Global threshold dynamics of a stochastic differential equation SIS model

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• 作者：Chuang Xu ^{cx1@ualberta.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：SIS epidemic model ; Global threshold dynamics ; Basic reproduction number ; Invariant density ; Stochastic differential equation ; Fokker&ndash ; Planck equation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：736-757
• 全文大小：799 K

文摘

In this paper, we further investigate the global dynamics of a stochastic differential equation SIS (Susceptible–Infected–Susceptible) epidemic model recently proposed in Gray et al. (2011) [8]. We present a stochastic threshold theorem in term of a stochastic basic reproduction number  height="18" width="23" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si1.gif">$R_0^S$: the disease dies out with probability one if height="18" width="52" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si2.gif">$R_0^S<1$, and the disease is recurrent if height="18" width="52" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si27.gif">$R_0^S⩾1$. We prove the existence and global asymptotic stability of a unique invariant density for the Fokker–Planck equation associated with the SDE SIS model when height="18" width="52" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si4.gif">$R_0^S>1$. In term of the profile of the invariant density, we define a persistence basic reproduction number  height="18" width="24" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si5.gif">$R_0^P$ and give a persistence threshold theorem: the disease dies out with large probability if height="18" width="54" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si6.gif">$R_0^P⩽1$, while persists with large probability if height="18" width="54" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si128.gif">$R_0^P>1$. Comparing the stochastic disease prevalence with the deterministic disease prevalence  , we discover that the stochastic prevalence is bigger than the deterministic prevalence if the deterministic basic reproduction number height="18" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306370-si193.gif">$R_0^D>2$. This shows that noise may increase severity of disease. Finally, we study the asymptotic dynamics of the stochastic SIS model as the noise vanishes and establish a sharp connection with the threshold dynamics of the deterministic SIS model in term of a Limit Stochastic Threshold Theorem.