• •

### 混合随机SIR传染病模型的动力学分析

1. 1. 山西财经大学应用数学学院 太原 030006;
2. 山西师范大学数学与计算机科学学院 临汾 041004
• 收稿日期:2021-05-06 修回日期:2021-11-01 出版日期:2022-04-25 发布日期:2022-06-18
• 基金资助:
山西省自然科学基金(201801D121011),晋财教2021-18号博士毕业生来晋科研项目(125/Z24179)资助课题.

GUO Xiaoxia, SUN Shulin. Dynamic Analysis of a Hybrid Stochastic SIR Epidemic Model[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(4): 992-1010.

### Dynamic Analysis of a Hybrid Stochastic SIR Epidemic Model

GUO Xiaoxia1, SUN Shulin2

1. 1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006;
2. School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004
• Received:2021-05-06 Revised:2021-11-01 Online:2022-04-25 Published:2022-06-18
In this paper, a hybrid stochastic SIR epidemic model with general incidence functional responses is proposed. First, we state that this model has a unique global positive solution for any initial value by constructing a suitable Lyapunov function. Then, we establish the sufficient and almost necessary condition for the extinction and permanence of the underlying system, and develop its ergordicity. Finally, a number of numerical examples are given to support our theoretical results

MR(2010)主题分类:

()
 [1] Cai Y L, Kang Y, Wang W M. A stochastic SIRS epidemic model with nonlinear incidence rate. Applied Mathematics and Computation, 2017, 305(772):221-240.[2] 陈易亮,滕志东.随机~SIVS传染病模型的持久性和灭绝性.东北师大学报(自然科学版), 2018, 50(1):47-53.(Chen Y L, Teng Z D. The extinction and persistence of stochastical perturbed SIVS epidemic models with general nonlinear incidence rate. Journal of Northeastern University (Natural Science), 2018, 50(1):47-53.)[3] Liu Y, Ruan S G, Yang L. Stability transition of persistence and extinction in an avian influenza model with Allee effect and stochasticity. Communications in Nonlinear Science and Numerical Simulation, 2020, 91:105416.[4] Wang Y M, Liu G R. Dynamics analysis of a stochastic SIRS epidemic model with nonlinear incidence rate and transfer from infectious to susceptible. Mathematical Biosciences and Engineering, 2019, 16(5):6047-6070.[5] 魏凤英,林青腾.非线性发病率随机流行病模型的动力学行为.数学学报, 2018, 61(1):155-166.(Wei F Y, Lin Q T. Dynamical behavior for a stochastic epidemic model with nonlinear incidence. Acta Mathematica Sinica, Chinese Series, 2018, 61(1):155-166).[6] Korobeinikov A, Maini P K. Nonlinear incidence and stability of infectious disease models. Mathematical Medicine and Biology-A Journal of the IMA, 2005, 22(2):113-128.[7] Jiang D Q, Yu J J, Ji J J, et al. Asymptotic behavior of global positive solution to a stochastic SIR model. Mathematical and Computer Modelling, 2011, 54(1-2):221-232.[8] 杨世新,刘贤宁.具有一般疾病发生率的~SIRS传染病模型分析.西南大学学报(自然科学版), 2013,35(1):1-5.(Yang S X, Liu X N. Analysis of an SIRS epidemic model with nonlinear incidence rate. Journal of Southwest University (Natural Science Edition), 2013, 35(1):1-5).[9] Liu W M, Levin S A, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. Journal of Mathematical Biology, 1986, 23(2):187-204.[10] 魏凤英,陈芳香.具有饱和发生率的随机SIRS流行病模型的渐近行为.系统科学与数学, 2016, 36(12):2444-2453.(Wei F Y, Chen F X, Asymptotic behaviors of a stochastic SIRS epidemic model with saturated incidence. Journal of Systems Science and Mathematical Sciences, 2016, 36(12):2444-2453.)[11] Du N H, Dieu N T, Nhu N N. Conditions for permanence and ergodicity of certain SIR epidemic models. Acta Applicandae Mathematicae, 2019, 160:81-99.[12] Du N H, Nhu N N. Permanence and extinction for the stochastic SIR epidemic model. Journal of Differential Equations, 2020, 269:9619-9652.[13] Li D, Liu S Q, Cui J A. Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching. Journal of Differential Equations, 2019, 266(7):3973-4017.[14] 张仲华,张倩.转换机制下具有非线性扰动的随机~SIVS传染病模型的定性分析.数学物理学报, 2021, 41A (4):1218-1234.(Zhang Z H, Zhang Q. Qualitative analysis of a stochastic SIVS epidemic model with nonlinear perturbations under regime switching. Acta Mathematica Scientia, 2021, 41A (4):1218-1234.)[15] 张向华. L\'{e}vy噪声驱动的传染病模型的动力学行为.哈尔滨:哈尔滨工业大学出版社, 2016.(Zhang X H. Dynamic Behavior of Epidemic Model Driven by Lévy Noises. Harbin:Harbin Institute of Technology Press, 2016.)[16] Zhao D L, Yuan S L. Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force. Journal of Applied Analysis&Computation, 2019, 9(2):440-451.[17] Zhou Y L, Zhang W G. Threshold of a stochastic SIR epidemic model with Lévy jumps. Physica A:Statistical Mechanics and Its Applications, 2016, 446(15):204-216.[18] Zhou Y L, Yuan S L, Zhao D L. Threshold behavior of a stochastic SIS model with Lévy jumps. Applied Mathematics and Computation, 2016, 275:255-267.[19] 孙树林,尹辉.具有不同时滞的捕食者-食饵恒化器模型的定性分析.系统科学与数学, 2016,36(12):1-17.(Sun S L, Yin H. Qualitative analysis of a predator-prey model with different delays in the chemostat. Journal of Systems Science and Mathematical Sciences, 2016, 36(12):1-17.)[20] 孙树林,晋丹慧.具有多个参数扰动的随机恒化器模型的持久性与灭绝性.系统科学与数学, 2017,37(1):277-288.(Sun S L, Jin D H. Exclusion and persistence in Chemostat model with stochastic perturbation multiple parameters. Journal of Systems Science and Mathematical Sciences, 2017, 37(1):277-288.)[21] Wang Z J, Deng M L, Liu M. Stationary distribution of a stochastic ratio-dependent predatorprey system with regime-switching. Chaos Solitons&Fractals, 2020, 142:110462.[22] Mao X R. Stochastic Differential Equations and Applications. Chichester:Horwood Publishing, 2007.[23] Liptser R. A strong law of large numbers for local martingales. Stochastics and Stochastics Reports, 1980, 3:217-228.[24] Applebaum D. Lévy Processes and Stochastic Calculus, 2nd Edition. Cambridge:Cambridge University Press, 2009.[25] Mao X R, Yuan C G. Stochastic Differential Equations with Markovs-Witching. London:Imperial College Press, 2006.[26] Phu N D, Donal O, Tuong T D. Long time characterization for the general stochastic epidemic SIS model under regime-switching. Nonlinear Analysis Hybrid Systems, 2020, 38:100951.[27] Dong Y. Ergodicity of stochastic differential equations driven by Lévy noise with local Lipschitz coefficients. Advances in Mathematics (China), 2018, 47:11-47.[28] Privault, N, Wang L. Stochastic SIR Lévy jump model with heavy-tailed increments. Journal of Nonlinear Science, 2021, 31:15.[29] Khasmiskii R Z. Stochastic Stability of Differential Equations. 2nd Edition. Berlin Heidelberg:Springer-Verlag, 2012.[30] Dieu N T, Fugo T, Du N H. Asymptotic behaviors of a stochastic epidemic models with jumpdiffusion. Applied Mathematical Modelling, 2020, 86:259-270.
 [1] 战琛祥, 王峰. 具有Holling-II型功能性反应函数的随机捕食者-食饵模型的平稳分布[J]. 系统科学与数学, 2021, 41(8): 2137-2148. [2] 张荣，孙树林. 一类具有变消耗率的随机恒化器模型的渐近行为[J]. 系统科学与数学, 2020, 40(12): 2237-2247. [3] 陈贤礼. 具有Allee效应及非线性扰动的随机单种群模型的平稳分布及灭绝性[J]. 系统科学与数学, 2019, 39(12): 2093-2104. [4] 王建军.  $d$-跟踪与遍历性及 proximality 的关系[J]. 系统科学与数学, 2019, 39(1): 30-36. [5] 魏凤英，陈芳香. 具有饱和发生率的随机SIRS流行病模型的渐近行为[J]. 系统科学与数学, 2016, 36(12): 2444-2453. [6] 徐立峰，张绍义. 不动点与Markov过程的稳定性[J]. 系统科学与数学, 2015, 35(2): 221-232. [7] 何泽荣，郑敏，周娟. 带有性别比和尺度结构的非线性种群的全局演化行为[J]. 系统科学与数学, 2013, 33(12): 1480-1490. [8] 李自然;成思危;祖垒. 基于格兰杰因果检验遍历性分析的中国股市和国际股市的时变联动特征研究[J]. 系统科学与数学, 2011, 31(2): 131-143. [9] 唐应辉. 推广的多重休假$M^X/G/1$排队系统[J]. 系统科学与数学, 2005, 25(1): 39-049. [10] 陈永义;傅自晦;张学显. 非齐次马尔科夫链遍历性的一些结果[J]. 系统科学与数学, 1996, 16(4): 311-317.