Topics in Bayesian inference and model assessment for partially observed stochastic epidemic models

Stochastic epidemic models can offer a vitally important public health tool for understanding and controlling disease progression. However, these models are of little practical use if they are not supported by data or are not applicable to efficient parameter inference methods. The peculiarities of the epidemic setting, where data are not independent and epidemic processes are rarely fully observed, complicate both model assessment and parameter inference for stochastic epidemic models. Methods for model assessment are not well-established and methods for inference, although more established, still remain inefficient for large-scale outbreaks. This thesis is concerned with the development of methods for both model assessment and inference for stochastic epidemic models. The methods are illustrated on continuous time SIR (susceptible → infective → removed) models and it is assumed that the available data consist only of the removal times of infected individuals with their infection times being unobserved. First, two novel model assessment tools are developed, based on the posterior predictive distribution of removal curves, namely the distance method and the position-time method. Both methods rely on the general idea of posterior predictive checking, where a model's fit is assessed by checking whether replicated data, generated under the model, look similar to the observed data. The distance method conducts the assessment by calculating distances between removal curves whereas the position-time method conducts the assessment pointwise, at a sequence of suitably chosen time points. Both methods provide visual and quantitative outputs with meaningful interpretation. The performance of the methods benefits from the development and application of a time shifting intervention, that horizontally (time) shifts each replicated removal curve by an appropriately chosen constant, so that the stages of each replicated curve better correspond to those of the observed. Extensive simulation studies suggest that both the distance and the position-time methods can successfully assess the infectious period distribution assumption and the infection rate form assumption of stochastic epidemic models. Then, the focus is placed on developing methods to assess the population mixing assumption of stochastic epidemic models, in the case that household information is available. To this end, a classical hypothesis test is developed for which the null hypothesis is that individuals mix in the population homogeneously. The test is based on household labels of individuals and relies on the idea that, in the presence of household effect, events of individuals belonging to the same household should occur closer in time rather than further apart. The key behind developing the test is that, under the null hypothesis of homogeneous mixing, the discrete random vector of household labels has a known sampling distribution that does not dependent on any model parameters. The test carries an ordinal interpretation, where the lower the observed value of the test statistic and its corresponding p-value are, the more the evidence against the null hypothesis and in favour of the hypothesis that there is a household effect in the spread of the outbreak. The test exhibits excellent performance when applied to both simulated data and to a widely studied real-life epidemic dataset. In the remainder of the thesis, attention is turned from model assessment to Bayesian inference. The relevant aim is to develop Markov chain Monte Carlo (MCMC) algorithms that can conduct more efficient updating of the unobserved infection times, than the currently existing algorithms. Initially, the problem of updating one infection time at a time is considered and a new 1-dimensional update algorithm is developed, namely the IS-1d MCMC algorithm. The main feature of the algorithm is the use of individual-specific parameters in the proposal distributions for the infection times. These parameters allow the proposal distributions to produce patterns of nonhomogeneity (among individuals) which are in some cases present in the target distribution. The IS-1d MCMC algorithm performs favourably when compared to currently existing 1-dimensional update algorithms. Subsequently, the more interesting problem of updating many infection times at a time is considered and a novel block update MCMC algorithm is developed, referred to as the DIS-block MCMC algorithm. Similar to the IS-1d algorithm, the proposal distributions of the DIS-block algorithm also have individual-specific parameters but they also have an additional parameter that induces dependency on the current state and makes the algorithm perform a dependent in nature exploration of the target space. The algorithm also benefits from another two features, parameter reduction and an automated method for optimally specifying the number of infection times to update. Simulation studies suggest that the DIS-block algorithm can offer a substantial improvement in mixing compared to the current optimally performing block update algorithm; for the considered datasets of the simulation study, the DIS-block algorithm is from 1.41 up to 6.57 times more efficient than its comparator, and 3.35 times on average.