一类具有空间结构的传染病模型
On a Vector-host Epidemic Model with Spatial Structure
摘要：
本文研究了一类反应-扩散向量-宿主传染病模型。我们定义了基本再生数$R_0$，证明了$R_0$是一个门限参数：如果$R_0\le 1$，则模型的无病稳态是全局稳定的;如果$R_0>1$，则模型存在唯一的全局稳定的正稳态.然后，我们将$R_0$写为一个乘法算子$R(x)$和两个谱半径等于1的紧算子的乘积的谱半径。这里，$R(x)$对应于没有扩散的模型的基本再生数，因此被称为局部基本再生数。我们研究了$R_0$和$R(x)$之间的关系作为扩散率的变化。
英文摘要：
In this paper, we study a reaction-diffusion vector-host epidemic model. We define the basic reproduction number $R_0$ and show that $R_0$ is a threshold parameter: if $R_0\le 1$ the disease free steady state is globally stable; if $R_0>1$ the model has a unique globally stable positive steady state. We then write $R_0$ as the spectral radius of the product of one multiplicative operator $R(x)$ and two compact operators with spectral radius equalling one. Here $R(x)$ corresponds to the basic reproduction number of the model without diffusion and is thus called local basic reproduction number. We study the relationship between $R_0$ and $R(x)$ as the diffusion rates vary.
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