Long-time behaviour of a perturbed SIR model by white noise

Yuguo Lin; Daqing Jiang

doi:10.3934/dcdsb.2013.18.1873

Article Contents

2013, Volume 18, Issue 7: 1873-1887. Doi: 10.3934/dcdsb.2013.18.1873

This issue Previous Article Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise Next Article Optimal stochastic differential games with VaR constraints

Long-time behaviour of a perturbed SIR model by white noise

Yuguo Lin^1, and
Daqing Jiang^1,

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024

Received: July 31, 2012

Revised: February 28, 2013

Published: August 2013

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Abstract

In this paper, we consider a stochastic SIR model with perturbed disease transmission coefficient. We present sufficient conditions for the disease to extinct exponentially. In the case of persistence, we analyze long-time behaviour of densities of the distributions of the solution. We will prove that the densities of the solution can converge in $L^1$ to an invariant density under appropriate conditions. Also we find the support of the invariant density. Specially, when the intensity of white noise is relatively small, we find a new threshold for an epidemic to occur.

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Mathematics Subject Classification: Primary: 47D07, 60H10; Secondary: 60J60, 92D30.

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