Bayesian Data Augmentation for Partially Observed Stochastic Compartmental Models

Deterministic compartmental models are predominantly used in the modeling of infectious diseases, though stochastic models are considered more realistic, yet are complicated to estimate due to missing data. In this paper we present a novel algorithm for estimating the stochastic SIR/SEIR epidemic model within a Bayesian framework, which can be readily extended to more complex stochastic compartmental models. Specifically, based on the infinitesimal conditional independence properties of the model, we are able to find a proposal distribution for a Metropolis algorithm which is very close to the correct posterior distribution. As a consequence, rather than perform a Metropolis step updating one missing data point at a time, as in the current benchmark Markov chain Monte Carlo (MCMC) algorithm, we are able to extend our proposal to the entire set of missing observations. This improves the MCMC methods dramatically and makes the stochastic models now a viable modeling option. A number of real data illustrations and the necessary mathematical theory supporting our results are presented.