A computational investigation of seasonally forced disease dynamics

Abstract

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods