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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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
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China |
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.
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 and Continuous Dynamical Systems - B,
2021, 26
(5)
: 2693-2719.
doi: 10.3934/dcdsb.2020201
![]()
References:
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doi: 10.1137/15M1024512. |
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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. |
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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.
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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.
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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.
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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
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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. |
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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. |
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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. |
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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. |
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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. |
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R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
| [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. |
| [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. |
| [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. |
| [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. |
| [25] |
A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
D. J. Higham,
An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
X. R. Mao, Stochastic Differential Equations and Applications, Second edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
| [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. |
| [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. |
| [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. |
| [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. |
| [25] |
A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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) $
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 $
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 $
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 $
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 $
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
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 $
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 $
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 $
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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