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Stabilization by intermittent control for hybrid stochastic differential delay equations
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 |
In this paper, we combined the previous model in [
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |










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