Bayesian computational methods for stochastic epidemics - Stockdale, Jessica E. (2019)
Mathematical modelling has become a useful and commonly-used tool in the analysis of infectious disease dynamics. Understanding disease spread is of considerable importance for public health planning and the prevention of future outbreaks, and mathematical analysis of disease outbreaks offers insight which may not be so easily obtained through direct biological study. One key aspect, in mathematical analysis of infectious diseases specifically, is that generally the epidemic process is only partially observed. We might be able to identify the time at which infective individuals become symptomatic or recover, but rarely are we able to observe when infection began, or from whom it was transmitted. This leads to a number of complications with analysis, which will be a focus of this work. The first part of this thesis describes a full Bayesian analysis for such an outbreak with only partial observation of the disease process. We will perform the first Bayesian analysis of the Abakaliki smallpox data, which have been widely cited within the infectious disease modelling literature, to include the full data. In order to do this, we use data augmented Markov Chain Monte Carlo (DA-MCMC) techniques to perform parameter estimation. Analysis involves interpretation of these parameter estimates as well as model assessment with simulation-based methods. We also compare our results to a previous analysis which used an approximate likelihood expression. The second part of this thesis describes novel approximate likelihood methods, motivated in part by the results of the Abakaliki study. Although DA-MCMC is generally considered the standard tool for analysis of partial epidemic data, it often struggles for large population sizes and large amounts of missing data, both through issues of highly correlated missing data and of potentially limiting computation times. We suggest that likelihood approximation methods are a useful tool for dealing with these issues. We develop a series of such methods, which essentially assume some independence in the outbreak population in order to obtain likelihood expressions which do not depend on any missing data. These methods will be motivated and developed, and then illustrated both by simulation study and by application to real data.