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A stochastic differential equation SIS epidemic model with regime switching. (English) Zbl 1464.92241
Summary: In this paper, we combined the previous model in [S. Cai et al., J. Math. Anal. Appl. 474, No. 2, 1536–1550 (2019; Zbl 1415.92173)] with A. Gray et al.’s work [J. Math. Anal. Appl. 394, No. 2, 496–516 (2012; Zbl 1271.92030)] 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 [X. Mao and C. Yuan, Stochastic differential equations with Markovian switching. Hackensack, NJ: World Scientific (2006; Zbl 1126.60002)] is fully applied. Computer simulations are clearly illustrated with different sets of parameters, which support our theoretical results. Compared to our previous work in [J. Math. Anal. Appl. 474, No. 2, 1536–1550 (2019; Zbl 1415.92173)], our threshold value are given based on the overall behaviour of the solution but not separately specified in every state of the Markov chain.
##### MSC:
 92D30 Epidemiology 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J28 Applications of continuous-time Markov processes on discrete state spaces
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##### References:
 [1] W. J. Anderson, Continuous-time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, New York, 1991. [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. · Zbl 1415.92173 [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. · Zbl 1415.92173 [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. · Zbl 07183295 [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. · Zbl 1437.37115 [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. [7] T. H. Fleming; J. N. Holland, The evolution of obligate pollination mutualisms: Senita cactus and senita moth, Oecologia, 114, 368-375 (1998) [8] A. Gray; D. Greenhalgh; X. Mao; J. Pan, The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394, 496-516 (2012) · Zbl 1271.92030 [9] A. Gray; D. Greenhalgh; L. Hu; X. Mao; J. Pan, A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71, 876-902 (2011) · Zbl 1263.34068 [10] D. Greenhalgh; Y. Liang, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A: Statistical Mechanics and its Applications, 462, 684-704 (2016) · Zbl 1400.92484 [11] J. D. Hamilton, Regime switching models, The New Palgrave Dictionary of Economics, 2016, 1-7. [12] A. Hening; D. H. Nguyen, Stochastic Lotka-Volterra food chains, Journal of Mathematical Biology, 77, 135-163 (2018) · Zbl 1392.92075 [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43, 525-546 (2001) · Zbl 0979.65007 [14] J. N. Holland; T. H. Fleming, Geographic and population variation in pollinating seed-consuming interactions between senita cacti (Lophocereus schottii) and senita moths (Upiga virescens), Oecologia, 121, 405-410 (1999) [15] J. N. Holland; D. L. DeAngelis; J. L. Bronstein, Population dynamics and mutualism: Functional responses of benefits and costs, The University of Chicago Press, 159, 231-244 (2002) [16] R. Khasminskii, Stochastic Stability of Differential Equations, 66. Springer, Heidelberg, 2012. [17] X. Li; D. Jiang; X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, Journal of Computational and Applied Mathematics, 232, 427-448 (2009) · Zbl 1173.60020 [18] H. Liu; X. Li; Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Systems & Control Letters, 62, 805-810 (2013) · Zbl 1281.93094 [19] Q. Luo; X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and applications, 334, 69-84 (2007) · Zbl 1113.92052 [20] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. [21] J. R. Norris, Markov Chains, Cambridge University Press, 1998. [22] S. Pang; F. Deng; X. Mao, Asymptotic properties of stochastic population dynamics, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis, 15, 603-620 (2008) · Zbl 1171.34038 [23] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59, 249-256 (1978) [24] L. S. Tsimring, Noise in biology, IOP Publishing, 77 (2014), 026601. [25] Y. Takeuchi; N. H. Du; N. T. Hieu; K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, Journal of Mathematical Analysis and Applications, 323, 938-957 (2006) · Zbl 1113.34042 [26] D. A. Vasseur; P. Yodzis, The color of environmental noise, Wiley Online Library, 85, 1146-1152 (2004) [27] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer, 37, 2012. [28] C. Zhu; G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46, 1155-1179 (2007) · Zbl 1140.93045
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