American Institute of Mathematical Sciences

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2010, 7(2): 347-361. doi: 10.3934/mbe.2010.7.347

Global stability for a class of discrete SIR epidemic models

Yoichi Enatsu 1, , Yukihiko Nakata 1, and  Yoshiaki Muroya 2,

1. 

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

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan

Received  July 2009 Revised  February 2010 Published  April 2010

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.
Keywords: globally asymptotic stability, distributed delays, Backward Euler method, Lyapunov functional., discrete SIR epidemic model.
Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D3.
Citation: Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347-361. doi: 10.3934/mbe.2010.7.347
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