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&ndash;Infected&ndash;Susceptible) epidemic model recently proposed in Gray et al. (2011) an id="bbr0080">[8]a>an>. We present a stochastic threshold theorem in term of a stochastic basic reproduction number  an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=351318e2457aaacc78bad5c46da5707e">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">row>Rrow>row>0row>row>Srow>ath>an>an>an>: the disease dies out with probability one if an id="mmlsi2" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=fb4675954d37c5c5448670a8373de3d0">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">row>Rrow>row>0row>row>Srow><1ath>an>an>an>, and the disease is recurrent if an id="mmlsi27" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si27.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=72c1f6643f810ef8c119addf31605d3d">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si27.gif" overflow="scroll">row>Rrow>row>0row>row>Srow>⩾1ath>an>an>an>. We prove the existence and global asymptotic stability of a unique invariant density for the Fokker&ndash;Planck equation associated with the SDE SIS model when an id="mmlsi4" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=e41c67f23f67ebb57c5b6202c6785cb9">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">row>Rrow>row>0row>row>Srow>>1ath>an>an>an>. In term of the profile of the invariant density, we define a persistence basic reproduction number  an id="mmlsi5" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=4ca269476239a7ba84f3d4b3a1b713d4">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">row>Rrow>row>0row>row>Prow>ath>an>an>an> and give a persistence threshold theorem: the disease dies out with large probability if an id="mmlsi6" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=37a58d7ebfda1eb52dde4a939f424dab">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">row>Rrow>row>0row>row>Prow>⩽1ath>an>an>an>, while persists with large probability if an id="mmlsi128" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si128.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=ea244a284fb789348a6851efe2b6083a">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si128.gif" overflow="scroll">row>Rrow>row>0row>row>Prow>>1ath>an>an>an>. 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 an id="mmlsi193" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306370&_mathId=si193.gif&_user=111111111&_pii=S0022247X16306370&_rdoc=1&_issn=0022247X&md5=42b7121d4b8eac06c47800ef08edcfe5">ass="imgLazyJSB inlineImage" 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">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si193.gif" overflow="scroll">row>Rrow>row>0row>row>Drow>>2ath>an>an>an>. 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.