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.
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)
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
- Linda J.S. Allen, An Introduction to mathematical biology.
Pearson/Prentice Hall, 2007.
Linda J.S. Allen, An introduction to stochastic processes with
applications to biology. Pearson Education, 2003.
L. Edelstein-Keshet, Mathematical Models in Biology. McGraw-Hill,
Samuel Karlin and Howard M. Taylor, A first course in stochastic
processes. Academic Press, 1975.
Samuel Karlin and Howard M. Taylor, A second course in stochastic
processes. Academic Press, 1981.
Howard M. Taylor and Samuel Karlin, An introduction to stochastic
modelling. Academic Press, 1998.
J. M. Steele, Stochastic Calculus and Financial
Applications. Springer, 2001.
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.