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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 |
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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