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Mathematical analysis for an age-structured SIRS epidemic model.

Kento Okuwa1, Hisashi Inaba1, Toshikazu Kuniya2

  • 1Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan.

Mathematical Biosciences and Engineering : MBE
|September 11, 2019
PubMed
Summary

This study introduces a new mathematical technique for age-structured SIRS epidemic models, crucial for understanding diseases where immunity wanes. It analyzes disease spread and the impact of mass vaccination strategies.

Keywords:
SIRS epidemicage structurebasic reproduction numbercompact attractorforward bifurcationpersistence

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Area of Science:

  • Mathematical epidemiology
  • Disease modeling
  • Population dynamics

Background:

  • The SIRS (Susceptible-Infectious-Recovered-Susceptible) model is a key framework for studying infectious diseases.
  • Age structure significantly influences disease transmission dynamics.
  • Reversion of susceptibility in recovered individuals presents unique modeling challenges.

Purpose of the Study:

  • To develop novel mathematical techniques for an age-structured SIRS epidemic model.
  • To analyze the existence and stability of endemic steady states.
  • To investigate the impact of mass vaccination on disease control.

Main Methods:

  • Well-posedness proof for the normalized age-structured SIRS model.
  • Fixed point arguments and bifurcation methods to determine endemic equilibria.
  • Introduction of the next generation operator and basic reproduction number (R₀).
  • Stability analysis using bifurcation calculations and comparison methods.
  • Population persistence theory for global behavior analysis.

Main Results:

  • Established well-posedness of the age-structured SIRS model.
  • Determined conditions for the existence of endemic steady states using R₀.
  • Demonstrated stability of steady states and identified a compact attractor.
  • Provided numerical examples illustrating disease dynamics and vaccination effects.

Conclusions:

  • The developed techniques effectively handle the complexities of age-structured SIRS models.
  • Mass vaccination can be optimized by considering the reinfection threshold and critical coverage.
  • The study provides a robust framework for understanding and controlling infectious diseases in structured populations.