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

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SIS Model
Age Structure
Endemic Equilibrium
Epidemic Model
Irreducibility
Partial differential equations
Stability of Equilibria
Infinitesimal Generator
Global Dynamics
Infectious Diseases
Systems of Partial Differential Equations
Infection
Existence and Uniqueness
Semigroup
Eigenvalue
Converge
Dependent
Modeling
Model

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

Global behavior of a multi-group SIS epidemic model with age structure. / Feng, Zhilan; Huang, Wenzhang; Castillo-Chavez, Carlos.

In: Journal of Differential Equations, Vol. 218, No. 2, 15.11.2005, p. 292-324.

Research output: Contribution to journalArticle

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N2 - 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.

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