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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 and 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), 1062-1084. doi: 10.1137/15M1043315.

[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), 187-202, Available from: http://projecteuclid.org/euclid.jap/1457470568. doi: 10.1017/jpr.2015.18.

[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), 876-902. doi: 10.1137/10081856X.

[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), 235-254; Corrections in 39 (1977), 81-84. doi: 10.1007/BF00533476.

[5]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland Publishing Co., Amsterdam, 1989.

[6]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067-5079. doi: 10.1016/j.apm.2014.03.037.

[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), 755-773. doi: 10.1080/07362994.2012.684319.

[8]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, J. IFAC, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.

[9]

R. Z. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[10]

W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15 (1987), 690-707. doi: 10.1214/aop/1176992166.

[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), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873.

[12]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Chichester, 1997.

[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), 518-548. doi: 10.2307/1427522.

[14]

J. Norris, Simplified Malliavin calculus, In: Séminaire de probabilitiés XX, Lecture Notes in Mathematics, Springer, New York, 1204 (1986), 101-130. doi: 10.1007/BFb0075716.

[15]

D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin Heidelberg, 2006.

[16]

H. Schurz and K. Tosun, Stochastic asymptotic stability of SIR model with variable diffusion rates, J. Dynam. Differential Equations, 27 (2015), 69-82. doi: 10.1007/s10884-014-9415-9.

[17]

L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report, No. 86-16, April 1986, Brown University, Providence. Available from: https://www.amazon.co.uk/existence-uniqueness-invariant-continuous-processes/dp/B000722C66

[18]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A, 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[19]

Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. Biosci. Eng., 11 (2014), 1003-1025. doi: 10.3934/mbe.2014.11.1003.

[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), 13-22.

[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), 118-131. doi: 10.1016/j.amc.2014.06.100.

show all references

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), 1062-1084. doi: 10.1137/15M1043315.

[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), 187-202, Available from: http://projecteuclid.org/euclid.jap/1457470568. doi: 10.1017/jpr.2015.18.

[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), 876-902. doi: 10.1137/10081856X.

[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), 235-254; Corrections in 39 (1977), 81-84. doi: 10.1007/BF00533476.

[5]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland Publishing Co., Amsterdam, 1989.

[6]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067-5079. doi: 10.1016/j.apm.2014.03.037.

[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), 755-773. doi: 10.1080/07362994.2012.684319.

[8]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, J. IFAC, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.

[9]

R. Z. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[10]

W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15 (1987), 690-707. doi: 10.1214/aop/1176992166.

[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), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873.

[12]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Chichester, 1997.

[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), 518-548. doi: 10.2307/1427522.

[14]

J. Norris, Simplified Malliavin calculus, In: Séminaire de probabilitiés XX, Lecture Notes in Mathematics, Springer, New York, 1204 (1986), 101-130. doi: 10.1007/BFb0075716.

[15]

D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin Heidelberg, 2006.

[16]

H. Schurz and K. Tosun, Stochastic asymptotic stability of SIR model with variable diffusion rates, J. Dynam. Differential Equations, 27 (2015), 69-82. doi: 10.1007/s10884-014-9415-9.

[17]

L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report, No. 86-16, April 1986, Brown University, Providence. Available from: https://www.amazon.co.uk/existence-uniqueness-invariant-continuous-processes/dp/B000722C66

[18]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A, 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[19]

Q. Yang and X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. Biosci. Eng., 11 (2014), 1003-1025. doi: 10.3934/mbe.2014.11.1003.

[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), 13-22.

[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), 118-131. doi: 10.1016/j.amc.2014.06.100.

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