MAGIC042: Stochastic mathematical modelling in biology

Course description: Immunological processes can span scales from handfuls of interacting signalling molecules within a cell to huge populations of proliferating lymphocytes. Thus, a range of deterministic and stochastic modelling approaches are required to describe them. Moreover, experimental advances are providing ever more refined tools with which to probe immune responses and constrain models of infectious disease. For example, recent advances in two-photon microscopy and cell labelling have made it possible to directly observe cells interacting in vivo and are opening new perspectives in Immunology by providing a wealth of quantitative data regarding immune responses in real time. Furthermore, the importance of mathematical modelling for infectious disease is widely recognised, with work on SARS, Foot-and-Mouth and Avian Influenza influencing government policies. Increasingly such modelling tries to take into account the stochastic nature of the transmission process. Those going on to work in this area will need to be aware of the underpinning probabilistic theory and techniques. The time is ripe to prepare the new generation of theoretical immunologists and/or to expose the wider community to the tools/techniques that are currently used in modelling immunological processes and infectious disease.

Aims: To introduce some areas of the biological and medical sciences in which mathematics can have a significant contribution to make. To present different stochastic modelling approaches to understand a wide variety of biological (immunological and infectious disease) phenomena.

Informal description: All the major developments in the physical sciences are underpinned by mathematics, both as (i) a framework (structure or language) for the concise statement of the laws of nature and as (ii) a tool for developing an understanding of new phenomena by modelling analysis. The introduction of mathematics to the biological and the medical sciences is still at an early stage, but it is becoming increasingly important in many areas. This module aims to introduce the student to some areas of mathematical biology that give rise to exciting new developments and to some of the current challenges for mathematical biology.

Basic bibliography

1. Linda J.S. Allen, An Introduction to mathematical biology. Pearson/Prentice Hall, 2007.
2. Linda J.S. Allen, An introduction to stochastic processes with applications to biology. Pearson Education, 2003.
3. L. Edelstein-Keshet, Mathematical Models in Biology. McGraw-Hill, 1987.
4. Samuel Karlin and Howard M. Taylor, A first course in stochastic processes. Academic Press, 1975.
5. Samuel Karlin and Howard M. Taylor, A second course in stochastic processes. Academic Press, 1981.
6. Howard M. Taylor and Samuel Karlin, An introduction to stochastic modelling. Academic Press, 1998.
7. J. M. Steele, Stochastic Calculus and Financial Applications. Springer, 2001.
8. Hakan Anderson and Tom Britton, Stochastic epidemic models and their statistical analysis. Springer, 2000.

There are no "formal" pre-requisites for this course. We expect the students to have a mathematical/theoretical physics background, in particular, calculus, vector calculus, elementary ODEs and elementary dynamical systems theory.

1-2

Introduction to "ordinary" mathematical biology: deterministic mathematical biology. Birth and death processes, populations and the chemostat (bacterial growth). (2 lectures)

3

Introduction to immunology, in particular T cell immunology: T cell receptor, antigen presenting cells, T cell activation, T cell homeostasis and T cell-dendritic cell interactions. (1 lectures)

4

Revision of probability and introduction to random variables: basic probability, discrete random variables, continuous random variables and generating functions. (1 lecture)

5-6

Discrete time Markov chains: definition, birth and death processes and extinction. (2 lectures)

7-8

Continuous time Markov chains: definition, birth and death processes and extinction: the quasi-stationary distribution. (2 lectures)

9

Multi-variate competition processes (1 lecture)

10

Applications to immunology I: T cell homeostasis and clonotype extinction, thymic output and receptor-ligand clustering. (1 lecture)

11

Continuous time: Brownian motion and stochastic calculus. The Ito formula. (1 lecture)

12

First passage and exit times: one dimension. First passage and exit times: multiple dimensions. (1 lecture)

13

Local time and excursions. Diffusion-limited reaction. (1 lecture)

14

Numerical methods for solutions of stochastic dynamical systems. (1 lecture)

15

Applications to immunology II: in vivo T cell-dendritic cell interactions. (1 lecture)

16-17

Stochastic models of infectious disease transmission. (2 lectures)

18-19

Threshold behaviour and diffusion limits for population models. (2 lectures)

20

Quasi-stationary behaviour of population models. (1 lecture)