Stochastic amplification in an epidemic model with seasonal forcing

Andrew J. Black (1) , Alan J. Mckane (1)

1 Respiratory Research Group

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.

Mots clés: Non-linear dynamics; Period doubling; Measles

Andrew J. Black, Alan J. Mckane. Stochastic amplification in an epidemic model with seasonal forcing. Journal of Theoretical Biology, 2010, 267 (1), pp.85. ⟨10.1016/j.jtbi.2010.08.014⟩. ⟨hal-00627148⟩