Qualitative analysis of an age-structured SEIR epidemic model with treatment

Mohammad A. Safi, Abba Gumel, Elamin H. Elbasha

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

A new age-structured model, which incorporates the use of treatment, is designed and qualitatively analysed. The model is, first of all, shown to be properly-posed mathematically by formulating it as an abstract Cauchy problem. For the case where the contact rate is separable (i.e., β(a,b)= β1(a) β2(b)), it is shown that the disease-free equilibrium of the model is locally- and globally-asymptotically stable whenever a certain epidemiological threshold, denoted by R0s, is less than unity. Furthermore, the model has a unique endemic equilibrium when the threshold exceeds unity (this equilibrium is shown to be locally-asymptotically stable if another condition holds). For the case where the natural death and contact rates are constant (i.e., independent of age), the unique endemic equilibrium of the resulting model is shown, using Lyapunov function theory and LaSalle's Invariance Principle, to be globally-asymptotically stable when it exists. Furthermore, for this reduced version of the model (with constant natural death and contact rates), it is shown that the use of treatment could offer positive or negative population-level impact, depending on the size of the parameter associated with the reduction of infectiousness of treated individuals.

Original languageEnglish (US)
Pages (from-to)10627-10642
Number of pages16
JournalApplied Mathematics and Computation
Volume219
Issue number22
DOIs
StatePublished - 2013
Externally publishedYes

Fingerprint

Age-structured Model
Epidemic Model
Qualitative Analysis
Endemic Equilibrium
Globally Asymptotically Stable
Contact
LaSalle's Invariance Principle
Abstract Cauchy Problem
Model
Asymptotically Stable
Lyapunov Function
Lyapunov functions
Invariance
Exceed

Keywords

  • Abstract Cauchy problem
  • Age-structure
  • C-semigroup
  • Equilibria
  • Stability

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

Qualitative analysis of an age-structured SEIR epidemic model with treatment. / Safi, Mohammad A.; Gumel, Abba; Elbasha, Elamin H.

In: Applied Mathematics and Computation, Vol. 219, No. 22, 2013, p. 10627-10642.

Research output: Contribution to journalArticle

Safi, Mohammad A. ; Gumel, Abba ; Elbasha, Elamin H. / Qualitative analysis of an age-structured SEIR epidemic model with treatment. In: Applied Mathematics and Computation. 2013 ; Vol. 219, No. 22. pp. 10627-10642.
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