Holmes University of Melbourne, Melbourne, VIC, Australia email: [email protected] A critical event in early mammalian embryo development is robust construction of a pluripotent inner cell mass surrounded by a trophoectoderm. Here, we utilize multi-scale, spatial stochastic modeling in conjunction with quantitative immunofluorescence imaging to uncover the design principles responsible for robust establishment of blastocysts. Results show that by controlling the timing and pace of cell fate specification, the embryo creates a crucial window of time where noise in gene regulation promotes accurate organization of these multicellular 61 structures. Imaging results further indicate different gene products (Oct4 and Cdx2) exhibit significantly different levels of noise variation. Surprisingly, this asymmetry is functional and provides a novel means to balance the positive and negative influence of stochasticity on organization. More generally, these results suggest noise has a crucial positive role in spatial organization, and that levels of stochasticity may be tuned much like expression levels themselves. On the Euclidean Dimension of Graphs Jin Hyup Hong Great Neck South High School, Great Neck, USA email: The Euclidean dimension of a graph G is defined to be the smallest integer d such that the vertices of G can be located in Rd in such a way that the two vertices are unit distance apart if and only if they are adjacent to G. In this paper we determine the Euclidean dimension for twelve well-known graphs. Five of these graphs, D¨ urer, Franklin, Desargues, Heawood and Tietze can be embedded in the plane, while the remaining graphs, Chv´ atal, Goldner-Harrary, Herschel, Fritschr, Gr¨otzsch, Hoffman and Soifer have Euclidean dimension 3. We also present explicit embeddings for all these graphs. A comparison of computational techniques of the key properties of Markov Chains Jeffrey Hunter Auckland University of Technology, Auckland, New Zealand email: [email protected] The presenter has recently been exploring the accurate computation of the stationary distribution for finite Markov chains based upon the Grassman, Taksar and Heyman (GTH) algorithm ([1]) with further extensions of this procedure, based upon the ideas of Kohlas ([2]), for finding the mean first passage time matrix. The methods are numerically stable as they do not involve subtraction. In addition, a number of perturbation techniques, where the rows of the transition matrix are sequentially updated, are also considered for computing these quantities. These techniques, together with some standard techniques using matrix inverses and generalized inverses, are compared for accuracy, using some test problems from the literature. References: [1]Grassman W.K., Taksar M.I., and Heyman D.P., Regenerative analysis and steady state distributions for Markov chains, Oper. Res. 33, (1985), 1107–1116. [2]Kohlas J. Numerical computation of mean first passage times and absorption probabilities in Markov and semi-Markov models, Zeit fur Oper Res, 30, (1986), 197–207. 62 Numerical Continuation of Equilibria of an Atherosclerosis Model Md Hamidul Islam Griffith University, Brisbane, QLD, Australia email: [email protected] Coauthors: Peter R. Johnston Atherosclerosis is a chronic inflammatory disease. Elevated concentration of low-density lipoprotein (LDL) and their subsequent modification in arterial walls leads to the initiation and hence overall progression of this disease. Endothelial dysfunction, resulting from modified lipoprotein in the arterial wall, permits accelerated transport of monocytes into the arterial wall, where they differentiate into macrophages. Modified LDLs are then taken up by the macrophages to form foam cells, which in turn initiate a series of intracellular events that may lead to a chronic inflammatory process. This auto-amplified process leads to the formation of an atherosclerotic plaque. We present a mathematical model of the early stages atherosclerosis obtained from a quasi-steady state approximation of a model proposed by Bulelzai and Dubbeldam (2012). We reduce the original model, consisting of four ordinary differential equations, to a system of two ODEs. The remaining system describes the time evolution of monocytes and macrophages. We perform a phase plane analysis on this reduced system corresponding to different parametric conditions to explore the role of monocytes/macrophages in the initiation of an inflammatory process. Monocytes induce the oxidation of low-density lipoprotein (ox-LDL), and ox-LDL in turn induces the migration of monocytes. Corresponding to different oxidation rates of LDL and migration rates of monocyte, we obtain a range of initial monocyte concentrations which lead to different outcomes. Any initial value chosen from this range leads to an inflammatory reaction, while the system remains healthy for an initial state outside of this range. We also find that this range increases with the increasing uptake rate of ox-LDL by the macrophage. Influence of homeostasis on the long-time-limit behaviour of an autoimmune disease Owen Jepps Griffith University, Brisbane, QLD, Australia email: [email protected] Coauthors: Lindsay Nicholson and David Nicholson We have recently developed ODE models for experimental autoimmune uveitis (EAU), a disease characterised by T-cell-mediated inflammation of the retina and choroid. Uveitis is the second commonest cause of blindness in the working population in the developed world. EAU serves not only as an important biological model for uveitis in humans, but more generally in developing our understanding of autoimmunity and disease in immuneprivileged environments, where immune-cell species are compartmentalised inside and outside the relevant organ (in our case the eye) during autoimmune disease. Interestingly, our mathematical models of EAU bear important similarities with epidemiological models of mosquito-borne diseases such as malaria and dengue fever. Despite their complexity, the long-time-limit behaviour of our models always leads to either a globally asymptotically stable (GAS) disease-free equilibrium (DFE), or an endemic equilibrium (EE) which is either GAS or admits globally attracting limit cycles. In this talk I will discuss recent work in which we have used Lyapunov functions to establish the global stability of the DFE and EE, under appropriate conditions on the parameters. I will also outline the crucial role played by the homeostasis terms, which describing the disease-free cell-population dynamics, in determining the nature of the endemic dynamics. 63 Incorporating the effects of chemotherapeutic drugs into a multiphase model of cancer spheroid growth Wang Jin Queensland University of Technology, Brisbane, Australia email: [email protected] Multiphase models of cancer spheroid growth have been studied for about two decades, yet most studies focus on models in one-dimensional Cartesian geometry. In this work we extend a spheroid model developed by Breward and co-workers [1]. We obtain numerical solutions in a spherical geometry, and we classify three different types of solutions: (i) travelling-wave-like, (ii) steady-state, and (iii) phase-separation solutions. We further extend Breward’s model to include the effect of chemotherapy drugs. Using our numerical model we study the responsiveness of the spheroid growth to three different classes of drugs, and we demonstrate how different boundary conditions can be applied to mimic different drug treatment protocols. 1. Breward CJW, Byrne HM, Lewis CE (2002) The role of cell-cell interactions in a two-phase model for avascular tumour growth. Journal of Mathematical Biology 45: 125–152. A monomial transformation for evaluating two-dimensional nearly singular boundary element integrals Barbara Johnston Griffith University, Nathan, Australia email: [email protected] When implementing the boundary element method, particular care must be taken with the evaluation of the resulting singular and nearly-singular integrals. The latter category arises when the source point is close to, but not on, the element of integration. This causes at sharp peak in the integrand because the source point is close to the element, making accurate evaluation difficult. A sinh transformation method, which automatically takes into account both the position of the projection of the source point onto the element and the distance b between the source point and the element, has previously been introduced. This method has been shown to be superior to existing methods in evaluating the 2D nearly singular potential and flux integrals that arise in the solution of Laplace’s equation in three dimensions via the boundary element method. Here a similar, but alternative method, based this time on a monomial transformation, is studied and compared with the sinh transformation method. It is demonstrated that the new monomial method is particularly effective for very small values of b (b ≤ 10−6 ), while remaining equally as easy to implement as the sinh transformation method. Aggression Model for Wolbachia Flies Peter Johnston Griffith University, Nathan, Queensland, Australia email: [email protected] Coauthors: Jeremy Brownlie Wolbachia is a mosquito bacteria used to control dengue fever and malaria. The effect of Wolbachia on the mosquitoes is to shorten their lifespan and change their reproduction patterns. It is also thought that the bacteria changes the way male and female insects interact with one another. Here we will develop a model for male and female fruit flies (because they are used for laboratory experiments) both with and without Wolbachia 64 Consider a population of male and female flies, some of which are infected with Wolbachia. The basic breeding model is as follows: ♂ × ♀ → 100% normal flies ♂w × ♀ → 0 flies ♂w × ♀w → ♂ × ♀w → 95% Wolbachia infected flies 5% normal flies 95% Wolbachia infected flies 5% normal flies where the subscript w indicates that the flies are infected with Wolbachia and × indicates a breeding event. We will derive a simple system of four ordinary differential equations that takes into account the above breeding rules and includes an “aggression” factor for mating relationships. For this system, we will examine the steady states and study their stability. How much information can be obtained from tracking the position of the leading edge in a scratch assay? Stuart Johnston Queensland University of Technology, Queensland, Australia email: [email protected] Coauthors: Matthew Simpson and Sean McElwain Collective cell behaviour is a key component of tissue repair and tumour spreading. Quantification of the processes behind collective cell behaviour, namely the cell diffusivity, D, and the cell proliferation, λ, is critical for the evaluation of putative treatments. We are therefore interested in developing mathematical techniques that can be applied to experimental data to estimate D and λ. Scratch assays are a simple and inexpensive method to observe collective cell behaviour. We describe a new method that combines scratch assay leading edge data, discrete simulations and edge detection methods, which results in robust estimates of D and λ. Steady Saturated-Unsaturated Water Flow in a Sloping Domain and its Application to Landslides. Laura Karantgis La Trobe University, Melbourne, VIC, Australia email: [email protected] Coauthors: Philip Broadbrige and Vincent Lemiale Landslide events, often caused by heavy rainfall, can have a devastating impact on communities and industries. Modelling these complex systems is valuable for predictive and preventative measures to assist in reducing the impact of these events. As the soil water content affects slope stability we will need to model the distribution of water in the soil and its effects on the soil strength parameters. In this talk we will construct a mathematical water infiltration model to predict the behaviour of rainfall through a soil slope to assist in analysing the effects of rainfall on the occurrence of landslides. Models for water infiltration flow through a porous medium have been constructed using Darcy’s law and the Richards equation. In this talk we will use these equations to develop a combined analytical and numerical approach to effectively create a two dimensional saturated-unsaturated water infiltration model that can predict the water table and flow of water through soil for varying parameters such as slope angle, length, rainfall rate and soil type. 65 Activating Lyapunov-based Feedback Design: Nonsmoothness and State Constraints Christopher Kellett University of Newcastle, Callaghan, NSW, Australia email: [email protected] In the 1990’s, significant progress was made in the use of Lyapunov-based techniques for design of feedback stabilisers for systems described by ordinary differential equations. However, in the late 1990’s it was discovered that for many important systems, these designs were not applicable. Feedback stabilisers applicable to a wider class of systems, based on so-called nonsmooth control Lyapunov functions, were proposed in the early 2000’s, though apparently never implemented. An additional drawback of both the original and newer Lyapunov-based designs is that they do not account for state constraints. Herein, we describe the impediment to the original Lyapunov-based designs. We also describe the newer designs and how they might be combined with ideas due to Chetaev from the 1950’s to overcome the problem of dealing with state constraints. Dynamics of systems with three timescales Vivien Kirk University of Auckland, Auckland, New Zealand email: [email protected] Coauthors: Pingyu Nan, Yangyang Wang and Jonathan Rubin Many physical systems have the property that some quantities evolve much faster than others. Dividing timescales into two classes and applying techniques that exploit the timescale separation can yield significant insights about the dynamics of such systems. However, some dynamical phenomena cannot be captured by a two timescale reduction and little is known about how to efficiently study systems with three or more timescales. Motivated by applications in neural dynamics, this talk will discuss an ordinary differential equation model with three timescales. We identify complex oscillations that appear to be intrinsically three timescale phenomena, and use geometric singular perturbation theory to explain the mechanisms underlying these solutions. Lewis Fry Richardson: pioneer of finite difference methods for partial differential equations John Knight University of Sydney, Sydney, Australia email: [email protected] Lewis Fry Richardson (1881-1953) was an English polymath who made important contributions to many fields including numerical weather prediction, finite difference solution of partial differential equations, turbulent flow and diffusion, fractals, and the causes of war. During World War I he invented the field of numerical weather prediction, although his methods were not successfully applied until 1950, after the invention of fast digital computers. Richardson’s first published papers in 1908 concerned the numerical solution of the free surface problem of unconfined saturated soil water flow, arising in the design of drain spacing in peat. He favoured finite difference rather than analytical solutions because he was attacking complicated real world problems which needed numerical answers. In his 1910 paper he mainly considered finite difference solutions for elliptic problems, but also developed finite difference methods for the heat or diffusion equation, which he called ”marching methods.” He developed a three time level explicit method, but needed to use an implicit two level method for the first time step. Unfortunately he stopped the calculations too early to discover that his explicit method for the diffusion equation was unstable. In 1947 Crank and Nicolson became famous by discovering that the Richardson explicit method was unstable, and publishing the Crank Nicolson method which became very widely used. In fact what Crank and Nicolson did was to use the same central difference method as Richardson for the first time step, and then use it for all subsequent time steps. In 1910 Richardson could easily have done the same, but presumably preferred to use a 66 fully explicit method which involved much less calculation. So there is a strong case for calling the 1947 method the Richardson-Crank-Nicolson method. Frequency-domain Monte Carlo method for linear oscillatory gas flows Daniel Ladiges The University of Melbourne, Victoria, Australia email: [email protected] Coauthors: John Sader Gas flows generated by resonating nanoscale devices inherently occur in the non-continuum, low Mach number regime. Numerical simulation of such flows presents a tremendous challenge, which has motivated the development of several Monte Carlo methods for low Mach number flows. We present a frequency-domain Monte Carlo method for oscillatory low Mach number gas flows, based on the linearised Boltzmann equation. This circumvents the need for temporal simulations, providing direct access to both amplitude and phase information using a pseudo-steady algorithm. The proposed method is demonstrated with several examples, and good agreement is found with both existing time-domain Monte Carlo simulations and accurate numerical solutions of the Boltzmann-BGK equation. Further, we present a rigorous statistical method for analysing the convergence of stochastic simulations. Using this approach, we show that simulations in the frequency-domain provide a significant improvement in computational speed compared to existing time-domain Monte Carlo methods. Exact derivation of a neural field model from a network of theta neurons Carlo Laing Massey University, Auckland, New Zealand email: [email protected] Neural field models are used to study macroscopic spatio-temporal patterns in the cortex. Their derivation from networks of model neurons normally involves a number of assumptions, which may not be correct. We present an exact derivation of a neural field model from an infinite network of theta neurons, the canonical form of a Type I neuron. Solving capacitated vehicle routing problems with time windows by goal programming approach Dwi Lestari Yogyakarta State University, Yogyakarta, Indonesia email: Coauthors: Eminugroho Ratnasari and Atmini Dhoruri Abstract. This paper presents how to build multiobjective linear programming model as solution of Capacitated Vehicle Routing Problem with Time Windows (CVRPTW). We use a goal programming approach to solve the model. We have discussed an objective function for two main goals: the first is to minimize the total number of vehicles and the second is to minimize the travelling time of the used vehicles. The proposed model is applied to a problem distribution of Liquefied Petroleum Gas (LPG). Computational results of the proposed model are discussed. 67 A Free Boundary Problem for Corporate Bond with Credit Rating Migration Jin Liang Tongji University, Shanghai, China email: liang [email protected] Coauthors: Bei Hu and Yuan Wu In this work, a free boundary model for pricing a corporate bond with credit rating migration is proposed. This is a new model for credit rating migration. The existence, uniqueness and regularity of the solution for the model are obtained together with some interesting properties. Furthermore, numerical examples are presented. Nanopterons in a granular chain Christopher Lustri University of Sydney, Sydney, Australia email: [email protected] Coauthors: We consider a simple model representing a chain of particles, each of which interacts only with its nearest neighbours. This interaction is governed by an potential, which determines the behaviour of the particles when they are disturbed. One particular model, known as the Toda chain, has been famously shown to admit soliton solutions. In this study, we consider the behaviour of soliton-type solutions in a periodic Toda chain with particles of alternating mass, where m1 and m2 represent the masses of the odd and even particles respectively. We are particularly interested in the asymptotic behaviour when one set of particles is significantly heavier than the other (ie. 0 < m1 /m2 1) . In this case, we find that the system no longer produces pure soliton solutions, but instead the solutions take the form of nanopterons, or weakly nonlocal solitary waves. Specifically, we find that the solitons produce an exponentially-small wavetrain in the far field of constant amplitude. We apply exponential asymptotic methods to determine the behaviour of these far-field waves. Pricing European options written on a hard to borrow stock Guiyuan Ma University of Wollongong, Wollongong, Australia email: [email protected] Coauthors: Songping Zhu In this talk, we shall demonstrate how a restrictive trading environment with stocks being hard to borrow would affect the price of an option with the introduction of a stochastic buy-in rate process. Following the framework of Avellaneda and Lipkin’s (2009) innovative work, which proposed a model with dynamically coupled systems between stock price and buy-in rate, we present two approaches to approximate the weight functions via the Monte Carlo method. Furthermore, we propose a PDE approach to solve the European option problem with a set of appropriate boundary conditions. Finally, we present some numerical results to show the quantitative price drop of an European option written on hard-to-borrow stocks with dividend payments taken into consideration. 68 The wave equation is Toeplitz plus Hankel Shev MacNamara UNSW, NSW, Australia email: [email protected] We begin with the wave equation, which we show has Toeplitz parts and Hankel parts. The Hankel part comes from the boundary, and we try to explain this. Examples with and without reflections are described. Reference: “Functions of Difference Matrices Are Toeplitz Plus Hankel,” Gilbert Strang and Shev MacNamara, SIAM Review (2014) Scattering of acoustic plane waves by obstacles with corners: the effect of rounding. Audrey Markowskei Macquarie University, Sydney, NSW, Australia email: [email protected] Coauthors: Paul Smith If an integral equation approach is employed as the basis of numerical studies of the scattering of plane waves by an obstacle, a common technique for dealing with domains with corners is to round the corners. Using an integral equation formulation designed for domains with smooth boundaries we examine the relationship between the radius of curvature and the convergence of the solution as the radius tends to zero. We then employ an integral equation formulation with a quadrature scheme using a graded mesh designed to achieve a rapidly convergent solution for a domain with a corner exactly represented. A comparison of the two studies allows us to examine the effect of rounding a corner, that is, the correlation between the radius of convergence of the rounded corner and the true results from the domain with a corner. In all cases three different boundary conditions are examined - Dirichlet, Neumann and an impedance condition. The effect of surface wettability on droplet dynamics Lisa Mayo Queensland University of Technology, Brisbane, Australia email: [email protected] Coauthors: Scott McCue and Timothy Moroney In agricultural spray applications, it is of great importance to maximise the retention of the spray formulation by plant foliage. Since many plant species are resistant to wetting by water-based formulations, a surfactant (surface active agent) is often added to the spray mixture in order to increase wettability by way of reducing the interfacial surface tension and thus contact angle between the fluid and leaf surface. In this study, a thin film model is used to simulate the dynamics of droplets in situations where the contact angle plays a major role, such as spray applications. We simulate droplet motion on a virtual leaf surface and observe coalescence behaviours which mimic experimental spray observations. We also consider how the wettability of a surface influences the oval-corner-pearling transition of a sliding drop. 69 Depicting the outbreak and spread of algal blooms in New South Wales Lagoons using NOVA Lynne McArthur RMIT University, Melbourne, Australia email: [email protected] This project is inspired by the infrequent and random occurrence of algal blooms on Avoca Lagoon in central New South Wales, Australia. The triggers for an algal event are poorly understood, as are the elements that drive the spread. There was an outbreak in spring/summer 2012, when the lagoon was rapidly overcome with the algae, which then dissipated by early January. The task here is to identify the conditions which trigger an outbreak and then to estimate the extent and coverage of the bloom.stems. In order to model the spread we have previously used MATLAB to program cellular automata models. This report describes the use of NOVA, which allows a greater capacity to describe the dynamics of each cell and thus more flexibility. The project was designed around identifying the parameters which contribute to the outbreak of a bloom, and those that drive the spread. NOVA had proved to be particularly suited to this kind of modelling. Exploring long-term drivers of pertussis resurgence and improved vaccine control strategies James McCaw University of Melbourne, Victoria, Australia email: [email protected] Coauthors: Patricia Campbell and Jodie McVernon Mass vaccination against pertussis, introduced in many countries in the 1950s, dramatically reduced the impact of this disease. However, several developed countries with longstanding immunisation programs, including Australia, have experienced pertussis resurgence during the past decade. We used mathematical models to identify the determinants of this increase, and define improved vaccine strategies to mitigate disease burden. We constructed a mathematical model of pertussis transmission explicitly capturing the uncertainty in biological and immunological mechanisms thought to be important for transmission and protection. Multiple parameter combinations were simulated, and those which reproduced key features of pertussis disease and infection in Australia were selected for use in predictive models. Simulated projections compared the impact of alternative vaccination schedules on infection incidence. Natural immunity lasting decades longer than vaccine immunity was needed to explain the initial control of pertussis followed by late resurgence. The mean duration of natural immunity exceeded 50 years in almost 90% of simulations matching observed epidemiologic patterns. Removal of a fourth (toddler) dose in Australia in favour of an adolescent dose in 2003 was found to have contributed to resurgence, resulting in a 40% increase in infections in the 18mth-¡4yr age group. Supplementing the existing five dose schedule with an 18mth dose was the best future strategy, resulting in a 43% reduction in incidence in toddlers and 8% reduction in infant incidence from 2014-2020. Long-lasting natural immunity is important in driving long-term trends in pertussis cycles, which must be considered when evaluating and designing vaccination strategies. Vaccine protection is comparatively shortlived, requiring multiple boosters over the first two decades of life for sustained direct and indirect protection. This requirement is particularly marked in contexts where pertussis circulation is low, and opportunities for natural boosting of immunity are limited. 70 Analytical expressions for infection path probabilities of an SIR model on small networks Karen McCulloch Massey University, Albany, New Zealand email: [email protected] A significant amount of effort has been directed at understanding how the structure of a contact network can impact the spread of an infection through a population. However, investigating how an infection spreads through large networks can be computationally expensive and is seldom mathematically tractable. This research is focused on finding tractable results to aid our understanding of how infections spread through networks. Given that there is one initial infectious individual (node) in a network, how many other individuals are likely to become infected and which ones? Here, we use an SIR (Susceptible-Infectious-Recovered) model for the spread of an infection to address questions like these for small networks of different topological structure. We also illustrate how we can use the results from small networks to analytically describe how the infection spreads through a larger network. The key here is to correctly decompose the larger network into an appropriate assemblage of small networks so that the results are exact. Derivation of Fractional SIR Model Anna McGann UNSW, Sydney, Australia email: [email protected] Coauthors: Chris Angstmann, Bruce Henry and James Nichols This talk will show the derivation for the governing equations that describe the evolution of an SIR model for the spread of a disease in which the probability of recovering from the disease is a function of the time since infection. The derivation is based on a stochastic process, a continuous time random walk, describing the motion of individuals through SIR compartments. If the probability of recovering is power law distributed then the governing equations involve fractional-order derivatives. It can be shown that the fractional order recovery model is consistent with the general age-structured Kermack-McKendrick SIR model. I will also discuss steady state solutions and numerical solutions of the governing equations. Erupting Dusts Mark McGuinness Victoria University of Wellington, Wellington, New Zealand email: [email protected] Coauthors: Harpreet Singh We present a new model for the initiation of high-speed eruptive two-phase dust flows in the laboratory. Shocktube experiments have been conducted on beds of solid particles in nitrogen under high pressure, which are suddenly decompressed. Our model is successful in explaining the slab-like structures that are often observed during initiation of bed movement, by considering the interaction between the compressible flow of gas through the bed and the stress field in the particle bed, which ruptures when bed cohesion is overcome by the effective stress in the bed generated by the gas flow. Our model includes the effects of overburden and wall friction, and predicts that all layered configurations will rupture initially in this fashion, consistent with experimental observation. We also find that the modelled dependence of layer size on particle size is a good match to experiment. 71 Wind power simulation using Correlated Innovation Matrix and Wavelet Multi-resolution Analysis approaches Dougal McQueen University of Canterbury, Canterbury, New Zealand email: [email protected] Coauthors: Alan Wood and Allan Miller To meet carbon emissions targets, increased demand, and replace retiring plant it will be necessary to construct new electricity generation plant in New Zealand. One of the least cost and carbon neutral methods of generating electricity is wind power. The intermittent and variable nature of wind coupled with the passive reaction of wind turbines ensures that wind power requires scheduled and spinning reserves. Reserve requirements can be alleviated, to an extent, by distributing wind farms throughout the country, exploiting the diversity of wind. However, spatial diversification may increase costs through reducing economies of scale in wind farm construction and increasing transmission requirements. Transmission expansion has long lead times, and needs good models of an uncertain future mix of generation size, type and location. There is insufficient measured wind power or wind speed data to assess the trade-offs for envisaged wind development scenarios hence a model of wind power is required. The model must be temporally and spatially congruent with respect to wind, demand, and other generation types. In this paper specific models are developed applying wind speed time-series derived from a Numerical Weather Prediction model, temporal interpolation methods, transformation to power using wind farm power curves, and assumptions concerning electrical and operational efficiencies. The temporal interpolation, or turbulence modeling, is achieved through two methods: a Correlated Innovation Matrix approach, and a Wavelet Multi-Resolution Analysis approach. The models are used to simulate power time-series for seven wind farms, and subsets of these used to assess a centralised scenario and a diversified scenario. Results are compared with aggregate measured power time-series demonstrating the benefit of spatial diversification and illustrating differences in the turbulence modeling approaches. Lastly progress toward a generalised model is discussed. Spark - a new research tool for investigating novel bushfire spread concepts Claire Miller CSIRO Digital Productivity Flagship, Melbourne, Victoria, Australia email: [email protected] Coauthors: James Hilton, Andrew Sullivan and Mahesh Prakash The improvement of faster than real-time bushfire spread models requires an increased understanding of underlying fire behaviour and spread mechanisms. Current simulation tools for operational use implement onedimensional rate of spread models extrapolated to two dimensions. This is done using methods such as the assumption of fire shape or the adjustment of rates according to the angle between the front normal and the wind direction. Environmental conditions are commonly predefined for computational efficiency and ease of implementation. The next steps for improving these tools are to extend fire spread rate models beyond spread in the direction of the wind and to investigate how the fire front affects its surroundings and, consequently, itself. Our bushfire spread prediction tool, Spark, has been developed with the functionality for use as a research tool. Spark is based around a level set method which makes it both computationally efficient and adaptable to complex fire behaviour and conditions. It can therefore effectively be used to investigate various phenomena and increase understanding of bushfire spread. Elements of spread, such as rate equations and environmental conditions, are all inputs with capabilities to provide dynamic calculations during simulation. Consequently, new fire behaviour concepts can be easily added and investigated. We provide examples of the use of Spark to study how different factors can impact the fire spread. The first example is the inclusion of curvature of the fire front in the spread function. The second example is the incorporation of variation in fuel and wind inputs. 72 Efficient and robust iterative solutions of the potential equation applied to modelling of electrochemical electrodes. Tony Miller Flinders University, Adelaide, Australia email: [email protected] The increasing use of battery technologies for remote energy storage and transport applications highlights the importance of battery management systems. While traditionally such systems have used simple equivalent circuits and empirical calibration factors, new approaches that make use of detailed electrochemical models are being developed. These require solving an electrical conduction (potential) equation with non-linear current sources terms at each step of a time-stepping approach. For this to be computationally feasible in a real time battery management setting, the solution of this potential equation needs to be both efficient and, most importantly, robust. This paper describes an iterative solution technique to do this. It can be thought of as a kind of approximate Newton method with dynamic updating of the iteration parameters. The order of convergence is one, however convergence can be proved under a wide range of conditions. A simple lumped circuit approximation is used to start the iteration. Examples are given which show how the technique performs in some extreme cases. The method is also applicable to other related electrochemical modelling problems. How much does your social network reveal about you? Predictability and social information flow Lewis Mitchell University of Adelaide, Adelaide, SA, Australia email: [email protected] Coauthors: James Bagrow The recent explosion in big data coming from online social networks has led to an increasing interest in bringing quantitative methods to bear on questions in social science. Examples such as the study of emotional contagion have led to substantial interest as well as controversy within this emerging field. In this talk we will discuss one of the processes underlying emotional contagion, namely the flow of information between individuals in a social network. Such an idea leads readily to the concept of predictability for an individual based upon their friend network, allowing us to study how predictability relates to various social characteristics. By analysing a massive data set of messages from Twitter using tools from information theory we present results relating individual predictability and effective vocabulary size to numbers of friends and followers, as well as to cases where one’s friend network is being monitored in order to predict that individual. I’ve Got Fauxs in Different Area Codes. John Mitry The University of Sydney, Sydney, NSW, Australia email: [email protected] Folded singularities are commonly found in mathematical models of cellular activity. The canards associated with these singularities distinguish between different solution behaviours. Thus canards can form physiologically significant boundaries between solutions which represent varied cellular activity. We here present an analysis of faux canards associated with folded saddle singularities. True canards pass from attracting manifolds to repelling manifolds via folded singularities, while faux canards travel from attracting manifolds to repelling manifolds also via folded singularities. There exists a 2-parameter family of faux canards in the vicinity of a folded saddle singularity, while there exists only one true canard. We demonstrate and analyse the oscillatory behaviour of faux canards about the primary faux canard, identifying a dependence on the ratio of the eigenvalues associated with the saddle singularity. Additionally we demonstrate the existence of and characterise the set of secondary faux canards, the number of which also depends on the eigenvalue ratio. Both of these form a novel contribution to the study of faux canards and canards in general. If time permits we shall also describe solutions associated with folded saddle singularities which follow one primary canard (true or faux) and then switch to the other. These switching solutions can also possess oscillatory behaviour, which is itself also a novel 73 contribution. Given these results we observe that the folded saddle singularity, far from being straight forward, is ludicrously complicated. Enclosing solutions of the delay eigenvalue problem Shinya Miyajima Gifu University, Gifu-shi, Japan email: [email protected] Mathematical models consisting of delay-differential equations (DDEs), in the simplest form dx(t) = Ax(t) + Bx(t − τ ), dt A, B ∈ Cn×n , τ ≥ 0, occur naturally in a wide variety of fields related to applied mathematics, such as engineering, control theory, biology, traffic modeling, neural networks, mechanics, and electronic circuits. One common approach for obtaining the solution of the DDEs is to solve the delay eigenvalue problem (DEP): find λ ∈ C and x ∈ Cn \ {0} such that (λI − A − Be−τ λ )x = 0, where I is the n × n identity matrix. If λ and x satisfy this equality, then cxeλt is a solution of the DDEs, where c ∈ C is an arbitrarily constant, which can be determined from initial conditions. From the above discussion, the DEP is important. Numerical methods for solving the DEP have been proposed by several researchers, e.g., Jarlebring, Sakurai, and their colleagues. Since the DEP is the special case of nonlinear eigenvalue problems, numerical methods for solving the nonlinear eigenvalue problems are also applicable. On the other hand, these numerical methods rely on floating point arithmetic and thus cannot provide an exact solution of the DEP. Indeed, they usually give only approximations to exact solutions. In this talk, we consider enclosing the exact solutions of the DEP, specifically, computing intervals which contain the solutions using floating point arithmetic. Such intervals are called “confidence intervals”, and give reliability to the obtained approximations. While there are well-established algorithms for enclosing solutions of the other problems, less attention has been paid to the DEP. When B = 0, the DEP reduces to the standard eigenvalue problem. For the standard eigenvalue problem, effective and efficient methods for enclosing solutions have been proposed in some literatures. On the other hand, these literatures do not mention how to extend these methods to the DEP. The purpose of this talk is to propose a method for enclosing the solutions of the DEP. This method computes the interval containing the solution taking all the possible errors into account. In this method, the DEP is reduced to a system of nonlinear equations, and the interval containing the solution of the nonlinear system is computed based on the Newton operator and Brouwer fixed point theorem. This method moreover verifies that the solution of the DEP contained in the interval is unique by checking the contraction property of the Newton operator. We finally report numerical results to observe the properties of the proposed method. Preconditioned finite volume methods on non-uniform grids for one-dimensional fractional diffusion equations Tim Moroney QUT, Brisbane, Australia email: [email protected] Coauthors: Alex Simmons and Qianqian Yang Fractional diffusion equations are used to model anomalous diffusion processes where the particle scale behaviour is not consistent with Brownian motion. Numerical methods for these equations must deal with fractional derivatives, whose discretisations tend to produce dense coefficient matrices. 74 In this talk I present some recent work on a finite volume method for one-dimensional fractional diffusion equations with Riemann-Liouville fractional derivatives on non-uniform grids. The method utilises a quadrature scheme to discretise the fractional derivatives at control volume faces. The finite volume formulation ensures mass conservation, while the non-uniform mesh allows for refinement in regions of interest. For efficient numerical solution, matrix-free iterative solvers are used, thereby avoiding the need to form the dense coefficient matrix. I discuss how an effective preconditioner can be constructed for this problem to accelerate the convergence of the iterative solver. Furthermore, this preconditioner performs well on other, different-butrelated diffusion equations with fractional Laplacians, for which effective preconditioning has previously been problematic. Epidemic detection and forecasting from surveillance data via Bayesian estimation Robert Moss Melbourne School of Population & Global Health, Melbourne, Australia email: [email protected] Coauthors: Peter Dawson and James McCaw Meteorological forecasting methods combine mathematical models of the weather system with statistical inference methods (such as Bayesian estimation) to determine which model realisations are most likely to yield observations consistent with those obtained from real-world surveillance. The application of these methods to early detection and forecasting of disease epidemics (e.g., seasonal influenza) is a recent innovation, and it remains unclear how extant surveillance systems and other rich data sources (e.g., social media) can be best used for such purposes. In addition, most of these studies have used non-mechanistic statistical models and have therefore sacrificed the ability to develop significant insight into the mechanics of the infection process. We have begun using Bayesian estimation (via the particle filter) to couple mechanistic models of infection with surveillance data from previous influenza seasons in order to evaluate the accuracy and precision of different forecasting strategies during the early stages of an influenza epidemic. Preliminary results include quantification of forecast accuracy as a function of forecasting date and surveillance characteristics (including observation frequency and background noise levels). It is our intention to apply these forecasting methods to near-real-time surveillance data during the 2015 Victorian influenza season and produce weekly forecasts, which we will then be able to evaluate against the true epidemic. A mechanism design approach to efficient dynamic market clearing Ellen Muir The University of Melbourne, Victoria, Australia email: [email protected] Coauthors: Peter Taylor and Simon Loertscher In markets with buyers and sellers that arrive over time, we wish to determine the optimal market clearing policy which maximises expected gains from trade. There is generally a cost associated with delaying trade while buyers and sellers accumulate. However, there is also a loss of efficiency associated with clearing markets that contain few buyers and sellers. We analyse several simple models which capture this tradeoff. Assuming buyers and sellers arrive according to independent random processes, a variety of simplifying behavioural assumptions may be introduced. Once a tractable model is fixed, all possible market clearing policies must be compared to determine the optimal policy. To accomplish this, coupling techniques and dynamic programming methods can be exploited. The results of this analysis have implications for simple dynamic matching markets, some classic static mechanism design setups and lay the groundwork for a mechanism design analysis of dynamic centralised exchanges. 75 Agent-based modelling of hepatitis B virus infection and clearance John Murray UNSW Australia, Sydney, NSW, Australia email: [email protected] Coauthors: Ashish Goyal Hepatitis B virus (HBV) infection and replication occurs in liver hepatocytes. This comprises a dynamic process within each infected cell describing the generation of new intracellular viral components, as well as infection and the immune response throughout the liver. To properly describe this we constructed an agent-based model where each agent represented an hepatocyte and where the state of each infected hepatocyte changed according to the HBV infection cycle and the effects of various components of the immune system. We used the model to test the relative roles of different mechanisms of the immune system in the clearance of acute HBV infection. The time between infection and viremia clearance as well as the amount of liver turnover (HT), assessed against literature estimates, were used as the two major criteria in determining reasonable outcomes. Modelling resulted in 90% of cells containing between 1 and 17 HBV cccDNA (the template of HBV replication) copies and normally distributed at the peak of infection, consistent with experimental findings. Variations in p36 levels, responsible for determining export of virions or recirculation to amplify cccDNA numbers, had a much greater impact on mean cccDNA levels/cell at peak viremia than virus infectivity and cccDNA half-life. Acute infection clearance was possible with HT in the desired range of 0.7 to 1 provided a combined cytolytic and non-cytolytic immune response occurred in conjunction with complete loss of intracellular viral components during cell proliferation. This in silico model provided an excellent basis for investigation of HBV infection. No animals were harmed during the course of these experiments. The Lax pairs of discrete Painlev´ e equations arising from the integer lattice: (A2 + A1 )(1) case Nobutaka Nakazono The University of Sydney, Sydney, NSW, Australia email: [email protected] Coauthors: Nalini Joshi and Yang Shi Construction of the Lax pairs of discrete Painlev´e equations from the Lax pairs of ABS equations are well investigated. In this talk, I will show new method to obtain the Lax pairs of discrete Painlev´e equations by using the integer lattice associated with ABS equations in detail by taking an example of q-Painlev´e equation with the affine Weyl group symmetry of type (A2 + A1 )(1) . This work supported by the Australian Research Council grant # DP130100967. The effects of Wolbachia on dengue transmission dynamics Meksianis Ndii University of Newcastle, Newcastle, Australia email: [email protected] Coauthors: David Allingham, Roslyn Hickson and Kathryn Glass The use of Wolbachia bacterium is a proposed new strategy against dengue. This can affect the dengue transmission dynamics, which is known to be seasonal. In the regions with strong variation in temperature and rainfall, the dengue epidemic generally takes off only in certain time of the year. The time period when imported cases enter the population determines whether the outbreak occurs. This work investigates the effect of Wolbachia on dengue transmission dynamics for different transmission rate, strength of seasonality. Our results show that Wolbachia reduces the time-window in which outbreak can occur. The benefits of Wolbachia depends on the transmission rate, with the bacteria most effective at moderate transmission rate. Also, as the seasonality increases, the Wolbachia is less effective. 76 Biogas production in anaerobic bioreactors Mark Nelson University of Wollongong, Wollongong, New South Wales, Australia email: [email protected] Anaerobic digesters provide an efficient waste treatment method, reducing the organic loading of the waste stream, whilst producing residues, both liquid and solid, that can be used as biofertilisers. A further by-product of the digestion process is a methane rich biogas which can be converted, via combustion, into electricity. It has been estimated that a substantial amount of the power requirements of an anaerobic digester, perhaps all of the power requirements, could be obtained by optimising the production of the biogas, which is a clean burning environmentally friendly fuel. The steady-state biogas production in an anaerobic digester is investigated using an engineering model consisting of species equations in liquid and gas phases. The model contains differential equations for eight substrates, five microbial species and four gases. There are six rate expressions, thirty-three yield values and thirty-eight other parameter values (including initial conditions). Some additional parameter values are required that are not specified in the paper. Can we say anything useful about this model, or should we just whack the model equations into a continuation package? Modelling reaction-diffusion systems with anomalous diffusion using a discrete time random walk, with examples in modelling of HIV James Nichols UNSW, Sydney, Australia email: [email protected] Coauthors: Chris Angstmann, Bruce Henry, John Murray and Isaac Donnelly In nature anomalous diffusion of particles often arises, for example in crowded environments or where trapping occurs. Modelling of this diffusion system typically involves reaction-diffusion PDEs with fractional derivatives (fPDEs). Solving these fPDEs, both analytically and numerically, is difficult. We proposed an numerical method, the Discrete Time Random Walk, which approximates the fPDE. By virtue of being derived as a physical stochastic process, conservation of mass is preserved and the method is stable. We present recent developments, including demonstrations of multiple species reaction sub-diffusion systems, and applications of these sorts of systems to early-infection dynamics in epithelial tissue of HIV. Modeling wound closure in an epethelial cell sheet using the Cellular Potts Model Adrian Noppe University of Queensland, QLD, Australia email: [email protected] Coauthors: Zoltan Neufeld and Anthony Roberts We use the Cellular Potts Model to simulate an epithelial cell layer and a microscopic wound, around one to five cells, in two dimensions. Using an energy function to describe properties of the cells we find qualitative results and insights into the wound closure process. The interaction between the contractile line tension due to the actin ring around the perimeter of the cells and adhesion between the cells appears to play an important role to determine whether a wound will open or close. This also suggests an active response, changing the balance between contraction and adhesion, is required for the wound closure process to occur. 77 (Re)wiring network models to understand the economics of innovation. Dion O’Neale University of Auckland, Auckland, New Zealand email: [email protected] Here we analyze over 30 years of patent data from the European Patent Office to investigate patterns of regional specialisation in different technical areas. We construct a bipartite network of over 4000 geographic regions and over 600 areas of technology. We find that those regions that exhibit a revealed comparative advantage in a larger number of technical areas (i.e. regions with high diversity) tend to, on average, have less ubiquitous technologies in their patent portfolio than regions with lower technical diversity. Furthermore, we find that this effect increases over time with low diversity regions holding patent portfolios whose contents become relatively more ubiquitous. We use a number of null models to test a variety of potential hypotheses that might explain the observed trend. The null models allow us to distinguish between intra-regional effects due to spillovers or agglomeration, and effects due to exogenous factors such as regional populations and the relative abundance of different technology codes. The null models involve re-wirings of the bipartite network according to different heuristics intended to capture the above effects. The null models reveal that while the co-occurrence of codes on patents (some inventions are commonly associated with multiple technologies) can account for the negative correlation between mean ubiquity and diversity of regions, it leads to higher levels of mean ubiquity for the region than are observed in the empirical data. This suggests that regions are exploiting spill-over and agglomeration effects to specialize in low ubiquity combinations of technical capabilities. Fast and stable spectral methods for PDEs Sheehan Olver The University of Sydney, Sydney, Australia email: [email protected] We describe a fast and stable approach to calculate the solution of general PDEs via global spectral methods, based on formulation using Chebyshev and Ultraspherical polynomials. Many simple examples can be written as generalized Sylvester equations solveable in O(n3 ) operations for n2 unknowns, which is a dramatic improvement over the O(n6 ) operations required by collocation methods. Other examples can be written as a Kronecker product of banded matrices, which also leads to an observed (but onproved) complexity of O(n3 ) operations. This approach is used in the ApproxFun package for Julia to provide a black-box PDE solver. Multiscale modelling of multicellular biological systems: mechanics, development and disease James Osborne University of Melbourne, Vic, Australia email: [email protected] When investigating the development and function of multicellular biological systems it is not enough to only consider the behaviour of individual cells in isolation. For example when studying tissue development, how individual cells interact, both mechanically and biochemically, influences the resulting tissues form and function. In this talk we present a multiscale modelling framework for simulating the development and function of multicellular biological systems (in particular tissues). Utilising the natural structural unit of the cell, the framework consists of three main scales: the tissue level (macro-scale); the cell level (meso-scale); and the sub-cellular level (micro-scale), with multiple interactions occurring between all scales. The cell level is central to the framework and cells are modelled as discrete interacting entities using one of a number of possible modelling paradigms, including lattice based models (cellular automata and cellular Potts) and off-lattice based models (cell centre and vertex based representations). The sub-cellular level concerns numerous metabolic and biochemical processes represented by interaction networks rendered stochastically or into ODEs. The outputs from such systems influence the behaviour of the cell level affecting properties such as adhesion and also influencing cell mitosis and apoptosis. At the tissue level we consider factors or restraints that influence the cells, for example the distribution of a nutrient or messenger molecule, which is represented by field equations, on a growing domain, 78 with individual cells functioning as sinks and/or sources. The modular approach taken within the framework enables more realistic behaviour to be considered at each scale. This framework is implemented within the Open Source Chaste library (Cancer Heart and Soft Tissue Environment, http://www.cs.ox.ac.uk/chaste/) and has been used to model biochemical and biomechanical interactions in various biological systems. In this talk we present the framework along with a demonstration of its applicability to modelling developmental processes. Singularities in diffusion-driven flows Michael Page Monash University, Victoria, Australia email: [email protected] Independent studies by Wunsch and Phillips in 1970 showed that steady flow can be generated in a stable density-stratified fluid simply due to the container having sloping insulated surface. Most of the analysis for this type of problem has been undertaken within two-dimensional containers with uniformly sloping planar surfaces in the small-R boundary-layer limit. However, external flows around objects in an unbounded fluid can also exhibit unexpected features in that regime, a simple example of which is the sloping finite-length plate considered by Woods (1991, J. Fluid Mech., 226, p625). Woods suggested a possible form of this flow but more-recent numerical and experimental results show additional flow features near the ends of the plate. In this talk, an asymptotic structure for Wood’s finite-length sloping plate problem is outlined in that limit. Leading-order solutions are described, including in regions that extend horizontally from the ends of the plate, which are present due to singularities at those points. This analytical structure is compared with both numerical calculations at small R (or large Rayleigh number) and also published experimental results. Dying in order: how crowding affects particle lifetimes Catherine Penington Queensland University of Technology, Brisbane, QLD, Australia email: [email protected] Coauthors: Matthew Simpson Suppose we have several agents on a line. They move around randomly, but once they reach an end they disappear permanently. How long will they survive? Without crowding effects, particles near the edges tend to leave sooner than those at the centre, but there is a lot of variation. When crowding is included, and agents cannot occupy the same position at the same time, both the mean time to disappear and its variance changes dramatically. In this presentation, we use simulation and analysis to discuss the effects of crowding on agent lifetimes. 79 The apparent wake angle of a ship travelling in a fluid of finite depth Ravindra Pethiyagoda Queensland University of Technology, Brisbane, Queensland, Australia email: [email protected] Coauthors: Scott McCue and Timothy Moroney For more than a century the characteristic wedge shape associated with the wake of a ship in infinitely deep water was accepted to have a half angle of arcsin(1/3) ≈ 19.47◦ , known as Kelvin’s angle. Over the past one or two years, however, this idea has been challenged by numerous papers documenting apparent wake angles less than Kelvin’s angle, at least for sufficiently fast-moving “ships”. One key observation is that the apparent angle we see in practice can be defined using the highest peaks of the wake. For finite depth flows there is an analogue of Kelvin’s angle, here referred to as the caustic angle, that varies with the Froude number, F , a nondimensional measure of speed. Using linear water wave theory, we calculate the apparent wake angles and the caustic angles for a variety of ship speeds and fluid depths and shed light on some seemingly contradictory results between these two measures. What’s the catch? Michael Plank University of Canterbury, Christchurch, New Zealand email: [email protected] Coauthors: Jeppe Kolding and Richard Law Regulations on minimum legal landing size are an almost universal tool in fisheries management worldwide. But do we need them? To investigate this question, we combine an agent-based model of fisher’s decisions about which fish sizes to target with a size-spectrum model of the population dynamics. The system settles to a Nash equilibrium in which each individual fisher obtains the same expected catch. We compare the size distribution of fish in the catch and the total yield with and without restrictions on the minimum size of fish that can be caught. The results have implications for how fisheries can best be managed. Stability of liquid films covered by a carpet of self-propelled surfactant particles Andrey Pototsky Swinburne University of Technology, Hawthorn, Victoria, Australia email: [email protected] Coauthors: Uwe Thiele and Holger Stark We consider a carpet of self-propelled particles (swimmers) that move along the liquid-gas interface of a liquid film on a solid substrate. The swimming direction of the swimmers changes in time due to rotational diffusion and due to the fluid motion. We study the intricate influence of the swimmers on the stability of the film surface and show that depending on the strength of in-surface rotational diffusion and the absolute value of the in-surface velocity several instability modes can occur. In particular, the rotational diffusion can have a stabilizing or destabilizing influence and may even suppress the instability entirely. 80 Numerical and analytical solutions of confined subdiffusion in three dimensions Shanlin Qin Queesland University of Technology, Brisbane, Australia email: [email protected] Coauthors: Fawang Liu and Ian Turner Fractional order diffusion equations with a time fractional derivative of order (0 < α < 1) have been widely applied to model the anomalous subdiffusive system. In this paper, we consider a fractional subdiffusion equation in three dimensions with the initial condition and reflecting boundaries. The fractional alternating direction implicit scheme (FADS) is proposed to solve the fractional subdiffusion equation with different reflecting boundaries. Analytic solution is giving by using separation of variables method and showing a good agreement with the numerical result. The stability and convergence of the method are proved. This initial-boundary problem is applied to model practical subdiffusive problems like the motion of the real biomolecules inside cells. The Evaluation of Faculty Employments Policies Using Markov Chain Model Rahela Abdul Rahim University Utara Malaysia, Kedah, Malaysia email: Coauthors: Syafawati Saad, Haslinda Ibrahim, Farah Adibah Adnan and Sahubar Ali Nadhar The approach to manpower policy in most Malaysian universities appears to be guided by the traditional method of putting the right number of people in the right place at the right time or arranging for suitable number of people to be allocated to various jobs usually in a hierarchical structure. The technique has been practiced for years. This traditional method is deficit in the sense that it neither offers computational tools that will enable managers to determine possible line of action to be taken nor provide tools to generate alternative policies and strategies. The objective of this study was to design a planning model for projecting university faculty employment under alternative policy suggested by government. The planning model was developed using Markov chain technique. Two scenarios were considered in the study; scenario 1 was based on historical data pattern between year 2005–2010 and scenario 2 was based on RMK 9 policies. Differences between actual and projected numbers of faculty by status were tested using chi-square goodness of fit tests. The results showed that there were no significance differences in the projected numbers of faculty by status for both scenarios. The projection for diversity of faculty status based on the the two scenarios for year 2015 was also presented. Effects of oblique magnetic field on mixed ferrofluid convection Md. Habibur Rahman Swinburne University of Technology, Melbourne, Victoria, Australia email: [email protected] Coauthors: Sergey A. Suslov Magnetic fluid, also known as ferrofluid, is a stable colloidal suspension consisting of the carrier liquid and magnetic nanoparticles. The behaviour of non-isothermal magnetic fluids strongly depends on external applied magnetic field, which can be used to control flow and heat transfer in fluid in various technological applications. Different mechanisms of convective instability manifest themselves in ferrofluid. In this talk we will present the results of linear stability analysis for convection flow in a layer of ferrofuid bounded by two vertical differentially heated plates placed in an external oblique magnetic field in the presence of gravity. The convective flows in ferrofluid are driven by the ponderomotive force due to the magnetic field effects and by the buoyancy. A set of characteristic ferrofluid parameters have been explored and thermomagnetic waves have been detected in a ferrofluid flow. The influence of the external magnetic field inclination angle on the flow structure is investigated. 81 The Probability of Bushfire Ignition Nicholas Read The University of Melbourne, VIC, Australia email: [email protected] Bushfire is a significant and increasing threat to Australia. It is an old threat and the scientific literature on forecasting bushfire is large although perhaps not as diverse as it could be. This talk will outline the practical problem, discuss the currently popular logistic regression models before proposing some point process models for lightning fire ignition. Hybrid Markov chain models for disease dynamics. Nicolas Rebuli The University of Adelaide, Adelaide, Australia email: [email protected] Coauthors: Nigel Bean and Joshua Ross Continuous-time Markov chains (CTMCs) continue to increase in popularity for modelling disease dynamics. They owe their popularity to often finding an appropriate balance between computational feasibility and sufficient realism. However, this balance is lost as the size of the population being modelled increases, due to a ‘curse of dimensionality’. In this talk I consider the susceptible–infectious–recovered (SIR) CTMC epidemic model and present two novel methods for overcoming this dimensionality problem. The so called SIR hybrid models approximate the SIR CTMC on a subset of its state space using either a deterministic, or a diffusion, approximation. I assess the accuracy of the hybrid models by comparing their final epidemic size and epidemic duration distributions to those of the SIR CTMC. Multiphase modelling of biological gel mechanics James Reoch University of Adelaide, SA, Australia email: [email protected] Cells are often grown within collagen gels in vitro for applications in tissue engineering. Since the behaviour of cells is regulated by their mechanical environment, we aim to gain more insight into the mechanics of these gels using mathematical modelling. In this talk, I will outline the modelling approaches being used to study the mechanical interactions between the cells and the gel. A novel aspect of the problem is the inclusion of chemical effects such as osmosis, which, together with the forces exerted by the cells, drive gel contraction and swelling. I will present our current multiphase model, discussing its steady state behaviour and numerical simulations of its time evolution. 82 Folded Saddle-Node Bifurcations Kerry-Lyn Roberts University of Sydney, NSW, Australia email: [email protected] Coauthors: Martin Wechselberger and Jonathan Rubin Folded saddle-node (FSN) bifurcations occur generically in one parameter families of singularly perturbed systems with at least two slow variables. Recently we identified a novel FSN bifurcation (FSN III) in a coupled neural model. In this talk we analyse a canonical model of the FSN III bifurcation. We combine techniques from geometric singular perturbation theory (the blow-up technique) and dynamic bifurcation theory (complex time path analysis) to understand the local dynamics and show the existence of canards. Understanding risk through virtual sensing: an application to the agricultural industry. Melanie Roberts IBM Research - Australia, Melbourne, Victoria, Australia email: [email protected] Crop yield is impacted by the weather. Accurate risk modelling of the yield across several geographies remains a challenge due to sparsity and access to yield and/or fine resolution weather data. The sparsity of fine resolution of weather data is addressed through dynamical downscaling and post-processing methods. The impact of the downscaled weather data on crop yield is studied using historical yield data. The weather data then forms the basis for developing an accurate crop yield risk model. Exponential growth and the final size of an epidemic Mick Roberts Massey University, Auckland, New Zealand email: The value of the basic reproduction number, R0 , may be estimated when the incidence of infection is growing exponentially in the early stages of an epidemic. For the Kermack-McKendrick model the final size of the epidemic - the proportion of the population that would be infected if no interventions were made - depends only on R0 and the initial proportion susceptible. This well-known result will be generalised to the situation where R0 has an uncertain estimate, specified as a probability distribution. A stochastic SIR model will then be described, where the contact rate fluctuates randomly, and the initial growth rate and final size determined. Epidemics in a heterogeneous population due to an infection spread by a vector or environmental contamination may be modelled with so-called separable mixing. Expressions for R0 and the final size will be obtained, and compared with those for an epidemic on a network. 83 High-order evolution PDEs model nonlinear dispersive waves over large scales Tony Roberts University of Adelaide, South Australia, Australia email: [email protected] Many practical approximations in science and engineering involve wave propagation over a relatively long physical domain. In this scenario we typically expect the waves to have structures that vary slowly in the long dimension. Extant approximation methodologies are typically self-consistency arguments. The proposed new approach is to analyse the dynamics based at each cross-section in a rigorous Taylor polynomial. Slow manifold theory supports the local modelling of wave modulation with coupling to neighbouring locales treated as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and relevance of a slow manifold, wave modulation. Our resolution of the coupling between neighbouring locales leads to novel quantitative estimates of the error. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling, and to open previously intractable modelling issues to new tools and insights. A nonlinear least squares approach to time of death estimation via body cooling Marianito Rodrigo University of Wollongong, Wollongong, NSW, Australia email: marianito [email protected] The problem of time of death estimation by body cooling is revisited by proposing a nonlinear least squares approach that takes as input a series of temperature readings only. Using a reformulation of the Marshall-Hoare double exponential formula and a technique for reducing the dimension of the state space, an error function that depends on the two cooling rates is constructed, with the aim of minimising this function. Then an explicit formula for the time of death is given. Results of numerical simulations using both theoretical and experimental data are presented, both yielding reasonable estimates. The proposed procedure does not require knowledge of the temperature at death nor the body mass. In fact, the method allows the estimation of the temperature at death once the cooling rates and the time of death have been calculated. The procedure requires at least three temperature readings, although more measured readings could improve the estimates. With the aid of computerised recording and thermocouple detectors, temperature readings spaced 10-15 minutes apart, for example, can be taken. Solutions of the discrete Painlev´ e equation q-P (A∗1 ) which are meromorphic at the origin or infinity. Pieter Roffelsen University of Sydney, Sydney, Australia email: [email protected] In a series of papers Kaneko and Ohyama classified all the meromorphic solutions of the continuous Painlev´e equations around fixed singularities of Briot-Bouquet type. Using a q-discrete analogue of the celebrated BriotBouquet Theorem we classify the solutions of the q-P (A∗1 ) equation which are meromorphic at the origin or infinity. For these special Painlev transcendents, we construct global solutions of the Lax pair of q-P (A∗1 ) around z = 0 and z = ∞ and consider the corresponding connection problem. 84 Condition numbers in conic feasibility problems Vera Roshchina University of Melbourne, Melbourne, Australia email: [email protected] Coauthors: Javier Pena Condition numbers in conic optimisation quantify the difficulty in solving a problem’s instance. There are different condition numbers which capture distinct geometric properties of the problem; some are better suited for the characterisation of complexity of numerical methods, others serve better for the analysis of the problem’s geometric properties. Of particular interest are the ill-posed problems, and the issues of preconditioning, particularly in the linear programming setting. The talk is based on collaborative work with Prof. Javier Pena (Carnegie-Mellon University) Computation of epidemic final size distributions Joshua Ross The University of Adelaide, Adelaide, South Australia, Australia email: [email protected] Coauthors: Andrew Black An important statistic associated with the outbreak of an infectious disease is the total number of individuals that contract the disease. Such final size data is highly informative to estimate the transmissibility of a disease. In these situations the accurate and efficient computation of the probability of each possible final size is paramount. A new method for the computation of the final size distribution for Markovian epidemic models will be presented. For the case of the S-I-R (susceptible-infectious-recovered) model, this is the most efficient algorithm produced. The method is also physically transparent, and hence allows relatively easy extension to a range of more complex models, such as those with a phase-type infectious period and/or with waning immunity. Complex Network Transformations of Time Series: the Ordinal Partitions Method Konstantinos Sakellariou University of Western Australia, Perth, WA, Australia email: Recently several methods which utilise complex network theory as a means of analysing time series have been developed. Nodes may represent regions in phase space, dynamical states or even time series points. Network connectivity is defined in such a way so as to capture specific information about the dynamical systems generating the time series. We explore the so-called ’ordinal partitions’ network transform, a transformation technique where nodes encode discrete dynamical states and network connectivity is defined by temporal succession. Consequently, dynamical information and modes of transition from state to state are the focus here - in contrast to the majority of the existing network transforms which concentrate on topological aspects. By applying this technique to data generated by numerical simulation of model chaotic dynamical systems (e.g. Lorenz system, Ikeda map), we perform a parameter investigation. We determine the ’interesting’ parameter regimes in terms of applicability and test the robustness of the method in the presence of noise. We then analyse the resulting networks and identify how their local and global statistical properties reflect the underlying dynamics governing the original time series. In particular, we examine the relation between network properties and important dynamical notions, such as unstable periodic orbits or recurrence of states on a chaotic attractor. 85 Fractional-in-space partial differential equations on finite intervals, boundary conditions, and associated stochastic processes Harish Sankaranarayanan University of Otago, Dunedin, New Zealand email: [email protected] Coauthors: Boris Baeumer and Mih´ aly Kov´ acs We present Gr¨ unwald approximations for fractional derivative operators on finite intervals, whose domains encode various boundary conditions. The well-posedness of the Cauchy problem associated with fractional derivative operators on C0 (Ω), L1 (Ω), Ω ⊂ R is established. The stochastic processes associated with fractional derivative operators are identified as limits of the processes associated with the respective Gr¨ unwald approximations. The probability of extreme rain on your parade given the El Ni˜ no Southern Oscillation Kate Saunders University of Melbourne, Melbourne, Australia email: [email protected] Courtesy of agricultural stakeholders, mean rainfall processes in Australia are fairly well understood; conversely the drivers and processes behind extreme rainfall still pose a significant question for researchers. We use extreme value theory to quantify how the large scale climate driver of the El Ni˜ no Southern Oscillation (ENSO) alters the distribution of extreme daily rainfall events in Australia. We do this by fitting a generalized Pareto distribution to the high-quality sites in the Bureau of Meteorology’s climate change network. The observational record for these sites spans approximately 100 years allowing us to make more statistically significant conclusions about the effect of ENSO on the distributions parameters than studies that have come before. For sites where significance is not detected, we are still faced with the question of whether given the size of our observational set, should we have reasonably been able to detect significance. We also aim to address this. Tractable Quadrature in Infinite Dimensions and Applications in Uncertainty Quantification Robert Scheichl University of Bath, Bath, UK email: [email protected] Coauthors: Frances Kuo, Christoph Schwab, Ian Sloan and Elisabeth Ullmann The coefficients in mathematical models describing physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. It is of great importance to quantify the uncertainty in the model outputs based on the (uncertain) information that is available on the model inputs. This invariably leads to very high dimensional quadrature problems associated with the computation of statistics of quantities of interest, such as the time it takes a pollutant plume in an uncertain subsurface flow problem to reach the boundary of a safety region or the buckling load of an airplane wing. Higher order methods, such as stochastic Galerkin or polynomial chaos methods, suffer from the curse of dimensionality and when the physical models themselves are complex and computationally costly, they become prohibitively expensive in higher dimensions. Instead, some of the most promising approaches to quantify uncertainties in continuum models are based on Monte Carlo sampling and the “multigrid philosophy”. Multilevel Monte Carlo (MLMC) Methods have been introduced recently and successfully applied to many model problems, producing significant gains. In this talk I want to recall the classical MLMC method and then show how the gains can be (significantly) improved further by using quasi-Monte Carlo (QMC) sampling rules. More importantly the dimension independence and the improved gains can be justified rigorously for an important model problem in subsurface flow. To achieve uniform bounds, independent of the dimension, it is necessary to work in infinite dimensions and to 86 study quadrature in sequence spaces. I will present the elements of this new theory for the case of lognormal random coefficients in a diffusion problem and support the theory with numerical experiments. A geometric construction of shock waves in a tumour growth model, incorporating the Allee effect. Lotte Sewalt Leiden University, Leiden, The Netherlands email: [email protected] Coauthors: Kristen Harley, Peter van Heijster and Sanjeeva Balasuriya We discuss the influence of growth thresholds on the existence of travelling shock wave solutions in a reactionadvection-diffusion model describing the invasion of malignant tumour cells. Using geometric singular perturbation theory (GSPT) and canard theory, the existence of travelling shock waves as a solution to this PDE system is proved. In earlier work, the spread of cancer cells was modelled as logistic growth. However, recent studies have shown that such processes are often characterized by growth thresholds, a phenomena known in ecology as the Allee effect and one that cannot be described by a logistic term. We show how incorporating this effect changes our existence results. An Individual-based model approach to analyse the spatio-temporal dynamics of Influenza in Melbourne Shrupa Shah The University of Melbourne, Melbourne, Australia email: [email protected] Influenza is a major public health concern as it causes significant morbidity in the population at large and mortality in the very young, elderly and in persons with chronic illnesses. The 1918–1919 Spanish flu and 2003 SARS (Severe Acute Respiratory Syndrome) pandemics are stark reminders of the potential consequences of infectious diseases. Although the SARS epidemic was not the world-wide pandemic that scientists feared, it still managed to spread to nearly every continent on Earth. This clearly points out how crucial it is to understand how, when and why epidemics spread across the landscape so that effective planning, preparation and control measures can be in place before a disaster occurs. Geographic models can help us understand the spatial spread from the epicentre and the rate at which the disease diffuses from the epicentre. This will then inform control strategies like contact tracing and quarantine during the initial phases of the outbreak and ring vaccination or some other control strategy at later phases of the epidemic. In the light of this, I will briefly review some of the mathematical, computational and network models from the literature which implicitly or explicitly consider space and the assumptions these models make. I will also review the applications of some spatio-temporal models successfully implemented. And finally conclude with, how these applications motivate the early framework for my project which is also strongly informed by a field study conducted in Melbourne where data has been collected at the individual level of granularity. With the proposed framework I hope to investigate how seasonal Influenza spreads in Melbourne. 87 Modelling the intrinsic dynamics of bushfire propagation using plane curvature flow Jason Sharples UNSW, Canberra, ACT, Australia email: [email protected] Coauthors: James Hilton Bushfires are inherently dynamic phenomena that consistently pose threats to society and the environment. Despite their dynamic nature, current operational approaches to predicting the spread of bushfires are based on first-order propagation models, which assume that fires spread at a quasi-steady state defined by the relevant environmental variables. In this work we report on initial investigations into the use of a second-order propagation model, which incorporates a functional of the fire line curvature to emulate intrinsic fire line dynamics. The model is implemented in the form of a curvature flow via a level set method. Application of the work to modelling fire coalescence will also be discussed. Modelling Tumour Treatment using the Single Species Gompertz Population Model John Shepherd RMIT, Melbourne, Victoria, Australia email: [email protected] Coauthors: Stuart E Roberts In 1825, Benjamin Gompertz proposed his well-known equation to model constrained human population growth. In the 1960s, A K Laird used this Gompertz equation to model the growth of tumours as growing cell populations in a confined space. This approach has been used and extended by many investigators. A logical consequence of this is that the treatment of tumour growth by chemical or radiative means could be modelled by a harvested Gompertz equation with density dependent harvesting. A complication occurs if the parameters defining the harvesting model are no longer constants, but vary with time, as might occur in a varying environment. Then, for completely arbitrary time variation, exact solution of the harvested equation is rarely possible, and we must resort to numerical solutions, which have the disadvantage of requiring explicit parameter values and are of limited use in studying general trends. However, when the model parameters are slowly varying functions of time, multitiming techniques may be used to obtain explicit and useful approximations for the variation of the tumour population. In this talk, we use these techniques to analyse the harvested Gompertz model and use the approximations obtained to make useful predictions about the evolving population and related quantities of interest. Symmetry and combinatorics of Coxeter groups and discrete integrable systems Yang Shi The University of Sydney, Sydney, NSW, Australia email: [email protected] Coauthors: Nalini Joshi, Nobutaka Nakazono From the geometric and combinatorial descriptions of the Coxeter groups we construct various discrete integrable systems and show that the relationships between these different systems can be naturally explained using such descriptions. 88 Chemotactic adhesion in bacterial flocs in shear flow: a multi-scale model Sarthok Sircar University of Adelaide, Adelaide, SA, Australia email: [email protected] In this talk, I present a model for the attachment/detachment dynamics of bacterial aggregates in a fluid subject to a homogeneous planar shear-flow. To understand the adhesion-fragmentation dynamics of these flocs, the aggregates are modeled as ligand-covered rigid spheres. The binding ligands on the surface of the flocs experience attractive and repulsive surface forces in an ionic medium and exhibit finite resistance to rotation (via bond tilting). For certain range of material and fluid parameters, the results predict a nonlinear or hysteretic relationship between the binding/unbinding of the floc surface and the net floc velocity (translational plus rotational velocity). I show that the surface adhesion is promoted by increased fluid flow until a critical value, beyond which the bonds starts to yield. Moreover, adhesion is promoted in a medium with high ionic strength, or flocs with small size or lower binder stiffness. The numerical simulations of floc-aggregate number density studies support these findings. Do pioneer cells exist? Matthew Simpson Queensland University of Technology, Brisbane, Australia email: [email protected] Most mathematical models of collective cell spreading make the standard assumption that the cell diffusivity and cell proliferation rate are constants that do not vary across the cell population. Here we present a combined experimental and mathematical modeling study which aims to investigate how differences in the cell diffusivity and cell proliferation rate amongst a population of cells can impact the collective behavior of the population. We present data from a three–dimensional transwell migration assay which suggests that the cell diffusivity of some groups of cells within the population can be as much as three times higher than the cell diffusivity of other groups of cells within the population. Using this information, we explore the consequences of explicitly representing this variability in a mathematical model of a scratch assay where we treat the total population of cells as two, possibly distinct, subpopulations. Our results show that when we make the standard assumption that all cells within the population behave identically we observe the formation of moving fronts of cells where both subpopulations are well-mixed and indistinguishable. In contrast, when we consider the same system where the two subpopulations are distinct, we observe a very different outcome where the spreading population becomes spatially organized with the more motile subpopulation dominating at the leading edge while the less motile subpopulation is practically absent from the leading edge. These modeling predictions are consistent with previous experimental observations and suggest that standard mathematical approaches, wherewe treat the cell diffusivity and cell proliferation rate as constants, might not be appropriate. This is joint work with Emeritus Professor Sean McElwain and Ms Parvathi Haridas. Modelling Overwash on Ice Floes by Water Waves David Skene The University of Adelaide, Adelaide, Australia email: [email protected] Ocean waves have a significant impact on the vast regions sea ice covering the surfaces of the high-latitude oceans. Contemporary mathematical models of wave-ice interactions are based on linear theory: using potential theory for the water and thin-elastic-plate theoryfor the ice floes (discrete chunks of sea ice). In reality, highly non-linear effects occurin wave-ice interactions. In particular, a phenomenon known as overwash occurs. Overwash refers to thin streams of water being forced over the top of floes as their edges dip in and out of the surrounding water waves. 89 I will present a mathematical model of overwash. The model uses linear theory for the motion of the floe and the water surrounding the overwash region. The surrounding water drives the overwash. The overwashed water itself is modeled using the nonlinear shallow-water equations. Model results arevalidated usingresults of an experimental model. The ANOVA decomposition of a non-smooth function of an infinite number of variables Ian Sloan UNSW, Sydney, Australia email: [email protected] In this joint work with Frances Kuo (UNSW) and Michael Griebel (Bonn) we extend our earlier work motivated by path-dependent option pricing problems, in which we tried to understand how it is that sparse grid and QMC methods can be applied successfully to option pricing problems, even though the integrands do not live in any mixed derivative smoothness class. That difficulty derives from the “max function” in the integrand, describing the fact that options are considered worthless if the payoff falls below the strike price. In a previous paper (Math. Comp. 82, 383–400, 2013) we showed that if the expected value is expressed as an integral over Rd then the classical ANOVA decomposition of the integrand for an arithmetic Asian option can have every term smooth except for the very highest term. That highest ANOVA term has many discontinuities in first partial derivatives, but in most cases is expected to be pointwise very small. In the present work we consider the ANOVA decomposition of the corresponding continuous problem in the Brownian bridge (or Levy-Ciesielski) formulation, and show that in this case every term in the (infinite) ANOVA decomposition is smooth. With this result we are preparing for an error analysis of the cubature problem for option pricing problem, in which the discrete-time problem is approximated by the continuous problem, and the error analysis then applied to the truncated infinite ANOVA expansion, in which every term is smooth. The (un)importance of the temperature gradient in fibre drawing. Yvonne Stokes The University of Adelaide, Adelaide, SA, Australia email: [email protected] An optical fibre is fabricated by feeding a preform into a heated neck-down region and pulling it at a higher speed by winding onto a spool some distance downstream beyond the neck-down region. The existence of a strong temperature gradient (50-100 ◦ C/cm) along the length of the neck-down region (typically 2-4 cm) is essential to this process and much work has been done by others on the difficult task of modelling to determine the temperature, and hence the viscosity, profile. I will, however, show that obtaining a desired fibre geometry essentially depends only on the harmonic mean of the temperature over the neck-down length and, indeed, that control of the fibre tension circumvents the need to know anything about the temperature profile. 90 Non-linear thermomagnetic instabilities in a vertical layer of a ferromagnetic fluid Sergey Suslov Swinburne University of Technology, Hawthorn, Victoria, Australia email: [email protected] Coauthors: Pinkee Dey Nonlinear instabilities have been studied to reveal the exact mechanism leading to the appearance of various convection patterns arising in a differentially heated vertical layer of non-conducting ferromagnetic nanofluid placed in an external uniform magnetic field normal to the layer. Depending on the governing parameters, developing instability patterns consist of vertical stationary magneto-convective rolls and vertically counterpropagating thermo-gravitational or oblique thermo-magnetic waves (Suslov, Phys. Fluids 20, pp. 1–18, 2008). Weakly nonlinear analysis based on combined amplitude and multiple time scale expansions is applied to investigate those interacting patterns. Squire’s transformation is extended to include nonlinear terms to reduce the full three dimensional problem to an equivalent two dimensional problem and to keep the computational cost down. The character of bifurcations is analysed in detail to provide parametric guidance for future experiments. Symmetric 4-body motions Winston Sweatman Massey University, Auckland, New Zealand email: [email protected] The gravitational N-body problem has a rich and varied dynamics. We consider 4-body systems which are further simplified by having a symmetric arrangement. There are families of orbits analogous to earlier families of orbits found in the 3-body problem. Random coefficient autoregressive model and Maximum quasi likelihood estimation Laleh Tafakori University of Melbourne, Melbourne,victoria, Australia email: [email protected] We consider the specific kind of random coefficient autoregressive model and their statistical properties. Further, Maximum quasi likelihood estimation (MQE) which has many of the desirable properties of MLE, without requiring the existence of an objective function too be maximized, is derived. Therefore, the difficulties arising from the discontinuous likelihood function of the mentioned model can be avoided by using MQE. Reflection methods for Euclidean distance matrix reconstruction Matthew Tam University of Newcastle, Newcastle, NSW, Australia email: [email protected] Coauthors: Francisco Arag´ on Artacho and Jonathan Borwein The Douglas-Rachford reflection method is a general purpose algorithm useful for solving the feasibility problem of finding a point in the intersection of finitely many sets. Despite a lack of theoretical justification, the method has recently been experimentally observed to successfully solve a variety of difficult non-convex optimisation and inverse problems including Sudoku puzzles, finding Hadamard matrices, and numerous image reconstruction problems. In this talk I will focus on application of the Douglas-Rachford method to the (non-convex) problem of reconstructing a Euclidean distance matrix from a priori knowledge, and a small subset of its entries. The framework is then applied to the problem of protein conformation determination. 91 On solutions of a functional PDE for cell growth and division Steve Taylor University of Auckland, Auckland, New Zealand email: [email protected] Coauthors: Susan Yang We study the existence of solutions of a functional PDE model for size-structured cell growth and division, introduced by Graeme Wake et al. In this model, the density of cells relative to cell size x at time t is denoted n(x, t). A cell of size x is assumed to divide into two new cells of size αx and βx . Taking into account growth rate g, splitting rate b and death rate µ results in the model ∂ ∂ n(x, t) + (g(x)n(x, t)) = −(b + µ)n(x, t) + bαn(αx, t) + bβn(βx, t), ∂t ∂x n(0, t) = 0, n(x, 0) = n0 (x), lim n(x, t) = 0. x→∞ A natural space for solutions is L1 because the total number of cells should be finite; ∞ n(x, t)dx < ∞. 0 In this talk, we give a simple proof of existence of solutions in Lp for p ≥ 1. How old is this bird? Peter Taylor University of Melbourne, Victoria, Australia email: [email protected] Coauthors: Sophie Hautphenne Motivated by studies of a population of black robins (Petroica traversi ) in the Chatham Islands, we consider the situation where an individual’s lifetime is modelled by a finite-state continuous-time Markov chain with one absorbing state. Under this model, the time of death follows a phase-type distribution. We then attempt to provide an answer to the simple question “What is the age distribution of the individual, given its current phase”? There are a number of ways to think about this question, which we shall discuss. In particular, we show that the answer depends on the observation scheme under consideration. Optimal vaccine allocation for structured populations Mingmei Teo University of Adelaide, Adelaide, SA, Australia email: [email protected] Coauthors: Nigel Bean and Joshua Ross Vaccination is the most effective method for preventing the spread of an infectious disease. In many scenarios, vaccines may be in short supply or may be very expensive, and hence determining their optimal deployment will be of great interest. Examples include in the early stages of vaccine production following the identification of a suitable vaccine during an outbreak, as anticipated to be the case for Ebola in West Africa; or, in the control of livestock diseases, where maintaining disease-free premises at minimal cost is desired. We consider dynamic programming approaches to determine the optimal allocation of vaccines across a small number of interacting populations/households in order to minimise the mean final epidemic size. 92 Towards an extended Navier-Stokes hydrodynamics at the nanoscale Billy Todd Swinburne University of Technology, Hawthorn, Victoria, Australia email: [email protected] In this presentation, both theoretical and simulation studies are highlighted that clearly demonstrate the importance of several non-classical phenomena fundamental to the extension of Navier-Stokes hydrodynamics for highly confinded fluids. These are: (1) the prevalence of slip, (2) the strong coupling of molecular spin to linear translational momentum, and (3) the non-locality of viscous transport at the nanoscale. In the first of these, we utilize a newly developed equilibrium based model1 to accurately predict the slip velocity and slip lengths of systems such as water or methane flowing in graphene nanochannels and carbon nanotubes2,3 . We demonstrate that traditional molecular dynamics simulations of such systems are far less efficient and accurate than the easily implemented model we propose. Next, we show that ignoring the coupling of spin angular momentum to linear translational motion of a highly confined fluid can lead to significant over-estimation of the predicted flow rates using conventional Navier-Stokes treatments. By including spin-coupling into the extended NavierStokes equations, hydrodynamic prediction is seen to be very accurate down to length scales of a few atomic diameters4 . We also demonstrate how this knowledge, coupled with our knowledge of slip, can be used to pump molecular fluids such as water via non-intrusive application of a rotating electric field5,6 . Finally, we show that a complete generalisation of Navier-Stokes hydrodynamics comes about in the realisation that at the nanoscale viscous transport is fundamentally non-local in nature. We explore this theme for homogeneous systems and discuss the ramifications and problems yet to overcome for nanofluidic applications7,8 . 1 J.S. Hansen, B.D. Todd and P.J. Daivis. Phys. Rev. E 84, 016313 (2011). S. K. Kannam, B.D. Todd, J.S. Hansen and P.J. Daivis. J. Chem. Phys. 136, 024705 (2012). 3 S.K. Kannam, B.D. Todd, J.S. Hansen and P.J. Daivis. J. Chem. Phys. 138, 094701 (2013). 4 J.S. Hansen, J.C. Dyre, P.J. Daivis, B.D. Todd and H. Bruus. Phys. Rev. E 84, 036311 (2011). 5 J. D. Bonthuis, D. Horinek, L. Bocquet, and R. R. Netz, Phys. Rev. Lett. 103, 144503 (2009). 6 S. De Luca, B.D. Todd, J.S. Hansen and P.J. Daivis. Langmuir 30, 3095-3109 (2014). 7 B.D. Todd, J.S. Hansen and P.J. Daivis. Phys. Rev. Lett. 100, 195901 (2008) 8 B.A. Dalton, P.J. Daivis, J.S. Hansen and B.D. Todd. Phys. Rev. E 88, 052143 (2013). 2 A New Approach For Solving A Sparse Linear System With Periodic Boundary Conditions Minh Tran Flinders University, Adelaide, Australia email: [email protected] Differential equations (DE) can be used to mathematical describe useful physical systems. These equations often have additional constraints (boundary conditions) imposed on by the physical systems. One type of boundary conditions is the periodic boundary conditions (PBC). Numerical solution of a DE system with PBC involves discretizing the system, resulting in a cyclic tridiagonal system. This presentation will present a new, efficient and robust parallel algorithm for solving such a system with time complexity of O(log(n)) on a parallel computing platform. Furthermore, this is the first robust algorithm capable of solving a cyclic tridiagonal system that is not diagonally dominant. Extruding Complicated Fluid Structures Hayden Tronnolone University of Adelaide, Adelaide, South Australia, Australia email: [email protected] Coauthors: Yvonne Stokes and Darren Crowdy The extrusion of a very viscous fluid through a die involves a range of physical processes; however, the relative importance of each of these is not well understood. As a first step towards a better understanding, this process is modelled as a Stokes flow driven by both surface tension and gravity. Applying a slenderness approximation 93 yields two coupled systems of equations applicable to dies of arbitrary connectivity that readily reveal the effects of the physical processes under consideration. These effects are analysed and demonstrated though examples, including the application to microstructured optical fibre fabrication. A new mode of instability in compressible boundary-layer flows Adam Tunney University of Auckland, Auckland, New Zealand email: [email protected] In low disturbance environments such as aerodynamic flight, the initial growth of disturbances that cause the laminar-turbulent transition process in a boundary layer can be investigated with linear stability theory (LST). A large collection of results using LST are available in the literature, however excluded are the class of boundary-layer flows with region of velocity overshoot. Using a compressible, heated-wall flat-plate boundary layer with a favourable pressure gradient as a prototype, the linear stability of this class of boundary layers is investigated numerically and analytically using LST. Along with the traditional Mack modes, a new mode of inviscid instability is found that is localised within the region of velocity overshoot. The interaction between the new mode and the Mack modes is investigated through viscous stability analysis. Developing a Model of Bird Navigation Rebecca Turner University of Auckland, Auckland, New Zealand email: [email protected] Our understanding of how birds successfully navigate great distances during migration or homing still includes many unconfirmed hypotheses. However, there are clues to the underlying mechanism of navigation in the systematic errors birds make in their initial headings. A simple model has been proposed to capture the initial orientation error made when a bird is attempting to navigate home from an unfamiliar location [Postlethwaite & Walker, Journal of Theoretical Biology, (2011 & 2014)]. When comparing the predictions of the model to real experimental data one must keep in mind the two types of assumptions present in the model. Firstly, the navigational mechanism is assumed to be a two coordinate cognitive map system based on two environmental gradients. Secondly, the environmental gradients considered are limited by the data available. I will discuss the comparison of the model to real experimental data in light of the above assumptions and suggest future directions for the development of the model. A Cell Growth Model Adapted for Minimum Cell Size Division Bruce van Brunt Massey University, Palmerston North, New Zealand email: [email protected] Coauthors: Saima Gul and Graeme Wake In this talk we examine a cell growth model with a division kernel that models cells dividing only after they have reached a certain minimum size. The model features a functional differential equation of the pantograph type. In contrast with the earlier cases, however, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. This, in turn, leads to the study of a certain class of Dirichlet series. We show that there is a steady size distribution solution to this problem. 94 A geometric approach to stationary defect solutions in one space dimension Peter van Heijster Queensland University of Technology, Brisbane, Queensland, Australia email: [email protected] Coauthors: Feng Xie and Arjen Doelman We analyze a weakly heterogeneously perturbed system of N first order ordinary differential equations, u˙ = f (u), f (u) + εg(u), t ≤ 0, t > 0, in a general setting. Under the assumptions that the unperturbed system is hyperbolic, possesses a heteroclinic orbit, and that the perturbation is generic, we determine conditions such that the heterogenous system supports a nearby defect solution. This study is motivated by previously observed defect solutions in a perturbed threecomponent FitzHugh-Nagumo equation. Modelling Growth Variability in Cell Populations Graeme Wake Massey University, Auckland, New Zealand email: [email protected] Coauthors: Ali Zaidi and Bruce van Brunt The study of cell population dynamics has become increasingly significant - in part because of the importance of understanding phenomena such as tumour growth driven by epigenetic effects. These models will lead to a better understanding of the progression of the disease. The resulting dynamical models provide a relatively simple method for determining parameters that both regulate and enhance growth, which can help quantify the effectiveness of cancer therapy drugs. Populations of cells that are simultaneously undergoing growth and division are considered when the growth is random and cells are dividing symmetrically into two or more daughter cells. Following earlier work by Wake, van-Brunt, Kim and Cooper (Comm. Appl. Anal. 4, 2000, pp 561-574), use is made of the FokkerPlanck formulation to incorporate the stochastic effects in the growth. These models have separation of variables solutions which suggest there is an asymptotically attracting steady-size distribution. In this work a constructive existence theorem is obtained for the linear non-local dispersion-growth equation now with an arbitrary initial size-distribution and with a no-flux boundary condition. This solution is unique. It is still an open question as to whether or not the solutions obtained by separation of variables form a complete spanning set. Inference Methods for First Few Hundred Studies James Walker University of Adelaide, South Australia, Australia email: [email protected] Coauthors: Joshua Ross and Andrew Black Infectious diseases are a major, continuing threat to our health and well-being. Of particular concern are pandemics, which involve a major outbreak of a novel pathogen, most commonly influenza (‘flu’). During the early stages of such an outbreak, government authorities may undertake intensive data collection studies often termed First Few Hundred studies (FF100 studies) - in which the household members, and possibly other contacts, of the first few hundred cases are monitored for signs of symptoms. In this talk I will discuss novel methods appropriate for FF100 studies based upon stochastic household models of epidemics. In particular, we will discuss the estimation of the household basic reproductive number, which determines the transmissibility of a disease. 95 The flux paradox in gravitational lensing Steve Walters University of Tasmania, Hobart, Tasmania, Australia email: [email protected] An early result in the development of gravitational lensing theory is that at least one image is always magnified due to the existence of a gravitational lens. This appears to contradict conservation of photon flux. That is, photons should be neither created nor destroyed due to the presence of the lens. We will re-examine the origin and nature of this paradox, and consider a new solution. Analytical and numerical solutions of the multi-term time-space fractionaldiffusion equations with a fractional Laplacian operator Hao Wang Queensland University of Technology, Brisbane, Australia email: [email protected] Coauthors: Fawang Liu, Ian Turner, Pinghui Zhuang and Shanzhen Chen In this paper, we consider the one-dimensional and two-dimensional multi-term time and space fractional diffusion equations (1D-MTTSFDE, 2D-MTTSFDE). The multi-term time-fractional derivatives are defined in the Caputo sense, whose order belongs to the interval (0,1), and the space-fractional derivative is referred to the fractional Laplacian operator. We derive the analytical solutions of the 1D-MTTSFDE and 2D-MTTSFDE based on the spectral representation of the fractional Laplacian operator with homogeneous boundary conditions. The nonhomogeneous boundary condition is considered as well. We propose a computationally effective fractional predictor-corrector. It has been applied in 1D-MTTSFDE and could be extended to MTTSFDE in higher dimensions. Finally, numerical results in both one dimensional and two dimensional are given, which are in good agreement with the analytic solutions. Discrete needlet approximation Yu Guang Wang School of Mathematics and Statistics, UNSW Australia, Sydney, NSW, Australia email: Coauthors: Quoc Le Gia, Ian Sloan and Robert Womersley Needlets are highly localised filtered radial polynomials on the sphere S d of Rd+1 , d ≥ 2, with centers at the nodes of a suitable quadrature rule. They provide a multiscale decomposition for real L2 functions on S d . The original needlet decomposition has its coefficients defined by an inner product integrals. In this paper, we use additional quadrature rules, to establish a fully discrete version of the original semidiscrete needlet approximation. We prove that the fully discrete needlet approximation is exactly equivalent to filtered hyperinterpolation, that is to a filter-modified Fourier-Laplace series partial sum with inner products replaced by appropriate quadrature sums. From this, we establish Lp -error bounds, 2 ≤ p ≤ ∞, for the fully discrete needlet approximation of functions in Sobolev spaces Wps on S d for s > d/p. In particular, the Lp error for the fully discrete approximation loses convergence order compared to the semidiscrete needlet approximation only by the exponent d/p + for > 0 expected from the embedding of Wps (S d ) in C(S d ). The theory is illustrated numerically for the approximation of a function of known smoothness, using symmetric spherical designs (for both the needlet quadrature and the inner product quadrature). The power of the needlet approximation for local approximation is shown by a numerical experiment that uses low-level needlets globally and high-level needlets in a local region. 96 DES simulation for modelling patient congestion within a SA metropolitan hospital Dale Ward Flinders University, South Australia, Australia email: [email protected] Coauthors: Jerzy Filar and Shaowen Qin The Australian public hospital system is struggling under the burden of increased demand for inpatient care. In South Australia alone the incidents of ambulance ramping at metropolitan hospitals has increased over the last few years and has become the focus of public attention. With the average age of the population on the rise, this issue is only set to worsen and hospitals will need to address these issues under tighter budget restraints, essentially doing more with less. Changes to the operating procedures of hospitals need to be made now as patient flow congestion within hospitals can be tied to poor health outcomes. As part of ARC linkage project, our research group is investigating the issue of patient flow congestion within Flinders Medical Centre. In particular, the research aims at identifying early warning signs of congestion and the experimentation of potential early interventions for avoidance or for reducing the impact. As part of this research a discrete event based simulation model of patient flow in Flinders Medical Centre is being developed with colleagues from the hospital. It is hoped that this simulation will assist in gaining a deeper understanding of the underlying hospital system as well as testing possible congestion relief strategies. However, the development of such a simulation is not straightforward due to the highly complex nature of the hospital system. Furthermore, despite previous attempts, uptake of modelling as part of normal hospital management practice has not been overly successful and there remains a gap between the development of models and the buy-in from health care professionals in using them for management decision-making. The vital role of animals in the transmission of water-borne disease in rural Australia Edward Waters University of Notre Dame Australia, NSW, Australia email: [email protected] Coauthors: Andrew Hamilton, Leesa Sidhu and Harvinder Sidhu Though waterborne diseases such as giardia and cryptosporidium can be carried by animals commonly found in rural locations, epidemiological studies have failed to show convincing evidence that these animals play a role in transmission. It is possible that this is because they have not explicitly modelled the risk of transmission from animals to humans via environmental reservoirs. Many rural Australian households rely on rainwater for their household water supply, and this resource is easily contaminated by the droppings of animals carrying giardia and cryptosporidium. We develop a mathematical model of the transmission of these diseases within and between human and animal populations, with transmission between populations occurring via the drinking of contaminated rainwater. Analysis of the model shows that endemic infection in the animal population is sufficient to cause infection in the human population. This research has important implications for assessing the risk of waterborne disease outbreaks in rural Australia, as current risk assessment protocols do not consider the dynamics of disease in animal populations when estimating risk. Long-term stochastic modelling to predict and prevent osteoarthritis Francis Woodhouse University of Western Australia, Perth, Australia email: [email protected] Coauthors: Bruce Gardiner and David Smith Osteoarthritis afflicts around ten percent of all Australians and Americans, costing hundreds of billions of dollars a year in treatment and lost workforce labour. Primary affecting the knees and hips, it manifests in the slow deterioration of the articular cartilage protecting bones within diarthrodial joints. As the cartilage breaks down, the joint becomes painful and stiff, eventually necessitating a full joint replacement if mobility is severely compromised. Though it is partly a disease of age and genetics, it is also a disease of lifestyle and behaviour, 97 where obesity or otherwise abnormal joint loading can accelerate the onset of tissue degeneration. This implies that at least some proportion of the onset risk can be mitigated with the right lifestyle changes. By integrating what is known about the process of biomechanical damage with models of long-term cartilage homeostasis and random daily human behaviour, we hope not only to compute per-patient likelihoods of osteoarthritis onset five or ten years into the future, but to use this system to propose individually tailored risk reduction techniques. Modelling the role of innate and adaptive immune responses in controlling influenza infection Ada Yan The University of Melbourne, Melbourne School of Population and Global Health, Parkville, Victoria, Australia email: [email protected] Coauthors: Pengxing Cao, Teagan Guarnaccia, Louise Carolan, Malet Aban, Robert Moss, Stephen Petrie, Sophie Zaloumis, Jodie McVernon, Jane Heffernan, Karen Laurie and James McCaw A study conducted by Laurie et al. has demonstrated that a primary influenza infection can protect against subsequent challenge with related or unrelated viruses. However, this protection is exquisitely sensitive to the time interval between exposures, and the order in which viruses are presented. We seek to understand the mechanisms underlying temporary immunity by constructing mathematical models to describe the observed viral dynamics. Our model shows that the initial non-specific immune response to the first influenza virus is able to slow down the rate of infection and death of susceptible cells, thereby preventing or delaying a second infection. The subsequent virus-specific ‘adaptive’ immune response contributes to the resolution of infections once the innate immune response wanes. Varying model parameters between viruses, such as the viral production rate, transmission rate and degree of innate/adaptive immune response allows us to create a hierarchy of the ability of infection with one influenza virus to block or delay infection with another. However, the model includes three possible mechanisms for the innate immune response, which lead to similar outcomes. Furthermore, by taking into account additional biological processes and/or different assumptions about how model components interact, we can create different models which also reproduce the delay and/or blocking of secondary infection as seen in the experimental data. Hypotheses and predictions can be constructed from the models, which can then be experimentally tested to enable us to select the most suitable model. A better understanding of interaction between successive infections at the within-host level will be valuable in improving population-level models of infection, which can then be used to inform public health policy. Rarefied gas flow generated by an oscillating sphere Ying Wan Yap University of Melbourne, Parkville, Victoria, Australia email: [email protected] Coauthors: John Sader Flow generated by an oscillating sphere in a quiescent fluid is a classical problem, whose solution in the continuum limit is obtained via the Navier-Stokes equations. For gas flows away from the continuum limit, the kinetic theory of gases provides a more rigorous description. In this talk, I will discuss the effects of gas rarefaction on the flow generated by an oscillating sphere within the framework of the Boltzmann-BGK equation. In comparison to the continuum limit, where the flow is isothermal, non-continuum effects lead to a temperature jump at the sphere surface and thus the flow is strongly non-isothermal. 98 Modelling the Motions of a Sea Ice Floe in Waves Lucas Yiew University of Adelaide, Adelaide, South Australia, Australia email: [email protected] Coauthors: Luke Bennetts and Mike Meylan The relationship between sea ice and our global climate system is highly complex and dynamic. The distribution and concentration of sea ice has been shown to affect large-scale oceanic and atmospheric processes. In order to improve our understanding of the current conditions in the Arctic and Antarctic, and predict future changes to the sea-ice cover, it is important to accurately model the physical and dynamic processes occurring in these regions. One of these processes is the interaction between ocean waves and ice floes (discrete pieces of sea ice). Contemporary sea ice and global oceanic circulation models do not include wave-ice interactions, and floe-floe interactions (collisions and raftings) caused by waves. To model these processes, we start by developing a mathematical model to predict the motions of a solitary ice floe in ocean waves, using linear potential flow theory. The predictions of the theoretical model are validated using experimental data collected from laboratory wave basin experiments. Solutions to an advanced functional partial differential equation of the pantograph-type Ali A. Zaidi1 Massey University, Auckland, New Zealand email: [email protected] Coauthors: Bruce van-Brunt and Graeme Wake ∂n(x, t) ∂n(x, t) +g = bα2 n(αx, t) − (b + µ)n(x, t), ∂t ∂x in {(x, t) : x, t > 0}, with an initial condition. This partial differential equation arises in cell growth models where n(x, t) is a measure of the number density of cells of size x at time t, α > 1 (in applications it is usually 2) is the number of daughter cells produced when a mother cell divides symmetrically, b is the frequency of division of cells, g is the growth rate and µ is the mortality of cells. The evolution of the number density n(x, t) of cells by size, in an unperturbed situation, is observed experimentally to asymptote to a constant shape, known as the steady size distribution (SSD). Mathematically, this means n(x, t) ∼ T (t)y(x) as t → ∞. Previous work on this cell growth partial differential equation deals with finding the SSD solutions. We extend the work to find the non-SSD solutions and recover the SSD solutions for large time. Tracking and Predicting Multiple Object Dynamics in a Complex Environment (Animal’s Behaviour) Ayham Zaitouny The University of Western Australia, Western Australia, Australia email: [email protected] Coauthors: Michael Small, Thomas Stemler and Kevin Judd Understanding the collective motion of animals has always been a topic for biologists. Recently, mathematicians have entered this field and explore this complex system. The collective motion refers to the movement of a group of animals interacting among them and with surrounding environment. This interaction ensures the cohesive form of the flock. Examples of this kind of motion are numerous and can be found in bird flocks, fish schools and groups of deer or sheep. Understanding animal movement is a challenging mathematical exercise, that is, this movement is driven by complex animal requirements, such as, interactions between the group individuals, seeking for food or even interactions with different animal group. Research has been done in this field and the specific question of understanding animal behaviour has been approached using various methods. 99 In this framework we report on the application of new mathematical techniques to explore and model the collective motion of animals. Using GPS data from individual pigeons in a flock, we model the dynamics based on forces resulting from the collective movement of the flock as well as forces that resulting from the interaction of the individual pigeons. Our nonlinear dynamical model can explain the real observations quite well and we point out some difficulties that arise when working with such data sets. The numerical simulation of a fractional Black-Scholes model for European call Hongmei Zhang Fuzhou University, Fuzhou, China email: [email protected] Coauthors: Fawang Liu and Ian Turner It is well known that the classical Black-Scholes (B-S) models puts a important role in pricing the financial derivatives. However, the empirical data shows that the B-S model cannot correctly capture the dynamics of the option prices such as large movement or jumps over small time step. To overcome these drawbacks other more realistic models, which follow a jump process or a Levy process, have been proposed to model the movements in the stock price. One of these models is FMLS (Finite Moment Log Stable) model, which can be written as fractional partial differential equation. For the fractional differential equation, it is difficult to obtain the exact solution or analysis solution. Therefore, in the paper, we mainly consider the numerical approximate solution of the FMLS model in a finite domain. Unlike the existing literature (Chen (2013)), which constructed the discrete scheme of the fractional derivative by conversing the derivative definition from the Riemann-Liouville (R-L) fractional derivative to the Caputo fractional derivative. In this paper, directly from the R-L derivative definition itself in the FMLS model, we construct a discrete implicit numerical scheme with the second order accuracy. Then we analyze and prove the stability and convergence of the implict numerical scheme. Finally, various numerical experiments suggest that the efficiency of the implicit numerical scheme. Based on the numerical data, we also analysis the characteristics of the parameters in the FMLS model. 100 Speaker index Alexander Gilbert , 23 Andrew Eberhard , 25 Azam Asanjarani , 20 Dale Ward , 23 Dion O’Neale , 24 Dwi Lestari , 27 Hongmei Zhang , 19 Jeffrey Hunter , 20 Jerzy Filar , 20 Jin Hyup Hong , 26 Kate Saunders , 25 Konstantinos Sakellariou , 24 Laleh Tafakori , 27 Lewis Mitchell , 24 Louis Bhim , 19 Mark Fackrell , 23 Maryam Alavi-Shoshtari , 25 Matthew Tam , 22 Michelle Dunbar , 21 Peter Taylor , 20 Phil Broadbridge , 26 Philipp Braun , 21 Pieter Roffelsen , 25 Pouya Baniasadi , 24 rahela abdul Rahim , 19 Robert Scheichl , 27 Soorena Ezzati , 21 Vera Roshchina , 28 yang shi , 26 Ada Yan, 20 Adam Ellery, 20 Adam Tunney, 19 Adelle Coster, 27 Adrian Grantham, 21 Adrian Noppe, 24 Aimin Chen, 25 Alexandra Hogan, 21 Ali A. Zaidi, 17 Ali Eshragh, 20 Andras Czirok, 25 Andrea Babylon, 18 Andrew Black, 21 Andrew Cramer, 19 Andrew Holder, 22 Andrey Pototsky, 26 Anna McGann, 25 Anne Juel, 22, 30 Ashish Goyal, 20 Audrey Markowskei, 25 Awad Al-Mohy, 27 Ayham Zaitouny, 19 Barbara Johnston, 20 Bob Anderssen, 28 Boris Baeumer, 28 Bronwyn Hajek, 20 Bruce Gardiner, 19 Bruce van Brunt, 17 Carlo Laing, 27 Carson Drummond , 17 Catherine Penington, 18 Catheryn Gray, 25 Chen Chen, 23 Christopher Angstmann, 19 Christopher Kellett, 24 Christopher Lustri, 26 Claire Miller, 23 Cris Hasan, 24 Daniel Ladiges, 25 David Arnold, 21 David Harman, 19 David Skene, 24 Debadi Chakraborty, 28 Dougal McQueen, 22 Duncan Farrow, 23 Dylan Lusmore, 17 Eamon Conway , 23 Edward Green, 19 Edward Waters, 28 Elena Vynogradova, 27 Ellen Muir, 17 Elliot Carr, 26 Emma Greenbank, 23 Francis Woodhouse, 24 Frank de Hoog, 18 Gary Froyland, 28, 29 Graeme Hocking, 26 Graeme Wake, 17 Guiyuan Ma, 18 Hao Wang, 20 Harish Sankaranarayanan, 19 Hayden Tronnolone, 20 Heather Davidson, 23 Hugh Possingham, 17, 32 Ian Sloan, 17 James McCaw, 21 James Nichols, 20 James Osborne, 25 James Reoch, 25 James Walker, 22 Jason Cosgrove, 19 Jason Sharples, 23 Jennifer Flegg, 27 Jeong Ryeol Choi, 24 Jerome Droniou, 17 Jesse Collis, 18 Jim Denier, 17 Jin Liang, 18 101 John Boland, 21 John Hearne, 18 John Knight, 26 John Mitry, 25 John Murray, 20 John Shepherd, 28 Josh Chopin, 18 Joshua Ross, 26 Karen McCulloch, 25 Kerry Landman, 27, 31 Kerry-Lyn Roberts, 24 Larry Forbes, 17 Laura Karantgis, 25 Leah Edelstein-Keshet, 21, 29 Lisa Mayo, 21 Lotte Sewalt, 23 Lucas Yiew, 24 Luigi Cirocco, 22 Luke Bennetts, 25 Luke Fullard, 26 Lynne McArthur, 26 Marianito Rodrigo, 26 Mark McGuinness, 26 Mark Nelson, 23 Mary Myerscough, 23, 31 Matthew Chan, 19 Matthew Simpson, 21 Md Hamidul Islam, 24 Md. Habibur Rahman , 18 Megan Farquhar, 17 Meksianis Ndii, 18 Melanie Roberts, 28 Michael Jackson, 22 Michael Page, 17 Michael Plank, 17 Michael Small, 25, 32 Mick Roberts, 26 Mike Chen, 20 Mingmei Teo, 21 Minh Tran, 18 Robert Moss, 26 Roslyn Hickson, 27 Rowena Ball, 26 Saber Dini, 20 Sarthok Sircar, 27 Scott McCue, 22 Sergey Suslov, 18 Shanlin Qin, 25 Sheehan Olver, 27 Shev MacNamara, 20 Shinya Miyajima, 28 Shrupa Shah, 20 Silvestru Sever Dragomir, 20 Steve Taylor, 21 Steve Walters, 24 Stuart Johnston, 18 Sue Ann Chen, 21 Thomas Witeklski, 19 Thomas Witelski, 33 Timothy Moroney, 19 Tom Dyer, 24 Tony Miller, 23 Tony Roberts, 24 Vivien Challis, 22 Vivien Kirk, 24 Wang Jin, 22 William Holmes, 26 Winston Sweatman, 24 Xin-Jiang He , 18 Yan Ding, 24 Ying Wan Yap, 25 Yvonne Stokes, 20 Zoltan Neufeld, 28 Ngamta Thamwattana, 18, 33 Nicholas Read, 23 Nick Fewster-Young, 25 Nicolas Rebuli, 22 Nobutaka Nakazono, 26 Noel Barton, 26 Owen Jepps, 26 Pascal Buenzli, 19 Peter Ballard, 25 Peter Johnston, 17 Peter van Heijster, 23 Rachael Griffiths, 21 Rachelle Binny, 18 Ravindra Pethiyagoda, 24 Rebecca Turner, 17 102
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