The early stage behaviour of a stochastic SIR epidemic with term-time forcing

Abstract

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An `environment' in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here `term-time forcing' refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with $k$ different states denoted {Λ(t)}_t𢙛0. Threshold conditions in terms of R_⋆ are obtained, such that R_⋆>1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.