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## Bayesian Inference for Stochastic Epidemic Models using Markov chain Monte Carlo Methods

Demiris, Nikolaos (2004) Bayesian Inference for Stochastic Epidemic Models using Markov chain Monte Carlo Methods. PhD thesis, University of Nottingham.

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## Abstract

This thesis is concerned with statistical methodology for the analysis of stochastic SIR (Susceptible->Infective->Removed) epidemic models. We adopt the Bayesian paradigm and we develop suitably tailored Markov chain Monte Carlo (MCMC) algorithms. The focus is on methods that are easy to generalise in order to accomodate epidemic models with complex population structures. Additionally, the models are general enough to be applicable to a wide range of infectious diseases.

We introduce the stochastic epidemic models of interest and the MCMC methods we shall use and we review existing methods of statistical inference for epidemic models. We develop algorithms that utilise multiple precision arithmetic to overcome the well-known numerical problems in the calculation of the final size distribution for the generalised stochastic epidemic. Consequently, we use these exact results to evaluate the precision of asymptotic theorems previously derived in the literature. We also use the exact final size probabilities to obtain the posterior distribution of the threshold parameter $R_0$.

We proceed to develop methods of statistical inference for an epidemic model with two levels of mixing. This model assumes that the population is partitioned into subpopulations and permits infection on both local (within-group) and global (population-wide) scales. We adopt two

different data augmentation algorithms. The first method introduces an appropriate latent variable, the \emph{final severity}, for which we have asymptotic information in the event of an outbreak among a population with a large number of groups. Hence, approximate inference can be performed conditional on a major'' outbreak, a common assumption for stochastic processes with threshold behaviour such as epidemics and branching processes.

In the last part of this thesis we use a \emph{random graph} representation of the epidemic process and we impute more detailed information about the infection spread. The augmented state-space contains aspects of the infection spread that have been impossible to obtain before. Additionally, the method is exact in the sense that it works for any (finite) population and group sizes and it does not assume that the epidemic is above threshold. Potential uses of the extra information include the design and testing of appropriate prophylactic measures like different vaccination strategies. An attractive feature is that the two algorithms complement each other in the sense that when the number of groups is large the approximate method (which is faster) is almost as accurate as the exact one and can be used instead. Finally, it is straightforward to extend our methods to more complex population structures like overlapping groups, small-world and scale-free networks

Item Type: Thesis (University of Nottingham only) (PhD) O'Neill, Philip Bayesian Inference, Stochastic Epidemic Models, Markov chain Monte Carlo Methods Q Science > QA Mathematics > QA276 Mathematical statistics UK Campuses > Faculty of Science > School of Mathematical Sciences 10078 EP, Services 05 Oct 2004 14 Oct 2017 16:48 https://eprints.nottingham.ac.uk/id/eprint/10078

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