The threshold  of a stochastic SIRS epidemic model in a population with varying size

Yanan Zhao; Daqing Jiang; Xuerong Mao; Alison Gray

doi:10.3934/dcdsb.2015.20.1277

Article Contents

2015, Volume 20, Issue 4: 1277-1295. Doi: 10.3934/dcdsb.2015.20.1277

This issue Previous Article New results of the ultimate bound on the trajectories of the family of the Lorenz systems Next Article

The threshold of a stochastic SIRS epidemic model in a population with varying size

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin
2.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024
3.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH

Received: July 31, 2013

Revised: January 31, 2014

Published: May 2015

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Abstract

In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold $\tilde{R}_0$ is identified which determines the outcome of the disease. When the noise is small, if $\tilde{R}_0<1$, the infected proportion of the population disappears, so the disease dies out, whereas if $\tilde{R}_0>1$, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when ${R}_0 > 1$ and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold ${R}_0$, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations.

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Mathematics Subject Classification: Primary: 60H10; Secondary: 92D25, 92D30.

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