SSP 1995 project summary: Spatial Aspects of Epidemics

There is an extensive literature on the dynamics of childhood infectious diseases such as measles and chickenpox, and an appropriate mathematical model for the effect of the disease in an isolated population is generally accepted. However, there is growing recognition that in the modern world even island populations cannot be considered as isolated. Hence the importance of spatial factors and in particular the linkage between populations caused by migration has been of recent interest to those modelling disease dynamics. Grenfell et al. investigate the effect of migration of a small number of infectious individuals between ten distinct populations. They found that the course of the disease in the various populations grow more similar in time. This increased synchrony means that the model predicts common (and unrealistic) die-out of the disease.

The proposed project would also investigate the course of the disease in a set of populations. However these populations would have defined spatial positions: i .e. they would be at the vertices of a square lattice. In contrast to Grenfell et al. however, a migrating person would not be equally likely to move to any of the other populations; moves to near neighbours will be more likely than long distance jumps. The initial object of the study will be to investigate whether Grenfell et al.'s qualitative observations are preserved in this modified model. Grenfell et al. also assume that all the populations were identical in size. We would also investigate whether relaxing this assumption changes model predictions. If time allowed, the additional synchronising effect of seasonal variation in some model parameters would also be investigated.

Technical aspects of the project can vary depending on the interests and ability of the student. The traditional model describing the dynamics of the disease in an isolated population is a set of four ordinary differential equations. Solving these (with migration terms added) simulataneously on each lattice node in the model would be relatively simple. However recent work on modelling the disease in isolated populations use stochastic models which generate events probabilistically. Implementing a network of these populations efficently in a parallel environment would be much more challanging. Both approaches could be justified and comparison between them would be of interest in itself.

worked on this project.

Compressed PostScript of the project's final report is available here (1045718 bytes) .

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