ANZIAM SS2013 subject: Mathematical epidemiology: stochastic models and their statistical calibration

Johua Ross

Synopsis

Mathematical models are increasingly used to inform governmental policy-makers on issues that threaten human health or which have an adverse impact on the economy. It is this real-world success combined with the wide variety of interesting mathematical problems which arise that makes mathematical epidemiology one of the most exciting topics in applied mathematics. During the summer school, you will be introduced to mathematical epidemiology and some fundamental theory required for studying and parametrising stochastic models of infection dynamics, which will provide an ideal basis for addressing key research questions in this area; several such questions will be introduced and explored in this course.

Course content:

• An introduction to mathematical epidemiology.
• Ordinary differential equations models of infectious disease.
• Chain-binomial models of infectious disease.
• Frequentist & Bayesian inference methods for parametrising stochastic infection models.
• Continuous-time Markov chain models of infectious disease.
• Optimal observation of stochastic infection models.

Contact hours

28 hours spread over the four weeks, with consultation as required.

Prerequisites

Ideally, some previous study of the qualitative analysis of ordinary differential equations models, a second (or even better third) year course in probability & statistics, and some prior experience of programming (ideally in MATLAB).

Background reading prior to the summer school is:

• Chapter 3 of Kreyszig; and,
• Chapters 1-4 & 6 of Grimmett & Stirzaker

As examples will be provided in MATLAB (this is because it is a relatively easy to use language which is ideally suited to dealing with matrices which form the basis of the models we will be considering), students wishing to take this course, who have no prior programming experience, should attempt to learn the basics of MATLAB prior to the summer school; Googling "Introduction to MATLAB" should provide several good sources.

Assessment

Three assignments worth 15% each, and a 3-hour exam worth 55%. Example exam questions.

Resources

Lecture notes are available for download. These will also be provided in hardcopy. Additional readings, including relevant journal articles, will be provided during the course.

There are no specific texts required for this course, though students are expected to supplement the lecture material with additional reading. Some relevant texts are:

1. Daley and Gani, Epidemic modelling: an introduction, Cambridge University Press, 2001.
2. Grimmett and Stirzaker, Probability and random processes,, Oxford University Press, 2001.
3. Keeling and Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2008.
4. Gilks, Richardson and Spiegelhalter, Markov Chain Monte Carlo in practice, Chapman and Hall/CRC, 1996.
5. Kreyszig, Advanced engineering mathematics, multiple years.

Code

The course does have a substantial component of implementing theory computationally. If students do have a laptop, then they should check with their home institution about installing MATLAB on it, or download and install the free Scilab software.

Prac 1:

Prac 2:

Prac 3:

Lecture 21: