SIR epidemics with stochastic infectious periods

Abstract

We consider an SIR model for the spread of an epidemic in a closed and homogeneously mixing population, where the infectious periods are represented by an arbitrary absorbing Markov process. A version of this process starts whenever an infection occurs, and the contamination rate of the newly infected individual is a function of its state. When his process is absorbed, the individual becomes a removed case. We use a martingale approach to derive the distribution of the final epidemic size and severity for this class of models. Next, we examine some special cases. In particular, we focus on situations where the infection processes are Brownian motions and where they are Markov-modulated fluid flows. In the latter case, we use matrix-analytic methods to provide more explicit results. We conclude with some numerical illustrations.