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Multiple endemic states in age-structured SIR epidemic models.

Andrea Franceschetti1, Andrea Pugliese, Dmitri Breda

  • 1Dept. Mathematics, Universita di Trento, Via Sommarive 14, 38123 Povo (TN), Italy. francesc@science.unitn.it

Mathematical Biosciences and Engineering : MBE
|August 14, 2012
PubMed
Summary

SIR age-structured models can have multiple equilibria, challenging epidemic spread predictions. This study demonstrates a 3x3 contact matrix allowing multiple endemic states, unlike simpler models, revealing complex disease dynamics.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Dynamical Systems

Background:

  • Age-structured SIR models are fundamental for understanding epidemic spread.
  • Previous analyses often assumed restrictive conditions for unique endemic equilibria.
  • The behavior of SIR models with generic contact rates between age classes remains incompletely understood.

Purpose of the Study:

  • To investigate the existence of multiple non-trivial steady states in age-structured SIR models.
  • To explore the conditions under which non-uniqueness of endemic equilibria arises.
  • To analyze the complex dynamical behaviors associated with multiple equilibria.

Main Methods:

  • Construction of a 3x3 contact matrix to exemplify non-uniqueness.
  • Analysis of bifurcations, specifically saddle-node bifurcations, of positive equilibria.
  • Numerical simulations to explore dynamical behaviors, including periodic, quasi-periodic, and chaotic attractors.

Main Results:

  • A 3x3 contact matrix was shown to permit multiple non-trivial endemic equilibria.
  • Non-uniqueness arose from saddle-node bifurcations, not backward transcritical bifurcations.
  • Complex dynamics observed include coexistence of multiple periodic solutions, quasi-periodic, and chaotic attractors.
  • A 2x2 WAIFW matrix guarantees uniqueness, indicating 3x3 is the minimum dimension for multiple equilibria.

Conclusions:

  • The minimum dimension for a contact matrix to exhibit multiple endemic equilibria in SIR models is 3x3.
  • Age-structured SIR models can display significantly complex dynamics beyond simple endemic states.
  • These findings highlight the importance of considering matrix dimension and contact structure in epidemic modeling.