媒體報道下隨機SIQS流行病模型的動態行為研究

胡晶晶 李彩鳳 韋煜明 彭華勤

摘 要： 研究了媒體報道干預策略下的隨機SIQS流行病模型.構造合適的 Lyapunov 函數，使用It公式和馬爾可夫半群理論，證明了基本再生數Rs0可用于控制隨機流行病模型的動態行為，即如果再生數Rs0<1，并且在其他條件下，疾病將消亡;如果再生數Rs0>1，并且在其他條件下，疾病是持久性的.結論表明：大的白噪聲可以抑制疾病的爆發，這為制定有用的控制策略來調節疾病的動態行為提供有效幫助.最后通過數值模擬驗證了這一結果.

關鍵詞： 媒體報道; 隨機SIQS流行病模型; 馬爾可夫半群; 基本再生數

1 Introduction

The spread of diseases seriously hinders the development of society and economy[1].Hence，it′s necessary to control the spread of infectious diseases.HETHCOTE et al.[2] considers the following classic SIQS epidemic model：

[1] SHARMA R.Stability analysis of infectious diseases with media coverage and poverty [J].Mathematical Theory and Modeling，2014，4（4）：107-113.

[2] HETHCOTE H，MA Z，LIAO S.Effects of quarantine in six endemic models for infectious diseases [J].Mathematical Bioscienxes，2002，180（1）：141-160.

[3] ZHAO M，ZHAO H.Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage [J].Advances in Difference Equations，2016，40（1）：149.

[4] GUO W，CAI Y，ZHANG Q，et al.Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage [J].Physica A：Statistical Mechanics and Its Applications，2018，492：2220-2236.

[5] GUO W，ZHANG Q.Dynamic behavior of a SIS epidemic model with media coverage [J].Journal of Henan Normal University（Natural Science Edition），2017，45（3）：42-47.

[6] GUMEL A B，RUAN S，DAY T，et al.Modelling strategies for controlling SARS outbreaks [J].Proceedings of the Royal Society B：Biological Sciences，2004，271：2223-2232.

[7] PANG Y，HAN Y，LI W.The threshold of a stochastic SIQS epidemic model [J/OL].Advances in Difference Equations，2014，38：320[2018-10-12].https：//doi.org/10.1186/1687-1847-2014-320.

[8] ZHOU Y，ZHANG W，YUAN S.Survival and stationary distribution of a SIR epidemic model with stochastic perturbations [J].Applied Mathematics and Computation，2014，244（10）：118-131.

[9] WEI F，CAI Y，ZHAO Y.The asymptotic behavior of a stochastic SIQS epidemic model with nonlinear incidence [J].Journal of Biomathematics，2016，31（1）：109-117.

[10] WEI F，LIN Q.Extinction and distribution for an SIQS epidemic model with quarantined-adjusted incidence [J].Acta Mathematica Scientia，2017，37A（6）：1148-1161.

[11] CAI Y，KANG Y，BANERJEE M，et al.A stochastic SIRS epidemic model with infectious force under intervention strategies [J].Journal of Differential Equations，2015，259（12）：7463-7502.

[12] LYAPUNOV A M.The general problem of the stability of motion [J].International Journal of Control，1992，55（3）：531-534.

[13] MAO X.Stochastic Differential Equations and Their Applications [M].Chichester：Ellis Horwood，1997.

[14] RUDNICKI R，PICHR K，TYRAN-KAMIN'SKA M.Markov semigroups and their applications [J].Lecture Notes in Physics，2002，597：215-238.

[15] STROOCK D W，VARADHAN S R S.On the support of diffusion processes with applications to the strong maximum principle [J].Regents of the University of California，1970，3：333-359.

[16] HIGHAM D J.An algorithmic introduction to numerical simulation of stochastic differential equations [J].SIAM Review，2001，43（3）：525-546.

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