# American Institute of Mathematical Sciences

2014, 11(4): 1003-1025. doi: 10.3934/mbe.2014.11.1003

## Stochastic dynamics of SIRS epidemic models with random perturbation

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

Received  February 2013 Revised  September 2013 Published  March 2014

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.
Citation: Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003
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