| Citation: |
Zhewen Chen, Ruimin Zhang, Jiang Li, Xiaohui Liu, Chunjin Wei. GLOBAL DYNAMICS OF DETERMINISTIC AND STOCHASTIC SIRS EPIDEMIC MODELS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2211-2229. doi: 10.11948/20190387
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GLOBAL DYNAMICS OF DETERMINISTIC AND STOCHASTIC SIRS EPIDEMIC MODELS
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Abstract
In this paper, we analyze the dynamic behavior of Heesterbeek et al. [
12 ] obtained saturating contact rate applied to SIRS epidemic model. We define two threshold values, the deterministic basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, by comparing the value with one to determine the persistence and extinction of the disease. For deterministic model, if $ R_0<1 $, we show that the disease-free equilibrium is globally asymptotically stable; while if $ R_0>1 $, the system admits a unique endemic equilibrium which is locally asymptotically stable. For stochastic model, we also establish the threshold value $ R_0^s $ for disease persistence and extinction. Finally, some numerical simulations are presented to illustrate our theoretical results. Our results prove that large stochastic perturbation will lead to the extinction of diseases with probability one, revealing the significant influence of stochastic perturbation on diseases and the importance of incorporating stochastic perturbation into deterministic model.-
Keywords:
- Epidemic /
- stochastic /
- stationary solution /
- extinction /
- noise
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References
[1] Y. Cai and W. Wang, Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Anal. RWA., 2016, 30, 99-125. doi: 10.1016/j.nonrwa.2015.12.002 [2] Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 2017, 305, 221-240. [3] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motion, J. Math. Anal. Appl., 2019, 472, 1536-1550. [4] B. Cao, M. Shan, Q. Zhang and W. Wang, A stochastic SIS epidemic model with vaccination, Physica A., 2017, 486, 127-143. doi: 10.1016/j.physa.2017.05.083 [5] Z. Chang, X. Meng and X. Lu, Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates, Physical A., 2017, 472, 103-116. doi: 10.1016/j.physa.2017.01.015 [6] Z. Chen, R. Zhang, J. Li and S. Zhang, C. Wei, A stochastic nutrient-phytoplankton model with viral infection and Markov switching, Chaos Soliton Fract., 2020, 140, 110109. doi: 10.1016/j.chaos.2020.110109 [7] Y. Deng and M. liu, Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations, Appl. Math. Model., 2020, 78, 482-504. doi: 10.1016/j.apm.2019.10.010 [8] K. Fan, Y. Zhang, S. Gao and X. Wei, A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A., 2017, 481, 198-208. doi: 10.1016/j.physa.2017.04.055 [9] T. Feng and Z. Qiu, Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate, Int. J. Biomath., DOI: 10.1016/j.physa.2019.01.014. [10] W. Guo, Y. Cai, Q. Zhang and W. Wang, Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage, Physical A., 2018, 492, 2220-2236. doi: 10.1016/j.physa.2017.11.137 [11] R. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [12] J. Heesterbeek and J. Metz, The saturating contact rate in marriage and epidemic models, J. Math. Biol., 1993, 31, 529-539. [13] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 2000, 42, 599-653. doi: 10.1137/S0036144500371907 [14] D. Higham, Analgorithmic introduction to numerical simulation of stochastic differential equations, SIAM review., 2001, 43, 525-546. doi: 10.1137/S0036144500378302 [15] Z. Hu, Z. Teng and H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear. Anal-Real., 2012, 13, 2017-2033. doi: 10.1016/j.nonrwa.2011.12.024 [16] G. Hu, M. Liu and K. Wang, The asymptotic behaviours of an epidemic model with two correlated stochastic perturbations, Appl. Math. Comput., 2012, 218, 10520-10532. [17] S. Jeffrey and K. Melvin, Absolute humidity modulates influenza survival, transmission, and seasonality, Proc. Natl. Acad. Sci., USA, 2009, 106, 3243-3248. doi: 10.1073/pnas.0806852106 [18] W. Ji, Z. Wang and G. Hu, Stationary distribution of a stochastic hybrid phytoplankton model with allelopathy, Adv. Differ. Equ-Ny., 2020, 2020(1), 632. doi: 10.1186/s13662-020-03088-9 [19] G. Lan, Z. Chen, C. Wei and S. Zhang, Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion, Physica A., 2018, 511, 61-77. doi: 10.1016/j.physa.2018.07.041 [20] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal-Model., 2011, 16, 59-76. doi: 10.15388/NA.16.1.14115 [21] F. Li and X. Meng, Analysis and numerical simulations of a stochastic SEIQR epidemic system with quarantine-adjusted incidence and imperfect vaccination, Comput. Math. Method. M., 2018, DOI: 10.1155/2018/7873902. [22] Z. Li, Y. Mu and H. Xiang, Mean persistence and extinction for a novel stochastic turbidostat model, Nonlinear Dynam., 2019, 97(1), 185-202. doi: 10.1007/s11071-019-04965-z [23] Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A., 2015, 428, 140-153. doi: 10.1016/j.physa.2015.01.075 [24] J. Liu and F. Wei, Dynamics of stochastic SEIS epidemic model with varying population size, Physica A., 2016, 464, 241-250. doi: 10.1016/j.physa.2016.06.120 [25] Q. Liu, D. Jiang, N. Shi and T. Hayat, Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence, Physica A., 2017, 476, 58-69. doi: 10.1016/j.physa.2017.02.028 [26] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic dengue epidemic model, J. Franklin. I., 2018, 355, 8891-8914. doi: 10.1016/j.jfranklin.2018.10.003 [27] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates, J. Franklin. I., 2019, 356, 2960-2993. doi: 10.1016/j.jfranklin.2019.01.038 [28] S. Liu, L. Zhang, X. Zhang and A. li, Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size, Int. J. Biomath., 2019, 12(1), 1950005. doi: 10.1142/S1793524519500050 [29] M. Lu, J. Huang, S. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differ. Equations., 2019, 267, 1859-1898. doi: 10.1016/j.jde.2019.03.005 [30] Z. Ma and J. Li, Dynamical modeling and analysis of epidemics, International Association of Geodesy Symposia, 2009. [31] X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997. [32] P. Mead, Epidemics of plague past, present, and future, Lancet Infect. Dis., 2019, 19, 459-460. doi: 10.1016/S1473-3099(18)30794-1 [33] X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 2016, 433, 227-242. doi: 10.1016/j.jmaa.2015.07.056 [34] E. Mohamed, P. Roger, S. Idriss, S. Idriss and T. Regragui, A stochastic analysis for a triple delayed SIQR epidemic model with vaccination and elimination strategies, J. Appl. Math. Comput., 2020, 64, 781-805. doi: 10.1007/s12190-020-01380-1 [35] H. Qi, L. Liu and X. Meng, Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis, Complexity, 2017, DOI: 10.1155/2017/4861391. [36] H. Qi, S. Zhang and X. Meng, Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems, Physica A., 2018, 508, 223-241. doi: 10.1016/j.physa.2018.05.075 [37] R. Shi, X. Jiang and L. Chen, The effect of impulsive vaccination on an SIR epidemic model, Appl. Math. Comput., 2009, 212, 305-311. [38] W. Wang, Y. Cai, Z. Ding and Z. Gui, A stochastic differential equation SIS epidemic model incorporating Ornstein-Uhlenbeck process, Physica A., 2018, 509, 921-936. doi: 10.1016/j.physa.2018.06.099 [39] C. Wei and L. Chen, A delayed epidemic model with pulse vaccination, Discrete Dyn. Nat. Soc., DOI: 10.1155/2008/746951. [40] Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 2014, 243, 718-727. [41] Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 2014, 244, 118-131. [42] M. Zhou, H. Xiang and Z. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Anal-real., 2019, 46(2019), 446-464. [43] G. Zhu, G. Chen, H. Zhang and X. Fu, Propagation dynamics of an epidemic model with infective media connecting two separated networks of populations, Commun. Nonlinear Sci., 2015, 20, 240-249. doi: 10.1016/j.cnsns.2014.04.023 -
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Zhewen Chen, Ruimin Zhang, Jiang Li, Xiaohui Liu, Chunjin Wei. GLOBAL DYNAMICS OF DETERMINISTIC AND STOCHASTIC SIRS EPIDEMIC MODELS[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2211-2229. doi: 10.11948/20190387
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Figure 1. (a), (b) and (c) are frequency histograms of susceptible, infected and removed, respectively with
$ \sigma = 0.1 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ . -
Figure 2. (a), (b) and (c) are the sample paths of susceptible, infected and removed, respectively with
$ \sigma = 0.1 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ . -
Figure 3. (a), (b) and (c) are frequency histograms of susceptible, infected and removed, respectively with
$ \sigma = 0.15 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ . -
Figure 4. (a), (b) and (c) are the sample paths of susceptible, infected and removed, respectively with
$ \sigma = 0.15 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ . -
Figure 5. That is the sample paths of infected with
$ \sigma = 0.9 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ . -
Figure 6. That is the sample paths of infected with
$ \sigma = 0.97 $ and initial values$ (S(0), I(0), R(0)) = (3.5, 2, 0.5) $ .