Abstract

We study global dynamics of a system of partial differential equations. The system is motivated by modelling the transmission dynamics of infectious diseases in a population with multiple groups and age-dependent transition rates. Existence and uniqueness of a positive (endemic) equilibrium are established under the quasi-irreducibility assumption, which is weaker than irreducibility, on the function representing the force of infection. We give a classification of initial values from which corresponding solutions converge to either the disease-free or the endemic equilibrium. The stability of each equilibrium is linked to the dominant eigenvalue s(A), where A is the infinitesimal generator of a "quasi-irreducible" semigroup generated by the model equations. In particular, we show that if s(A) < 0 then the disease-free equilibrium is globally stable; if s(A) > 0 then the unique endemic equilibrium is globally stable.

Original languageEnglish (US)
Pages (from-to)292-324
Number of pages33
JournalJournal of Differential Equations
Volume218
Issue number2
DOIs
StatePublished - Nov 15 2005

Keywords

  • Epidemic model
  • Global stability
  • Partial differential equations
  • Quasi-irreducibility
  • Threshold conditions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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