具有非线性发生率的SIRS反应扩散传染病模型的动力学研究
首发时间:2021-04-08
摘要:本文主要研究了常系数情形下,具有一般发生率的SIRS反应扩散传染病模型。首先定义了基本再生数$\mathcal{R}_0$,说明了平衡点的存在唯一性。接下来,我们通过$\mathcal{R}_0$和1之间的关系来说明平衡点的稳定性:一般情形下,当$\mathcal{R}_0<1$时,无病平衡点$E_0$局部渐近稳定;特别地,发生率为标准发生率时,通过构造李雅普诺夫函数,我们说明了$\mathcal{R}_0<1$时,无病平衡点$E_0$全局渐近稳定;$\mathcal{R}_0>1$时,地方病平衡点$E^*$全局渐近稳定,并通过数值模拟验证了结论。
关键词: 反应扩散 SIRS传染病模型 平衡点 渐近稳定 基本再生数
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Dynamics of a diffusive SIRS epidemic model with nonlinear incidence rate
Abstract:We focus on a diffusive SIRS epidemic model with nonlinear incidence rate under constant coefficients in the paper. Firstly, we define the basic reproduction number and show the existence and uniqueness of the equilibrium for the system. Next, we expose the stability of equilibrium in terms of the relationship between $\mathcal{R}_0$ and 1: in a normal case, disease-free equilibrium $E_0$ is locally asymptotically stale; especially, when the incidence is the standard incidence, by constructing the Lyapunov function, we show disease-free equilibrium $E_0$ and the epidemic equilibrium $E^*$ are globally asymptotically stable for $\mathcal{R}_0<1$ and $\mathcal{R}_0>1$, res\\-pectively. Moreover, we support our conclusions through numerical simulation.
Keywords: Reaction-diffusion SIRS epidemic model Equilibrium Asymptotically stable Basic reproduction number
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具有非线性发生率的SIRS反应扩散传染病模型的动力学研究
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