Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes

C. Connell McCluskey; C. Connell McCluskey

doi:10.3934/mbe.2012.9.819

Mathematical Biosciences and Engineering

2012, Volume 9, Issue 4: 819-841. doi: 10.3934/mbe.2012.9.819

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Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes

C. Connell McCluskey

1.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario

Received: 01 November 2011 Accepted: 29 June 2018 Published: 01 October 2012
MSC : Primary: 34K20, 92D30; Secondary: 34D20.

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals. The model is very appropriate for tuberculosis. Key theorems, including asymptotic smoothness and uniform persistence, are proven by reformulating the system as a system of Volterra integral equations. The basic reproduction number $\mathcal{R}_{0}$ is calculated. For $\mathcal{R}_{0}<1$, the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_{0}>1$, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present. Finally, some special cases are considered.
- global stability,
- tuberculosis,
- Age-structure,
- latency,
- epidemiology.,
- Lyapunov functional
Citation: C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes[J]. Mathematical Biosciences and Engineering, 2012, 9(4): 819-841. doi: 10.3934/mbe.2012.9.819

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Abstract

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals. The model is very appropriate for tuberculosis. Key theorems, including asymptotic smoothness and uniform persistence, are proven by reformulating the system as a system of Volterra integral equations. The basic reproduction number $\mathcal{R}_{0}$ is calculated. For $\mathcal{R}_{0}<1$, the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_{0}>1$, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present. Finally, some special cases are considered.
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