[1]
|
Pruss-Ustun, A., Wolf, J., Corvalan, C., Bos, R. and Neira, M. (2016) Preventing Disease through Healthy Environments: A Global Assessment of the Burden of Disease from Environ-mental Risks. World Health Organization.
|
[2]
|
May, R.M. (1973) Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton.
|
[3]
|
Du, N.H. and Nhu, N.N. (2020) Permanence and Extinction for the Stochastic SIR Epidemic Model. Journal of Di.erential Equations, 269, 9619-9652.
https://doi.org/10.1016/j.jde.2020.06.049
|
[4]
|
Gray, A., Greenhalgh, D., Hu, L., Mao, X. and Pan, J. (2011) A Stochastic Di.erential Equations SIS Epidemic Model. SIAM Journal on Applied Mathematics, 71, 876-902.
https://doi.org/10.1137/10081856X
|
[5]
|
Du, N.H. and Nhu, N.N. (2017) Permanence and Extinction of Certain Stochastic SIR Models Perturbed by a Complex Type of Noises. Applied Mathematics Letters, 64, 223-230.
https://doi.org/10.1016/j.aml.2016.09.012
|
[6]
|
Chen, Q. and Liu, Q. (2015) Analysis of the Deterministic and Stochastic SIRS Epidemic Models with Nonlinear Incidence. Physica A: Statistical Mechanics and Its Applications, 428, 140-153. https://doi.org/10.1016/j.physa.2015.01.075
|
[7]
|
Han, Z. and Zhao, J. (2013) Stochastic SIRS Model under Regime Switching. Nonlinear Analysis: Real World Applications, 14, 352-364. https://doi.org/10.1016/j.nonrwa.2012.06.008
|
[8]
|
Tuerxun, N., Wen, B. and Teng, Z. (2021) The Stationary Distribution in a Class of Stochastic SIRS Epidemic Models with Non-Monotonic Incidence and Degenerate Di.usion. Mathematics and Computers in Simulation, 182, 888-912. https://doi.org/10.1016/j.matcom.2020.03.008
|
[9]
|
Cai, S., Cai, Y. and Mao, X. (2019) A Stochastic Di.erential Equation SIS Epidemic Model with Two Independent Brownian Motions. Journal of Mathematical Analysis and Applications, 474, 1536-1550. https://doi.org/10.1016/j.jmaa.2019.02.039
|
[10]
|
Cai, S., Cai, Y. and Mao, X. (2019) A Stochastic Di.erential Equation SIS Epidemic Model with Two Correlated Brownian Motions. Nonlinear Dynamics, 9, 2175-2187.
https://doi.org/10.1007/s11071-019-05114-2
|
[11]
|
Greenhalgh, D., Liand, Y. and Mao, X. (2015) Demographic Stochasticity in the SDE SIS Epidemic Model. Discrete and Continuous Dynamical Systems: Series B, 20, 2859-2884.
https://doi.org/10.3934/dcdsb.2015.20.2859
|
[12]
|
Greenhalgh, D., Liang, Y. and Mao, X. (2016) SDE SIS Epidemic Model with Demographic Stochasticity and Varying Population Size. Applied Mathematics and Computation, 276, 218-238. https://doi.org/10.1016/j.amc.2015.11.094
|
[13]
|
Fan, D., Wang, K. and Hong, L. (2009) The Complete Parameters Analysis of the Asymptotic Behaviour of a Logistic Epidemic Model with Two Stochastic Perturbations. Mathematical Problems in Engineering, 2009, Article ID: 904383. https://doi.org/10.1155/2009/904383
|
[14]
|
Cai, Y., Kang, Y., Banerjee, M. and Wang, W. (2015) A Stochastic SIRS Epidemic Model with Infectious Force under Intervention Strategies. Journal of Di.erential Equations, 259, 7463-7502. https://doi.org/10.1016/j.jde.2015.08.024
|
[15]
|
Liu, Q. and Jiang, D. (2020) Threshold Behavior in a Stochastic SIR Epidemic Model with Logistic Birth. Physica A: Statistical Mechanics and Its Applications, 540, Article ID: 123488.
https://doi.org/10.1016/j.physa.2019.123488
|
[16]
|
Lahrouz, A., Kiouach, D. and Omari, L. (2011) Global Analysis of a Deterministic and Stochastic Nonlinear SIRS Epidemic Model. Nonlinear Analysis Modelling and Control, 16, 59-76. https://doi.org/10.15388/NA.16.1.14115
|
[17]
|
Chang, Z., Meng, X. and Zhang, T. (2018) A New Way of Investigating the Asymptotic Behaviour of a Stochastic SIS System with Multiplicative Noise. Applied Mathematics Letters, 87, 80-86. https://doi.org/10.1016/j.aml.2018.07.014
|
[18]
|
Liu, Q. and Chen, Q. (2016) Dynamics of a Stochastic SIR Epidemic Model with Saturated Incidence. Applied Mathematics and Computation, 282, 155-166.
https://doi.org/10.1016/j.amc.2016.02.022
|
[19]
|
Li, D., Liu, S. and Cui, J.A. (2017) Threshold Dynamics and Ergodicity of an SIRS Epidemic model with Markovian Switching. Journal of Di.erential Equations, 263, 8873-8915.
https://doi.org/10.1016/j.jde.2017.08.066
|
[20]
|
Liu, J., Chen, L. and Wei, F. (2018) The Persistence and Extinction of a Stochastic SIS Epidemic Model with Logistic Growth. Advances in Di.erence Equations, 2018, Article No.
68. https://doi.org/10.1186/s13662-018-1528-8
|
[21]
|
Zhao, S., Yuan, S. and Wang, H. (2020) Threshold Behavior in a Stochastic Algal Growth Model with Stoichiometric Constraints and Seasonal Variation. Journal of Di.erential Equations, 268, 5113-5139. https://doi.org/10.1016/j.jde.2019.11.004
|
[22]
|
Imhof, L. and Walcher, S. (2005) Exclusion and Persistence in Deterministic and Stochastic Chemostat Models. Journal of Di.erential Equations, 217, 26-53.
https://doi.org/10.1016/j.jde.2005.06.017
|
[23]
|
Zhao, Y., Jiang, D., Mao, X. and Gray, A. (2015) The Threshold of a Stochastic SIRS Epidemic Model in a Population with Varying Size. Discrete and Continuous Dynamical Systems: Series B, 20, 31-35.
|
[24]
|
Dieu, N.T., Nguyen, D.H., Du, N.H. and Yin, G. (2016) Classi.cation of Asymptotic Behavior in a Tochastic SIR Model. SIAM Journal on Applied Dynamical Systems, 15, 1062-1084.
https://doi.org/10.1137/15M1043315
|
[25]
|
Du, N.H., Dieu, N.T. and Nhu, N.N. (2019) Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models. Acta Applicandae Mathematicae, 160, 81-99.
https://doi.org/10.1007/s10440-018-0196-8
|
[26]
|
肖燕妮,周义仓,唐三一.生物数学原理[M].西安:西安交通大学出版社,2012.
|
[27]
|
王克.随机生物数学模型[M].北京:科技出版社,2010.
|
[28]
|
胡适耕,黄乘明,吴付科.随机微分方程[M].北京:科学出版社,2008.
|
[29]
|
马知恩,周义仓,李承志.常微分方程定性与稳定性方法[M].北京:科学出版社,2001.
|
[30]
|
Mao, X. (2006) Stochastic Di.erential Equations and Applications. 2nd Edition, Academic Press, Cambridge, MA.
|
[31]
|
Tuong, T.D., Nguyen, D.H., Dieu, N.T. and Tran, K. (2020) Extinction and Permanence in a Stochastic SIRS Model in Regime-Switching with General Incidence Rate. Nonlinear Analysis: Hybrid Systems, 34, 121-130. https://doi.org/10.1016/j.nahs.2019.05.008
|
[32]
|
Alexandru, H. and Nguyen, D.H. (2018) Coexistence and Extinction for Stochastic Kolmogorov Systems. The Annals of Applied Probability, 28, 1893-1942.
https://doi.org/10.1214/17-AAP1347
|
[33]
|
Meyn, S.P. and Tweedie, R.L. (1993) Stability of Markovian Processes II: Continuous-Time Processes and Sampled Chains. Advances in Applied Probability, 25, 487-517.
https://doi.org/10.2307/1427521
|
[34]
|
Mao, X.R. (2007) Stochastic Di.erential Equations and Applications. Elsevier, Amsterdam.
|