  Stochastic epidemic models provide an interpretable probabilistic description
of the spread of a disease through a population. Yet, fitting these models to
partially observed data is a notoriously difficult task due to intractability
of the likelihood for many classical models. To remedy this issue, this article
introduces a novel data-augmented MCMC algorithm for exact Bayesian inference
under the stochastic SIR model, given only discretely observed counts of
infection. In a Metropolis-Hastings step, the latent data are jointly proposed
from a surrogate process carefully designed to closely resemble the SIR model,
from which we can efficiently generate epidemics consistent with the observed
data. This yields a method that explores the high-dimensional latent space
efficiently, and scales to outbreaks with hundreds of thousands of individuals.
We show that the Markov chain underlying the algorithm is uniformly ergodic,
and validate its performance via thorough simulation experiments and a case
study on the 2013-2015 outbreak of Ebola Haemorrhagic Fever in Western Africa.
