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December  2016, 21(10): 3429-3440. doi: 10.3934/dcdsb.2016105

## Long-time behavior of an SIR model with perturbed disease transmission coefficient

 1 Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam 2 Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam

Received  December 2015 Revised  June 2016 Published  November 2016

In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used to classify the extinction and permanence of the disease. Precisely, $\lambda<0$ implies that the disease-free $(\frac{\alpha}{\mu}, 0, 0)$ is globally asymptotic stable, i.e., the disease will disappear and the entire population will become susceptible individuals. If $\lambda>0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].
Citation: Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105
##### References:
 [1] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of Asymptotic Behavior in a Stochastic SIR Model,, SIAM J. Appl. Dyn. Syst., 15 (2016), 1062. doi: 10.1137/15M1043315. Google Scholar [2] N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, J. Appl. Probab., 53 (2016), 187. doi: 10.1017/jpr.2015.18. Google Scholar [3] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model,, SIAM J. Appl. Math., 71 (2011), 876. doi: 10.1137/10081856X. Google Scholar [4] K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization,, Z. Wahrsch. Verw. Gebiete, 30 (1974), 235. doi: 10.1007/BF00533476. Google Scholar [5] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, second edition, (1989). Google Scholar [6] C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model,, Appl. Math. Model., 38 (2014), 5067. doi: 10.1016/j.apm.2014.03.037. Google Scholar [7] C. Y. Ji, D. Q. Jiang and N. Z. Shi, The behavior of an SIR epidemic model with stochastic perturbation,, Stochastic Anal. Appl., 30 (2012), 755. doi: 10.1080/07362994.2012.684319. Google Scholar [8] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation,, J. IFAC, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044. Google Scholar [9] R. Z. Khas'minskii, Stochastic Stability of Differential Equations,, Springer-Verlag Berlin Heidelberg, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar [10] W. Kliemann, Recurrence and invariant measures for degenerate diffusions,, Ann. Probab., 15 (1987), 690. doi: 10.1214/aop/1176992166. Google Scholar [11] Y. G. Lin and D. Q. Jiang, Long-time behaviour of a perturbed SIR model by white noise,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1873. doi: 10.3934/dcdsb.2013.18.1873. Google Scholar [12] X. Mao, Stochastic Differential Equations and Their Applications,, Horwood Publishing Chichester, (1997). Google Scholar [13] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes,, Adv. Appl. Prob., 25 (1993), 518. doi: 10.2307/1427522. Google Scholar [14] J. Norris, Simplified Malliavin calculus,, In: Séminaire de probabilitiés XX, 1204 (1986), 101. doi: 10.1007/BFb0075716. Google Scholar [15] D. Nualart, The Malliavin Calculus and Related Topics,, Springer-Verlag, (2006). Google Scholar [16] H. Schurz and K. Tosun, Stochastic asymptotic stability of SIR model with variable diffusion rates,, J. Dynam. Differential Equations, 27 (2015), 69. doi: 10.1007/s10884-014-9415-9. Google Scholar [17] L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes,, LCDS Report, (1986), 86. Google Scholar [18] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system,, Phys. A, 354 (2005), 111. doi: 10.1016/j.physa.2005.02.057. Google Scholar [19] Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation,, Math. Biosci. Eng., 11 (2014), 1003. doi: 10.3934/mbe.2014.11.1003. Google Scholar [20] X. Zhong and F. Deng, Extinction and persistent of a stochastic multi-group SIR epidemic model,, Journal of Control Science and Engineering, 1 (2013), 13. Google Scholar [21] Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations,, Appl. Math. Comput., 244 (2014), 118. doi: 10.1016/j.amc.2014.06.100. Google Scholar

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##### References:
 [1] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of Asymptotic Behavior in a Stochastic SIR Model,, SIAM J. Appl. Dyn. Syst., 15 (2016), 1062. doi: 10.1137/15M1043315. Google Scholar [2] N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, J. Appl. Probab., 53 (2016), 187. doi: 10.1017/jpr.2015.18. Google Scholar [3] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model,, SIAM J. Appl. Math., 71 (2011), 876. doi: 10.1137/10081856X. Google Scholar [4] K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization,, Z. Wahrsch. Verw. Gebiete, 30 (1974), 235. doi: 10.1007/BF00533476. Google Scholar [5] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, second edition, (1989). Google Scholar [6] C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model,, Appl. Math. Model., 38 (2014), 5067. doi: 10.1016/j.apm.2014.03.037. Google Scholar [7] C. Y. Ji, D. Q. Jiang and N. Z. Shi, The behavior of an SIR epidemic model with stochastic perturbation,, Stochastic Anal. Appl., 30 (2012), 755. doi: 10.1080/07362994.2012.684319. Google Scholar [8] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation,, J. IFAC, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044. Google Scholar [9] R. Z. Khas'minskii, Stochastic Stability of Differential Equations,, Springer-Verlag Berlin Heidelberg, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar [10] W. Kliemann, Recurrence and invariant measures for degenerate diffusions,, Ann. Probab., 15 (1987), 690. doi: 10.1214/aop/1176992166. Google Scholar [11] Y. G. Lin and D. Q. Jiang, Long-time behaviour of a perturbed SIR model by white noise,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1873. doi: 10.3934/dcdsb.2013.18.1873. Google Scholar [12] X. Mao, Stochastic Differential Equations and Their Applications,, Horwood Publishing Chichester, (1997). Google Scholar [13] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes,, Adv. Appl. Prob., 25 (1993), 518. doi: 10.2307/1427522. Google Scholar [14] J. Norris, Simplified Malliavin calculus,, In: Séminaire de probabilitiés XX, 1204 (1986), 101. doi: 10.1007/BFb0075716. Google Scholar [15] D. Nualart, The Malliavin Calculus and Related Topics,, Springer-Verlag, (2006). Google Scholar [16] H. Schurz and K. Tosun, Stochastic asymptotic stability of SIR model with variable diffusion rates,, J. Dynam. Differential Equations, 27 (2015), 69. doi: 10.1007/s10884-014-9415-9. Google Scholar [17] L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes,, LCDS Report, (1986), 86. Google Scholar [18] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system,, Phys. A, 354 (2005), 111. doi: 10.1016/j.physa.2005.02.057. Google Scholar [19] Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation,, Math. Biosci. Eng., 11 (2014), 1003. doi: 10.3934/mbe.2014.11.1003. Google Scholar [20] X. Zhong and F. Deng, Extinction and persistent of a stochastic multi-group SIR epidemic model,, Journal of Control Science and Engineering, 1 (2013), 13. Google Scholar [21] Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations,, Appl. Math. Comput., 244 (2014), 118. doi: 10.1016/j.amc.2014.06.100. Google Scholar
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