American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions
  • DCDS-B Home
  • This Issue
  • Next Article
    Infinitely many solutions of the nonlinear fractional Schrödinger equations
December  2016, 21(10): 3429-3440. doi: 10.3934/dcdsb.2016105

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

Nguyen Huu Du 1, and  Nguyen Thanh Dieu 2,

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].
Keywords: SIR model, stationary distribution, ergodicity., permanence, extinction.
Mathematics Subject Classification: 34C12, 60H10, 92D2.
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

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. 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

[1]

Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119
[2]

Bum Il Hong, Nahmwoo Hahm, Sun-Ho Choi. SIR Rumor spreading model with trust rate distribution. Networks & Heterogeneous Media, 2018, 13 (3) : 515-530. doi: 10.3934/nhm.2018023
[3]

Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783
[4]

Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051
[5]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
[6]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069
[7]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121
[8]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[9]

Alan J. Terry. Pulse vaccination strategies in a metapopulation SIR model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 455-477. doi: 10.3934/mbe.2010.7.455
[10]

Tomás Caraballo, Renato Colucci. A comparison between random and stochastic modeling for a SIR model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 151-162. doi: 10.3934/cpaa.2017007
[11]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059
[12]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101
[13]

Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001
[14]

Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
[15]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[16]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699
[17]

Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743
[18]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[19]

Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219
[20]

Arturo Hidalgo, Lourdes Tello. On a climatological energy balance model with continents distribution. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1503-1519. doi: 10.3934/dcds.2015.35.1503

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences