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September  2021, 26(9): 4887-4905. doi: 10.3934/dcdsb.2020317

## A stochastic differential equation SIS epidemic model with regime switching

 1 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK 2 School of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, 315100, China

* Corresponding author: yongmei.cai@nottingham.edu.cn

Received  February 2020 Revised  September 2020 Published  September 2021 Early access  November 2020

In this paper, we combined the previous model in [2] with Gray et al.'s work in 2012 [8] to add telegraph noise by using Markovian switching to generate a stochastic SIS epidemic model with regime switching. Similarly, threshold value for extinction and persistence are then given and proved, followed by explanation on the stationary distribution, where the $M$-matrix theory elaborated in [20] is fully applied. Computer simulations are clearly illustrated with different sets of parameters, which support our theoretical results. Compared to our previous work in 2019 [2, 3], our threshold value are given based on the overall behaviour of the solution but not separately specified in every state of the Markov chain.

Citation: Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4887-4905. doi: 10.3934/dcdsb.2020317
##### References:
 [1] W. J. Anderson, Continuous-time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [2] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536–1550, http://www.sciencedirect.com/science/article/pii/S0022247X19301635. doi: 10.1016/j.jmaa.2019.02.039. [3] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two correlated Brownian motions, J. Math. Anal. Appl., 474 (2019), 1536–1550, https://doi.org/10.1007/s11071-019-05114-2. doi: 10.1016/j.jmaa.2019.02.039. [4] Y. Cai and X. Mao, Stochastic prey-predator system with foraging arena scheme, Applied Mathematical Modelling, 64 (2018), 357–371, http://www.sciencedirect.com/science/article/pii/S0307904X18303500. doi: 10.1016/j.apm.2018.07.034. [5] Y. Cai, S. Cai and X. Mao, Stochastic delay foraging arena predator–prey system with Markov switching, Stochastic Analysis and Applications, 38 (2020), 191–212, https://doi.org/10.1080/07362994.2019.1679645. doi: 10.1080/07362994.2019.1679645. [6] Y. Cai, S. Cai and X. Mao, Analysis of a stochastic predator-prey system with foraging arena scheme, Stochastics, 92 (2020), 193–222. https://doi.org/10.1080/17442508.2019.1612897. doi: 10.1080/17442508.2019.1612897. [7] T. H. Fleming and J. N. Holland, The evolution of obligate pollination mutualisms: Senita cactus and senita moth, Oecologia, 114 (1998), 368-375.  doi: 10.1007/s004420050459. [8] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [9] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71 (2011), 876-902.  doi: 10.1137/10081856X. [10] D. Greenhalgh and Y. Liang, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A: Statistical Mechanics and its Applications, 462 (2016), 684-704.  doi: 10.1016/j.physa.2016.06.125. [11] J. D. Hamilton, Regime switching models, The New Palgrave Dictionary of Economics, 2016, 1–7. [12] A. Hening and D. H. Nguyen, Stochastic Lotka–Volterra food chains, Journal of Mathematical Biology, 77 (2018), 135-163.  doi: 10.1007/s00285-017-1192-8. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [14] J. N. Holland and T. H. Fleming, Geographic and population variation in pollinating seed-consuming interactions between senita cacti (Lophocereus schottii) and senita moths (Upiga virescens), Oecologia, 121 (1999), 405-410.  doi: 10.1007/s004420050945. [15] J. N. Holland, D. L. DeAngelis and J. L. Bronstein, Population dynamics and mutualism: Functional responses of benefits and costs, The University of Chicago Press, 159 (2002), 231-244. [16] R. Khasminskii, Stochastic Stability of Differential Equations, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [17] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka–Volterra system under regime switching, Journal of Computational and Applied Mathematics, 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021. [18] H. Liu, X. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka–Volterra mutualistic system with regime switching, Systems & Control Letters, 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002. [19] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and applications, 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [20] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [21] J. R. Norris, Markov Chains, Cambridge University Press, 1998. doi: 10.1017/CBO9780511810633. [22] S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis, 15 (2008), 603-620. [23] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370. [24] L. S. Tsimring, Noise in biology, IOP Publishing, 77 (2014), 026601. doi: 10.1088/0034-4885/77/2/026601. [25] Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment, Journal of Mathematical Analysis and Applications, 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009. [26] D. A. Vasseur and P. Yodzis, The color of environmental noise, Wiley Online Library, 85 (2004), 1146-1152. [27] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer, 37, 2012. doi: 10.1007/978-1-4614-4346-9. [28] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179.  doi: 10.1137/060649343.

show all references

##### References:
 [1] W. J. Anderson, Continuous-time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [2] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536–1550, http://www.sciencedirect.com/science/article/pii/S0022247X19301635. doi: 10.1016/j.jmaa.2019.02.039. [3] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two correlated Brownian motions, J. Math. Anal. Appl., 474 (2019), 1536–1550, https://doi.org/10.1007/s11071-019-05114-2. doi: 10.1016/j.jmaa.2019.02.039. [4] Y. Cai and X. Mao, Stochastic prey-predator system with foraging arena scheme, Applied Mathematical Modelling, 64 (2018), 357–371, http://www.sciencedirect.com/science/article/pii/S0307904X18303500. doi: 10.1016/j.apm.2018.07.034. [5] Y. Cai, S. Cai and X. Mao, Stochastic delay foraging arena predator–prey system with Markov switching, Stochastic Analysis and Applications, 38 (2020), 191–212, https://doi.org/10.1080/07362994.2019.1679645. doi: 10.1080/07362994.2019.1679645. [6] Y. Cai, S. Cai and X. Mao, Analysis of a stochastic predator-prey system with foraging arena scheme, Stochastics, 92 (2020), 193–222. https://doi.org/10.1080/17442508.2019.1612897. doi: 10.1080/17442508.2019.1612897. [7] T. H. Fleming and J. N. Holland, The evolution of obligate pollination mutualisms: Senita cactus and senita moth, Oecologia, 114 (1998), 368-375.  doi: 10.1007/s004420050459. [8] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [9] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71 (2011), 876-902.  doi: 10.1137/10081856X. [10] D. Greenhalgh and Y. Liang, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A: Statistical Mechanics and its Applications, 462 (2016), 684-704.  doi: 10.1016/j.physa.2016.06.125. [11] J. D. Hamilton, Regime switching models, The New Palgrave Dictionary of Economics, 2016, 1–7. [12] A. Hening and D. H. Nguyen, Stochastic Lotka–Volterra food chains, Journal of Mathematical Biology, 77 (2018), 135-163.  doi: 10.1007/s00285-017-1192-8. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [14] J. N. Holland and T. H. Fleming, Geographic and population variation in pollinating seed-consuming interactions between senita cacti (Lophocereus schottii) and senita moths (Upiga virescens), Oecologia, 121 (1999), 405-410.  doi: 10.1007/s004420050945. [15] J. N. Holland, D. L. DeAngelis and J. L. Bronstein, Population dynamics and mutualism: Functional responses of benefits and costs, The University of Chicago Press, 159 (2002), 231-244. [16] R. Khasminskii, Stochastic Stability of Differential Equations, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [17] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka–Volterra system under regime switching, Journal of Computational and Applied Mathematics, 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021. [18] H. Liu, X. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka–Volterra mutualistic system with regime switching, Systems & Control Letters, 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002. [19] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and applications, 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [20] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [21] J. R. Norris, Markov Chains, Cambridge University Press, 1998. doi: 10.1017/CBO9780511810633. [22] S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis, 15 (2008), 603-620. [23] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370. [24] L. S. Tsimring, Noise in biology, IOP Publishing, 77 (2014), 026601. doi: 10.1088/0034-4885/77/2/026601. [25] Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment, Journal of Mathematical Analysis and Applications, 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009. [26] D. A. Vasseur and P. Yodzis, The color of environmental noise, Wiley Online Library, 85 (2004), 1146-1152. [27] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer, 37, 2012. doi: 10.1007/978-1-4614-4346-9. [28] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179.  doi: 10.1137/060649343.
Extinction with $I(0) = 90$
Extinction with $I(0) = 10$
Persistence Case 1 with $I(0) = 90$
Persistence Case 1 with $I(0) = 10$
Persistence Case 2 with $I(0) = 90$
Persistence Case 2 with $I(0) = 10$
Stationary Distribution Case 1 with $I(0) = 90$
Stationary Distribution Case 1 with $I(0) = 10$
Stationary Distribution Case 2 with $I(0) = 90$
Stationary Distribution Case 2 with $I(0) = 10$
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