Bayesian Data Augmentation for Partially Observed Stochastic Compartmental Models

Abstract

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 Susceptible-Infected-Recovered and Susceptible-Exposed-Infected-Recovered (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. In particular, it acts as a very good proposal for the unknown number of events, such as the number of infected individuals. Finding such a good proposal for this has historically been a difficult problem. As a consequence, rather than perform a Metropolis step updating one missing data point at a time, we are able to have a single proposal for the entire set of missing observations. A number of real data illustrations and the necessary mathematical theory supporting our results are presented.