Stochastic dynamics of SIRS epidemic models withrandom perturbation
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 solut...
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AIMS Press
2014-02-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1003 |
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| author | Qingshan Yang Xuerong Mao |
| author_facet | Qingshan Yang Xuerong Mao |
| author_sort | Qingshan Yang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-79acd437c55a4c84ac7618945ece65b0 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-79acd437c55a4c84ac7618945ece65b02025-01-24T02:28:19ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-02-011141003102510.3934/mbe.2014.11.1003Stochastic dynamics of SIRS epidemic models withrandom perturbationQingshan Yang0Xuerong Mao1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XHIn 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.https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1003extinctionergodic propertypositive recurrence. |
| spellingShingle | Qingshan Yang Xuerong Mao Stochastic dynamics of SIRS epidemic models withrandom perturbation Mathematical Biosciences and Engineering extinction ergodic property positive recurrence. |
| title | Stochastic dynamics of SIRS epidemic models withrandom perturbation |
| title_full | Stochastic dynamics of SIRS epidemic models withrandom perturbation |
| title_fullStr | Stochastic dynamics of SIRS epidemic models withrandom perturbation |
| title_full_unstemmed | Stochastic dynamics of SIRS epidemic models withrandom perturbation |
| title_short | Stochastic dynamics of SIRS epidemic models withrandom perturbation |
| title_sort | stochastic dynamics of sirs epidemic models withrandom perturbation |
| topic | extinction ergodic property positive recurrence. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1003 |
| work_keys_str_mv | AT qingshanyang stochasticdynamicsofsirsepidemicmodelswithrandomperturbation AT xuerongmao stochasticdynamicsofsirsepidemicmodelswithrandomperturbation |