Permanence for a Delayed Nonautonomous SIR Epidemic Model with Density-Dependent Birth Rate

Li Yingke,Chen Liang,Wang Kai

doi:10.1155/2011/350892

Abstract

Based on some well-known SIR models, a revised nonautonomous SIR epidemic model with distributed delay and density-dependent birth rate was considered. Applying some classical analysis techniques for ordinary differential equations and the method proposed by Wang (2002), the threshold value for the permanence and extinction of the model was obtained.

Highlights

For understanding the spread of infectious diseases in population, mathematical models that use the theories of ordinary differential equations in epidemiology have been developed rapidly
S θ φ1 θ, I θ φ2 θ, R θ φ3 θ, −∞ < θ ≤ 0, 2.2 where φ φ1, φ2, φ3 T ∈ C such that φi θ ≥ 0 i 1, 2, 3 for all −∞ < θ ≤ 0, and C denotes the Banach space C −∞, 0, R of bounded continuous functions mapping the interval −∞, 0 into R with the toplogy of uniform convergence, that is, for φ ∈ C, we designate the norm of an element φ ∈ C by φ sup−∞
We investigate the permanence of system 1.1 and demonstrate how the disease will be extinct under some conditions

Summary

Introduction

For understanding the spread of infectious diseases in population, mathematical models that use the theories of ordinary differential equations in epidemiology have been developed rapidly. Epidemic models with delay, including either the autonomous continuous systems or the discrete ones, were discussed by many authors 1–13. The important subjects for this models are looking for the threshold value that determines whether the infectious disease will be permanent or extinct. It is well known that models with distributed delay are more appropriate than the discrete ones because it is considered more realistic to assume the infectivity to be a function of the duration since infection and up to some maximum duration. Song and Ma in 4 and Song et al in 5 discussed the permanence of disease in a generalized autonomous SIR epidemic model with density dependent birth rate. Discrete Dynamics in Nature and Society consider the following nonautonomous delayed systems with density-dependent birth rate: St h. The function η s : 0, h → 0, ∞ is nondecreasing and has hounded variation such that h dη s η h − η 0 1

Preliminaries

Permanence and Extinction

Discussion