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2011, 2011(Special): 1119-1128. doi: 10.3934/proc.2011.2011.1119

On the global stability of an SIRS epidemic model with distributed delays

Yukihiko Nakata 1, , Yoichi Enatsu 2, and  Yoshiaki Muroya 3,

1. 

Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio
2. 

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

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

Received  July 2010 Revised  August 2010 Published  October 2011

In this paper, we establish the global asymptotic stability of an endemic equilibrium for an SIRS epidemic model with distributed time delays. It is shown that the global stability holds for any rate of immunity loss, if the basic reproduction number is greater than 1 and less than or equals to a critical value. Otherwise, there is a maximal rate of immunity loss which guarantees the global stability. By using an extension of a Lyapunov functional established by [C.C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RWA. 11 (2010) 55-59], we provide a partial answer to an open problem whether the endemic equilibrium is globally stable, whenever it exists, or not.
Keywords: global asymptotic stability, distributed delays, SIRS epidemic model, Lyapunov functional.
Mathematics Subject Classification: Primary: 34K20 and 34K25; Secondary: 92D30.
Citation: Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119
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