Stochastic dynamics of SIRS epidemic models withrandom perturbation

Qingshan Yang; Xuerong Mao

doi:10.3934/mbe.2014.11.1003

Article Contents

2014, Volume 11, Issue 4: 1003-1025. Doi: 10.3934/mbe.2014.11.1003

This issue Previous Article A note on global stability for malaria infections model with latencies Next Article

Stochastic dynamics of SIRS epidemic models with random perturbation

Qingshan Yang^1, and
Xuerong Mao^2,

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

Received: February 2013

Revised: September 2013

Published: March 2014

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Abstract

In this paper, we consider a stochastic SIRS model with parameter perturbation, which is a standard technique in modeling population dynamics. In our model, the disease transmission coefficient and the removal rates are all affected by noise. We show that the stochastic model has a unique positive solution as is essential in any population model. Then we establish conditions for extinction or persistence of the infectious disease. When the infective part is forced to expire, the susceptible part converges weakly to an inverse-gamma distribution with explicit shape and scale parameters. In case of persistence, by new stochastic Lyapunov functions, we show the ergodic property and positive recurrence of the stochastic model. We also derive the an estimate for the mean of the stationary distribution. The analytical results are all verified by computer simulations, including examples based on experiments in laboratory populations of mice.

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Mathematics Subject Classification: 37H10, 37H15, 60J60.

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