Existence of periodic solutions for the periodically forced SIR model

TL;DR

This paper proves the existence of T-periodic solutions in a seasonally-forced SIR epidemic model with R0>1 using Leray-Schauder degree theory, and demonstrates numerical computation methods for these solutions.

Contribution

It establishes the existence of periodic solutions in the forced SIR model and introduces numerical techniques for their computation.

Findings

Periodic solutions exist when R0>1

Numerical methods successfully compute T-periodic solutions

The approach applies to subharmonic and chaotic behaviors

Abstract

We prove that the seasonally-forced SIR model with a T-periodic forcing has a periodic solution with period T whenever the basic reproductive number R0>1. The proof uses the Leray-Schauder degree theory. We also describe some numerical results in which we compute the T-periodic solution, where in order to obtain the T-periodic solution when the behavior of the system is subharmonic or chaotic, we use a Galerkin scheme.