Global threshold dynamics of a stochastic differential equation SIS model

详细信息    查看全文

• 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) id="bbr0080">[8]. We present a stochastic threshold theorem in term of a stochastic basic reproduction number  id="mmlsi1" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="23" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si1.gif">ipt>iner hidden">i1.gif" overflow="scroll">i>Ri>0i>Si>: the disease dies out with probability one if id="mmlsi2" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="52" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si2.gif">ipt>iner hidden">i2.gif" overflow="scroll">i>Ri>0i>Si><1, and the disease is recurrent if id="mmlsi27" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="52" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si27.gif">ipt>iner hidden">i27.gif" overflow="scroll">i>Ri>0i>Si>⩾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 id="mmlsi4" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="52" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si4.gif">ipt>iner hidden">i4.gif" overflow="scroll">i>Ri>0i>Si>>1. In term of the profile of the invariant density, we define a persistence basic reproduction number  id="mmlsi5" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="24" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si5.gif">ipt>iner hidden">i5.gif" overflow="scroll">i>Ri>0i>Pi> and give a persistence threshold theorem: the disease dies out with large probability if id="mmlsi6" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="54" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si6.gif">ipt>iner hidden">i6.gif" overflow="scroll">i>Ri>0i>Pi>⩽1, while persists with large probability if id="mmlsi128" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="54" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si128.gif">ipt>iner hidden">i128.gif" overflow="scroll">i>Ri>0i>Pi>>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 id="mmlsi193" class="mathmlsrc">itle="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"><img class="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">ipt><img height="18" border="0" style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306370-si193.gif">ipt>iner hidden">i193.gif" overflow="scroll">i>Ri>0i>Di>>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.