  Stochastic epidemic models describe the dynamics of an epidemic as a disease
spreads through a population. Typically, only a fraction of cases are observed
at a set of discrete times. The absence of complete information about the time
evolution of an epidemic gives rise to a complicated latent variable problem in
which the state space size of the epidemic grows large as the population size
increases. This makes analytically integrating over the missing data infeasible
for populations of even moderate size. We present a data augmentation Markov
chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic
epidemic model parameters, in which measurements are augmented with
subject-level disease histories. In our MCMC algorithm, we propose each new
subject-level path, conditional on the data, using a time-inhomogeneous
continuous-time Markov process with rates determined by the infection histories
of other individuals. The method is general, and may be applied, with minimal
modifications, to a broad class of stochastic epidemic models. We present our
algorithm in the context of multiple stochastic epidemic models in which the
data are binomially sampled prevalence counts, and apply our method to data
from an outbreak of influenza in a British boarding school.
