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May  2021, 26(5): 2693-2719. doi: 10.3934/dcdsb.2020201

Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

Shangzhi Li and  Shangjiang Guo ,

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  January 2020 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The second author is supported by NSFC (Grant No. 11671123)

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold $ \lambda $, it is proved that if $ \lambda>0 $, the disease is permanent and there is a stationary distribution. And when $ \lambda<0 $, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

Keywords: Stochastic SIS model, threshold, permanence, extinction.
Mathematics Subject Classification: Primary: 34K20; Secondary: 92C60.
Citation: Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
References:
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[2]

S. Y. Cai, Y. M. Cai and X. R. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039.  Google Scholar

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Y. L. Chen, B. Y. Wen and Z. D. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.  Google Scholar

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N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Appl. Math., 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8.  Google Scholar

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N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 64 (2017), 223-230.  doi: 10.1016/j.aml.2016.09.012.  Google Scholar

[6]

J. P. Gao and S. J. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28 pp doi: 10.1142/S0218127420500741.  Google Scholar

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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.  Google Scholar

[8]

A. Gray, D. Greenhalgh, X. R. Mao and J. F. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029.  Google Scholar

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D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

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N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[11]

C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.  Google Scholar

[12]

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A. Lahrouz, L. Omari and D. Kioach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115.  Google Scholar

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A. Lahrouz, L. Omari, D. Kioach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Applied Mathematics and Computation, 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024.  Google Scholar

[15]

G. J. Lan, Y. L. Huang, C. J. Wei and S. W. Zhang, A stochastic SIS epidemic model with saturating contact rate, Physica A, 529 (2019), 121504, 14 pp. doi: 10.1016/j.physa.2019.121504.  Google Scholar

[16]

S. Z. Li and S. J. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164. doi: 10.1016/j.physa.2020.124164.  Google Scholar

[17]

X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376 (1) (2011), 11-28.  doi: 10.1016/j.lmaa.2010.10.053.  Google Scholar

[18]

Q. Liu and Q. M. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Applied Mathematics and Computation, 282 (2016), 155-166.  doi: 10.1016/j.amc.2016.02.022.  Google Scholar

[19]

Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate, Physica A, 526 (2019), 120946. doi: 10.1016/j.physa.2019.04.182.  Google Scholar

[20]

X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[21]

D. H. Nguyen, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4.  Google Scholar

[22]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.  Google Scholar

[23]

H. H. Qiu, S. J. Guo and S. Z. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224.  Google Scholar

[24]

J. H. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electronic Journal of Probability, 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018.  Google Scholar

[25]

A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989.  Google Scholar

[26]

B. Sounvoravong, S. J. Guo and Y. Z. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), 16 pp.  Google Scholar

[27]

Z. D. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084.  Google Scholar

[28]

B. Y. Wen, R. Rifhat and Z. D. Teng, The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence, Physica A, 524 (2019), 258-271.  doi: 10.1016/j.physa.2019.04.049.  Google Scholar

[29]

C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, J. Math. Anal. Appl., 447 (2017), 736-757.  doi: 10.1016/j.jmaa.2016.10.041.  Google Scholar

[30]

Y. Zhang, Y. Li, Q. L. Zhang and A. H. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187.  doi: 10.1016/j.physa.2018.02.191.  Google Scholar

[31]

X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar

[32]

Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution in a stochastic SIS model, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 592821, 12 pp. doi: 10.1155/2013/592821.  Google Scholar

[33]

Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100.  Google Scholar

[34]

C. J. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012.  Google Scholar

[35]

R. Zou and S. J. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, (2020). doi: 10.3934/dcdsb.2020093.  Google Scholar

show all references

References:
[1]

J. H. Bao and J. H. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM Journal on Mathematical Analysis, 48 (2016), 725-739.  doi: 10.1137/15M1024512.  Google Scholar

[2]

S. Y. Cai, Y. M. Cai and X. R. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039.  Google Scholar

[3]

Y. L. Chen, B. Y. Wen and Z. D. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.  Google Scholar

[4]

N. H. Du, N. T. Dieu and N. N. Nhu, Conditions for permanence and ergodicity of certain SIR epidemic models, Acta Appl. Math., 160 (2019), 81-99.  doi: 10.1007/s10440-018-0196-8.  Google Scholar

[5]

N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 64 (2017), 223-230.  doi: 10.1016/j.aml.2016.09.012.  Google Scholar

[6]

J. P. Gao and S. J. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28 pp doi: 10.1142/S0218127420500741.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Gray, D. Greenhalgh, X. R. Mao and J. F. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029.  Google Scholar

[9]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[10]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981.  Google Scholar

[11]

C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.  Google Scholar

[12]

R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[13]

A. Lahrouz, L. Omari and D. Kioach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115.  Google Scholar

[14]

A. Lahrouz, L. Omari, D. Kioach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Applied Mathematics and Computation, 218 (2012), 6519-6525.  doi: 10.1016/j.amc.2011.12.024.  Google Scholar

[15]

G. J. Lan, Y. L. Huang, C. J. Wei and S. W. Zhang, A stochastic SIS epidemic model with saturating contact rate, Physica A, 529 (2019), 121504, 14 pp. doi: 10.1016/j.physa.2019.121504.  Google Scholar

[16]

S. Z. Li and S. J. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164. doi: 10.1016/j.physa.2020.124164.  Google Scholar

[17]

X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376 (1) (2011), 11-28.  doi: 10.1016/j.lmaa.2010.10.053.  Google Scholar

[18]

Q. Liu and Q. M. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Applied Mathematics and Computation, 282 (2016), 155-166.  doi: 10.1016/j.amc.2016.02.022.  Google Scholar

[19]

Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate, Physica A, 526 (2019), 120946. doi: 10.1016/j.physa.2019.04.182.  Google Scholar

[20]

X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[21]

D. H. Nguyen, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4.  Google Scholar

[22]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.  Google Scholar

[23]

H. H. Qiu, S. J. Guo and S. Z. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25 pp. doi: 10.1142/S0218127420500224.  Google Scholar

[24]

J. H. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electronic Journal of Probability, 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018.  Google Scholar

[25]

A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989.  Google Scholar

[26]

B. Sounvoravong, S. J. Guo and Y. Z. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), 16 pp.  Google Scholar

[27]

Z. D. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084.  Google Scholar

[28]

B. Y. Wen, R. Rifhat and Z. D. Teng, The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence, Physica A, 524 (2019), 258-271.  doi: 10.1016/j.physa.2019.04.049.  Google Scholar

[29]

C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, J. Math. Anal. Appl., 447 (2017), 736-757.  doi: 10.1016/j.jmaa.2016.10.041.  Google Scholar

[30]

Y. Zhang, Y. Li, Q. L. Zhang and A. H. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187.  doi: 10.1016/j.physa.2018.02.191.  Google Scholar

[31]

X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar

[32]

Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution in a stochastic SIS model, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 592821, 12 pp. doi: 10.1155/2013/592821.  Google Scholar

[33]

Y. L. Zhou, W. G. Zhang and S. L. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100.  Google Scholar

[34]

C. J. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012.  Google Scholar

[35]

R. Zou and S. J. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, (2020). doi: 10.3934/dcdsb.2020093.  Google Scholar

Figure 1.  Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = b = c = 0.1 $
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Figure 2.  Stationary distribution of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = b = c = 0.1 $: (a) the graph of the relative frequency densities of $ S $ and $ I $; (b) the joint density distribution of solution $ (S,I) $
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Figure 3.  Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 1 $, $ b = 0.1 $, $ c = 0.1 $
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Figure 4.  Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 2 $, $ b = 0.1 $, $ c = 0.1 $
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Figure 5.  Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 1 $, $ c = 0.1 $
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Figure 6.  Trajectories of solutions of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.31 $
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Figure 7.  Numerical simulations of model (26) with initial value $ (S(0),I(0)) = (1.9,1.9) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.31 $: (a) Trajectories of the solution $ S(t) $ of (26) and the solution $ \varphi $ of (10) shows the convergence of $ S $ to a boundary distribution; (b) The dynamics of $ S(t) $ and $ I(t) $ in time average
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Figure 8.  Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 0.1 $
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Figure 9.  Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 1 $, $ b = 0.1 $, $ c = 0.1 $
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Figure 10.  Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 1 $, $ c = 0.1 $
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Figure 11.  Trajectories of solution $ (S(t),I(t)) $ of model (27) with initial value $ (S(0),I(0)) = (1,1) $ and parameters $ a = 0.1 $, $ b = 0.1 $, $ c = 1 $
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