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 Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary ∂Ω$\partial \Omega$; b,m,k,c$b,m,k,c$ and δ$\delta$ are positive constants; $\beta \left(x\right)\in C\left(\bar{\Omega }\right)$ and θ(x)$\theta \left(x\right)$ is a smooth positive function in $\bar{\Omega }$ within ${\partial }_n\theta \left(x\right)=0$ on ∂Ω$\partial \Omega$. 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 a:=b−m$a:=b-m$ of S$S$ is sufficiently small, the disease-induced death rate δ$\delta$ is slightly small, and the cross-diffusion coefficient c$c$ is sufficiently large, we show that the model admits a bounded branch Γ$\Gamma$ of positive solutions, which is a monotone $S$-type or fish-hook-shaped curve with respect to the bifurcation parameter δ$\delta$. 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.