Related Experiment Video
Updated: May 14, 2026

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
Published on: December 9, 2015
Seasonal dynamics in an SIR epidemic system
1Laboratoire de Mathématiques, Image et Applications (MIA), Université de La Rochelle, 17042 , La Rochelle, France, eaugerau@univ-lr.fr.
Abstract:
We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain D containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain K, included in D, for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.
Related Concept Videos
Infectious Diseases and Their Occurrence
Steps in Outbreak Investigation
Modeling with Differential Equations
Population Growth
Exponential Equations for Modeling Growth
Influenza
