Stochastic amplification in an epidemic model with seasonal forcing

TL;DR

This paper analyzes how stochastic effects interact with seasonal forcing in an epidemic SIR model, providing a systematic method to predict epidemic oscillation periods like those in whooping cough and measles.

Contribution

It introduces an analytic approach to disentangle stochasticity and external forcing in a time-dependent SIR model, unifying previous findings and enabling period predictions.

Findings

Analyzes fluctuations around limit cycles induced by seasonal forcing.

Provides a systematic method for predicting epidemic oscillation periods.

Unifies past work on stochastic epidemic models with seasonal forcing.

Abstract

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.