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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.
92D30 Epidemiology
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J28 Applications of continuous-time Markov processes on discrete state spaces
Full Text: DOI
[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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