A multi-group SIR epidemic model with age structure

Toshikazu  Kuniya; Jinliang  Wang; Hisashi Inaba

doi:10.3934/dcdsb.2016109

Article Contents

2016, Volume 21, Issue 10: 3515-3550. Doi: 10.3934/dcdsb.2016109

This issue Previous Article The vanishing surface tension limit for the Hele-Shaw problem Next Article Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes

A multi-group SIR epidemic model with age structure

1.
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-0067
2.
School of Mathematical Science, Heilongjiang University, Harbin 150080
3.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received: September 2013

Revised: August 2016

Published: December 2016

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Abstract

This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations. The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant. To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.

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Mathematics Subject Classification: Primary: 35Q92, 92D30; Secondary: 37N25.

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