Global stability for a class of discrete SIR epidemic models

Yoichi Enatsu; Yukihiko Nakata; Yoshiaki Muroya; Yoichi Enatsu; Yukihiko Nakata; Yoshiaki Muroya

doi:10.3934/mbe.2010.7.347

Mathematical Biosciences and Engineering

2010, Volume 7, Issue 2: 347-361. doi: 10.3934/mbe.2010.7.347

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Global stability for a class of discrete SIR epidemic models

1.
Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555
2.
Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555

Received: 01 July 2009 Accepted: 29 June 2018 Published: 01 April 2010
MSC : Primary: 34K20, 34K25; Secondary: 92D30.

In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.
- globally asymptotic stability,
- distributed delays,
- Backward Euler method,
- Lyapunov functional.,
- discrete SIR epidemic model
Citation: Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models[J]. Mathematical Biosciences and Engineering, 2010, 7(2): 347-361. doi: 10.3934/mbe.2010.7.347

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Abstract

In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.
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