Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion

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• 作者：Yongli Cai^a ; ^b ; ^{caiyongli06@163.com" class="auth_mail" title="E-mail the corresponding author} ; Weiming Wang^a ; ^{weimingwang2003@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cross-diffusion ; Heterogeneity ; Stationary solution ; Fish-hook ; Bifurcation branch
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：30
• 期：Complete
• 页码：99-125
• 全文大小：781 K

文摘

In this paper, we consider the following strongly coupled epidemic model in a spatially heterogeneous environment with Neumann boundary condition: where pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si3.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=dcf1dc2630f0ada010af0e614f7223f0" title="Click to view the MathML source">Ω⊂Rⁿpan>pan class="mathContainer hidden">pan class="mathCode">$\Omega \subset {\mathbb{R}}^n$pan>pan>pan> is a bounded domain with smooth boundary pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si4.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=b8d9cf44e91c5666dbc3b068d919c439" title="Click to view the MathML source">∂Ωpan>pan class="mathContainer hidden">pan class="mathCode">$\partial \Omega$pan>pan>pan>; pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si5.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=256cd5c252d8f65e19f4ba6c4fd0cb78" title="Click to view the MathML source">b,m,k,cpan>pan class="mathContainer hidden">pan class="mathCode">$b,m,k,c$pan>pan>pan> and pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si6.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=74c6c4a807e83f04e7f4b431ee2947a2" title="Click to view the MathML source">δpan>pan class="mathContainer hidden">pan class="mathCode">$\delta$pan>pan>pan> are positive constants; pan id="mmlsi7" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$\beta \left(x\right)\in C\left(\bar{\Omega }\right)$pan>pan>pan> and pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si8.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=ed0734c8786df9c87c8649c82b5d9e8f" title="Click to view the MathML source">θ(x)pan>pan class="mathContainer hidden">pan class="mathCode">$\theta \left(x\right)$pan>pan>pan> is a smooth positive function in pan id="mmlsi9" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$\bar{\Omega }$pan>pan>pan> within pan id="mmlsi10" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">${\partial }_n\theta \left(x\right)=0$pan>pan>pan> on pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si4.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=b8d9cf44e91c5666dbc3b068d919c439" title="Click to view the MathML source">∂Ωpan>pan class="mathContainer hidden">pan class="mathCode">$\partial \Omega$pan>pan>pan>. The main result is that we have derived the set of positive solutions (endemic) and the structure of bifurcation branch: after assuming that the natural growth rate pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si12.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=c8507b3c8bd64dc54b7c2c863e5f8b88" title="Click to view the MathML source">a:=b−mpan>pan class="mathContainer hidden">pan class="mathCode">$a:=b-m$pan>pan>pan> of pan id="mmlsi13" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si13.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=3eb340c014c3e5eb3de598b9dca8f668" title="Click to view the MathML source">Span>pan class="mathContainer hidden">pan class="mathCode">$S$pan>pan>pan> is sufficiently small, the disease-induced death rate pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si6.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=74c6c4a807e83f04e7f4b431ee2947a2" title="Click to view the MathML source">δpan>pan class="mathContainer hidden">pan class="mathCode">$\delta$pan>pan>pan> is slightly small, and the cross-diffusion coefficient pan id="mmlsi15" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si15.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=7e7ece1156f8873958e1a5387d8aa92b" title="Click to view the MathML source">cpan>pan class="mathContainer hidden">pan class="mathCode">$c$pan>pan>pan> is sufficiently large, we show that the model admits a bounded branch pan id="mmlsi16" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si16.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=73fa17088da943823127cdfda9565ca0" title="Click to view the MathML source">Γpan>pan class="mathContainer hidden">pan class="mathCode">$\Gamma$pan>pan>pan> of positive solutions, which is a monotone pan id="mmlsi1" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$S$pan>pan>pan>-type or fish-hook-shaped curve with respect to the bifurcation parameter pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121815001613&_mathId=si6.gif&_user=111111111&_pii=S1468121815001613&_rdoc=1&_issn=14681218&md5=74c6c4a807e83f04e7f4b431ee2947a2" title="Click to view the MathML source">δpan>pan class="mathContainer hidden">pan class="mathCode">$\delta$pan>pan>pan>. One of the most interesting findings is that the multiple endemic steady-states are induced by the cross-diffusion and the spatial heterogeneity of environments together.