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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

Siyang Cai 1, , Yongmei Cai 2,, and  Xuerong Mao 1,

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.

Keywords: SIS model, Markovian switching, $ M $-matrix, extinction, stationary distribution.
Mathematics Subject Classification: Primary: 60H30, 92D25; Secondary: 37H05.
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.

Figure 1.  Extinction with $ I(0) = 90 $
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Figure 2.  Extinction with $ I(0) = 10 $
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Figure 3.  Persistence Case 1 with $ I(0) = 90 $
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Figure 4.  Persistence Case 1 with $ I(0) = 10 $
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Figure 5.  Persistence Case 2 with $ I(0) = 90 $
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Figure 6.  Persistence Case 2 with $ I(0) = 10 $
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Figure 7.  Stationary Distribution Case 1 with $ I(0) = 90 $
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Figure 8.  Stationary Distribution Case 1 with $ I(0) = 10 $
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Figure 9.  Stationary Distribution Case 2 with $ I(0) = 90 $
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Figure 10.  Stationary Distribution Case 2 with $ I(0) = 10 $
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