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

Pages: 1119 - 1128, Issue Special, September 2011

 Abstract        Full Text (381.9K)              

Yukihiko Nakata - Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio, Spain (email)
Yoichi Enatsu - 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 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:  SIRS epidemic model, global asymptotic stability, distributed delays, Lyapunov functional
Mathematics Subject Classification:  Primary: 34K20 and 34K25; Secondary: 92D30

Received: July 2010;      Revised: August 2010;      Published: October 2011.