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Stochastic Transport Modelling

Lecturer

Professor Matthew Simpson, Queensland University of Technology

Synopsis

This four-week course will introduce students to a range of lattice-based stochastic transport models that have applications in a range of areas including biology, ecology and traffic flow. Taking a hands-on approach, we will build and explore a range of lattice-based interacting random walk models that will include unbiased migration, biased migration, birth-death phenomena and models of complex populations with interactions between different subpopulations. Students will learn how to build simple simulation tools, to visualise a range of computational experiments, and to extract averaged data from simulations to characterise the expected outcomes of these simulations.

To provide mathematical insight into the stochastic simulations we will introduce coarse-graining methods to extract approximate differential equation descriptions of the mean behaviour of the simulations. These differential equations will involve both ordinary differential equations and partial differential equations and we will use a range of exact and numerical solutions of these differential equations to explore how well the stochastic models can be modelled using simpler differential equations.

This course will proceed by presenting information about how we build and analyse simulation together with plenty of time for assignment-based assessment where students will build and explore stochastic simulation tools, as well as using mathematical descriptions of these simulations. The assessment can be undertaken in small groups or individually.

Course Overview

  • Week 1: Stochastic simulation algorithms, exclusion processes, random sequential update simulation algorithm, unbiased random walks, biased random walks, random walks and birth/death processes.
  • Week 2: Numerical solutions of ODE and PDE models.
  • Week 3: Coarse-graining methods for approximating stochastic simulations using differential equations. Comparing averaged data from simulation models and exact solutions of ODE/PDE models.
  • Week 4: Extensions include: two-dimensional stochastic transport phenomena, transport phenomena involving multiple subpopulations, exploring when ODE/PDE models are poor approximations.

This syllabus will be sported by a series of four weekly assignments to reinforce these concepts.

This will be a foundational course that requires basic skills in programming.

Prerequisites

Students should be confident in programming (e.g. MATLAB, C++, Python, Julia or equivalent). Students will not be required to work in a particular programming language, but students will be expected to build relatively simple algorithms with plenty of hands-on help. Sample code will be provided to get students started.

Some background knowledge of ordinary differential equations and partial differential equation will be an advantage. We will work with a mixture of exact solutions of simple differential equations (logistic equation, linear heat equation) but we will also develop numerical solutions of some nonlinear partial differential equations (Fisher’s equation). Skills in numerical methods will be developed during the course.

Assessment

  • Final assessment details to be confirmed

Attendance requirements

  • For those completing the subject for their own knowledge/interest, final attendance requirements to be confirmed

Resources/pre-reading

https://royalsocietypublishing.org/doi/10.1098/rsif.2008.0014

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Professor Matthew Simpson, Queensland University of Technology

I am professor of Mathematics and Queensland University of Technology, Australia.  Formerly, I held positions as an Australian Research Council Future Fellow (2014-2018) and as an Australian Research Council Postdoctoral Research Fellow (2006-2009).  In 2012 I was awarded the JH Michell Medal for excellence in Research by ANZIAM (Australian and New Zealand Industrial and Applied Mathematics), which is a decision of the Australian Mathematical Society.  In 2020 I was awarded the EO Tuck Medal for outstanding research and distinguished service by ANZIAM.  My research focuses on mathematical biology, mainly developing stochastic mathematical models of cell biology experiments and understanding how to construct approximate continuum limit descriptions of these stochastic models.  Applications of my work involves deploying these mathematical tools to understand various biological and biophysical applications that include studying tissue growth on 3D-printed scaffolds, in vitro observations of cell migration and proliferation using fluorescent cell cycle labels, as well as tumour spheroid experiments. Full details of my research group is available at www.mj-simpson.com