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Mathematical Biosciences and Engineering (MBE)
 

Global stability for a class of discrete SIR epidemic models

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

 
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Yoichi Enatsu - Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan (email)
Yukihiko Nakata - Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan (email)
Yoshiaki Muroya - Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan (email)

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.

Keywords:  Backward Euler method, discrete SIR epidemic model, distributed delays, globally asymptotic stability, Lyapunov functional.
Mathematics Subject Classification:  Primary: 34K20, 34K25; Secondary: 92D30.

Received: July 2009;      Accepted: February 2010;      Published: April 2010.