Stochastic approaches to infectious disease in heterogeneous populations

Infectious diseases represent a leading cause of human mortality, and have a substantial social and economic impact. In the last hundred years, a rich mathematical literature has developed which seeks to capture the dynamics of infectious diseases for the purpose of understanding, predicting, and controlling them. Mathematical models of infectious disease transmission often include heterogeneities along spatial, behavioural, or physiological lines, and these individual-level differences can have important effects on the population-level dynamics of infection. In this thesis we will draw on this background to propose some new approaches to modelling infectious diseases in both the short-timescale epidemic setting and the long-timescale endemic setting. We begin by considering the early stages of outbreaks, which are often characterised by the presence of “superspreading” individuals who produce a much higher than average number of subsequent cases. We introduce a beta-Poisson mixture model for the distribution of these early secondary case numbers, in which infectious individuals make contacts at the points of a Poisson process and then transmit infection along these contacts with a beta-distributed probability. By fitting our model to early outbreak data from several different locations and infectious agents, we find that while the beta-Poisson mixture can achieve a closer fit to data than the negative binomial distribution, it is consistently outperformed by the commonly-used negative binomial in terms of Akaike Information Criterion. We then shift our attention to the long-term dynamics of endemic infections. Pre-vaccination, childhood infections such as measles and mumps often circulate over long timescales which allow individuals to age out of risk groups and move between households. The resulting changes in household-level immunity and susceptibility make it difficult to predict the impact of public health interventions such as vaccination, suggesting the need for infectious disease models with underlying transmission structures which evolve dynamically. With this in mind we develop an approach which combines a demographic model describing the life cycle of a family unit with an age- and household-structured model of infectious disease dynamics. Using data from demographic studies and contact surveys, we model UK-like and Kenya-like populations in order to understand the impact of demography on infectious disease dynamics. We find that age and household structure act in tandem to concentrate infection within households containing school-age children and their younger siblings. Our comparisons between UK- and Kenya-like populations demonstrate the importance of demography to infectious disease transmission, with larger household sizes allowing the Kenya-like population to support a substantially higher burden of disease than the UK-like population. We use our model to study two possible control measures: vaccination and school closure. In particular, we make a direct comparison between the distribution of vaccines on a uniform individual basis and on a household basis, finding that uniform vaccination consistently achieves herd immunity at a lower rate of vaccination than household-based vaccination