Asymptotic analysis of periodic solutions of the seasonal SIR model

Abstract

Epidemiological models of diseases with periodically varying infection rates are known to possess a multiplicity of subharmonic resonant modes. With increasing period, these modes are known to be characterized by outbreaks of dramatically increased severity. Primarily through direct numerical simulation, prior studies have provided insight into their detailed structure and parametric dependence. The hallmark of these periodic modes are relatively short, rapidly varying outbreaks, interspersed by longer, slowly varying inter-epidemic periods. By leveraging this dualistic structure, we develop a hierarchical framework to construct approximate solutions to the seasonal-SIR model. Comparisons with numerical simulation of the full model show a strong level of agreement, not only in the limit that the period is long but also for moderate to shorter periods as well. The underlying and explicit analytic matching conditions provide direct insight into the existence of individual harmonic modes within a given parameter regime and detailed connection to the model parameters.