Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

Yoichi Enatsu; Yukihiko Nakata; Yoshiaki Muroya

doi:10.3934/dcdsb.2011.15.61

Article Contents

2011, Volume 15, Issue 1: 61-74. Doi: 10.3934/dcdsb.2011.15.61

This issue Previous Article Finite to infinite steady state solutions, bifurcations of an integro-differential equation Next Article The dynamics of a low-order model for the Atlantic multidecadal oscillation

Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

1.
Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555
2.
Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio
3.
Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555

Received: September 30, 2009

Revised: March 31, 2010

Published: June 2011

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Abstract

In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number $R_0$ and the distributed delays do not influence the global dynamics of the model.

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Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D30.

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