Inference for Ecological Dynamical Systems: A Case Study of Two Endemic Diseases

Abstract

A Bayesian Markov chain Monte Carlo method is used to infer parameters for an open stochastic epidemiological model: the Markovian susceptible-infected-recovered (SIR) model, which is suitable for modeling and simulating recurrent epidemics. This allows exploring two major problems of inference appearing in many mechanistic population models. First, trajectories of these processes are often only partly observed. For example, during an epidemic the transmission process is only partly observable: one cannot record infection times. Therefore, one only records cases (infections) as the observations. As a result some means of imputing or reconstructing individuals in the susceptible cases class must be accomplished. Second, the official reporting of observations (cases in epidemiology) is typically done not as they are actually recorded but at some temporal interval over which they have been aggregated. To address these issues, this paper investigates the following problems. Parameter inference for a perfectly sampled open Markovian SIR is first considered. Next inference for an imperfectly observed sample path of the system is studied. Although this second problem has been solved for the case of closed epidemics, it has proven quite difficult for the case of open recurrent epidemics. Lastly, application of the statistical theory is made to measles and pertussis epidemic time series data from 60 UK cities.