GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

doi:10.1016/S0252-9602(12)60066-6

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (3): 851-865.doi: 10.1016/S0252-9602(12)60066-6

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GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

Yoichi Enatsu¹, Yukihiko Nakata², Yoshiaki Muroya³

1.Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan|2.Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao, Spain|3.Department of Mathematics, Waseda University 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received:2009-11-18 Revised:2011-01-23 Online:2012-05-20 Published:2012-05-20
Supported by:
The authors' work was supported in part by JSPS Fellows, No.237213 of Japan Society for the Promotion of Science to the first author, by the Grant MTM2010-18318 of the MICINN, Spanish Ministry of Science and Innovation to the second author, and by Scientific Research (c), No.21540230 of Japan Society for the Promotion of Science to the third author.

Abstract

Abstract:

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin-ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci-dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.

Key words: SIRS epidemic model, nonlinear incidence rate, global asymptotic stability, distributed delays, Lyapunov functional

CLC Number:

34K20

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS[J].Acta mathematica scientia,Series B, 2012, 32(3): 851-865.

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GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

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