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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.
Journal of Mathematical Biology
|February 14, 2013
Summary
This study analyzes a SIR epidemic model with seasonal variations, proving the existence of periodic solutions and identifying characteristics of epidemic outbreaks. The research offers insights into predicting epidemic dynamics and their periodicity.
Area of Science:
- Epidemiology
- Mathematical Biology
- Dynamical Systems
Background:
- Seasonal variations in contact rates, such as school terms and holidays, influence epidemic dynamics.
- Switched systems are used to model the complex, time-varying nature of infectious disease transmission.
Purpose of the Study:
- To analytically prove the existence of periodic solutions in a seasonally forced SIR epidemic model.
- To establish the existence of a macroscopic attractor domain for the switched dynamics.
- To characterize epidemic outbreaks and predict their periodicity.
Main Methods:
- Analytical proof of an invariant domain containing all periodic orbits.
- Transformation of the SIR model into a slow-fast dynamical system using different time and variable scales.
- Establishment of a macroscopic attractor domain for the switched dynamics.
Main Results:
- The existence of a unique harmonic solution (annual infection) is proven for any seasonal forcing magnitude.
- Subharmonic solutions, representing epidemic outbreaks, are identified.
- Quantitative characteristics like maximal period between outbreaks and maximal prevalence are derived.
Conclusions:
- The study provides a theoretical framework for understanding seasonal influences on epidemic spread.
- The findings enable quantitative predictions of epidemic behavior, including outbreak frequency and severity.
- This research contributes to the predictive modeling of infectious diseases.
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