Stochastic dynamics of SIRS epidemic models withrandom perturbation
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School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024
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Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH
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Received:
01 February 2013
Accepted:
29 June 2018
Published:
01 March 2014
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MSC :
37H10, 37H15, 60J60.
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In this paper, we consider a stochastic SIRS model with parameterperturbation, which is a standard technique in modeling populationdynamics. In our model, the disease transmission coefficient and theremoval rates are all affected by noise. We show that the stochasticmodel has a unique positive solution as is essential in anypopulation model. Then we establish conditions for extinction orpersistence of the infectious disease. When the infective part isforced to expire, the susceptible part converges weakly to aninverse-gamma distribution with explicit shape and scale parameters.In case of persistence, by new stochastic Lyapunov functions, weshow the ergodic property and positive recurrence of the stochasticmodel. We also derive the an estimate for the mean of the stationarydistribution. The analytical results are all verified by computersimulations, including examples based on experiments in laboratorypopulations of mice.
Citation: Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models withrandom perturbation[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 1003-1025. doi: 10.3934/mbe.2014.11.1003
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Abstract
In this paper, we consider a stochastic SIRS model with parameterperturbation, which is a standard technique in modeling populationdynamics. In our model, the disease transmission coefficient and theremoval rates are all affected by noise. We show that the stochasticmodel has a unique positive solution as is essential in anypopulation model. Then we establish conditions for extinction orpersistence of the infectious disease. When the infective part isforced to expire, the susceptible part converges weakly to aninverse-gamma distribution with explicit shape and scale parameters.In case of persistence, by new stochastic Lyapunov functions, weshow the ergodic property and positive recurrence of the stochasticmodel. We also derive the an estimate for the mean of the stationarydistribution. The analytical results are all verified by computersimulations, including examples based on experiments in laboratorypopulations of mice.
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