一类非局部时滞的SVIR反应扩散模型的全局吸引性

一类非局部时滞的SVIR反应扩散模型的全局吸引性

杨瑜^,

^{上海立信会计金融学院统计与数学学院上海 201209}

Global Attractivity of a Nonlocal Delayed and Diffusive SVIR Model

Yang Yu^,

^{School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209}

收稿日期: 2021-01-11

Received: 2021-01-11

作者简介 About authors

杨瑜,E-mail:yangyu@lixin.edu.cn , E-mail：yangyu@lixin.edu.cn

Abstract

In this paper, by using Lyapunov functional, we prove the global attractivity of the endemic equilibrium for a nonlocal delayed and diffusive SVIR model when $\mathcal{R}_{0}>1$ , which cover and improve some known results.

Keywords： SVIR model ; Spatial heterogeneity ; Lyapunov functional ; Global attractivity

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本文引用格式

杨瑜. 一类非局部时滞的SVIR反应扩散模型的全局吸引性. 数学物理学报[J], 2021, 41(6): 1864-1870 doi:

Yang Yu. Global Attractivity of a Nonlocal Delayed and Diffusive SVIR Model. Acta Mathematica Scientia[J], 2021, 41(6): 1864-1870 doi:

1 引言

最近, 文献[1]考虑了如下一类非局部时滞的SVIR反应扩散模型:

$\begin{eqnarray} \left\{ \begin{array}{ll} S_{t} = \nabla\cdot(d_{S}(x)\nabla S)+\lambda(x)-\beta_1(x)SI-(\mu(x)+\alpha(x))S, \ t>0, \ x\in\Omega, \\ V_{t} = \nabla\cdot(d_{V}(x)\nabla V)+\alpha(x)S-\beta_2(x)VI-(\mu(x)+\gamma(x))V, \ t>0, \ x\in\Omega, \\ { } I_{t} = \nabla\cdot(d_{I}(x)\nabla I)+\int_{\Omega}G(\tau, x, y)[\beta_1(y)S(t-\tau, y)+\beta_2(y)V(t-\tau, y)]I(t-\tau, y){\rm d}y \\ \qquad-(\mu(x)+\delta_{I}(x))I, \ t>0, \ x\in\Omega, \\ R_{t} = \nabla\cdot(d_{R}(x)\nabla R)+\gamma(x)V+\delta_{I}(x)I-\mu(x)R, \ t>0, \ x\in\Omega, \\ {}[d_{S}(x)\nabla S]\cdot{\bf \nu} = [d_{V}(x)\nabla V]\cdot{\bf \nu} = [d_{I}(x)\nabla I]\cdot{\bf \nu} = [d_{R}(x)\nabla R]\cdot{\bf \nu} = 0, \ t>0, \ x\in\partial\Omega, \end{array} \right. \end{eqnarray}$

(1.1)

其中 $S(t, x)$ , $V(t, x)$ , $I(t, x)$ 和 $R(t, x)$ 分别表示时刻 $t$ 和空间位置 $x$ 处易感个体、疫苗接种个体、染病个体和康复个体的密度. $\nabla$ 是梯度算子. $G$ 是与 $\nabla\cdot(d_{E}(\cdot)\nabla)-(\mu(\cdot)+\delta_{E}(\cdot))$ 和Neumann边值条件相关的格林函数. $\Omega\in{{\Bbb R}} ^{n}$ 是具有光滑边界 $\partial \Omega$ 的有界区域.所有的参数都是连续和严格正的.

由于系统(1.1)的第四个方程与前三个方程是解耦的, 故只需考虑如下系统:

$\begin{eqnarray} \left\{ \begin{array}{ll} S_{t} = \nabla\cdot(d_{S}(x)\nabla S)+\lambda(x)-\beta_1(x)SI-(\mu(x)+\alpha(x))S, \ t>0, \ x\in\Omega, \\ V_{t} = \nabla\cdot(d_{V}(x)\nabla V)+\alpha(x)S-\beta_2(x)VI-(\mu(x)+\gamma(x))V, \ t>0, \ x\in\Omega, \\ { } I_{t} = \nabla\cdot(d_{I}(x)\nabla I)+\int_{\Omega}G(\tau, x, y)[\beta_1(y)S(t-\tau, y)+\beta_2(y)V(t-\tau, y)]I(t-\tau, y){\rm d}y \\ \qquad-(\mu(x)+\delta_{I}(x))I, \ t>0, \ x\in\Omega, \\ {}[d_{S}(x)\nabla S]\cdot{\bf \nu} = [d_{V}(x)\nabla V]\cdot{\bf \nu} = [d_{I}(x)\nabla I]\cdot{\bf \nu} = 0, \ t>0, \ x\in\partial\Omega. \end{array} \right. \end{eqnarray}$

(1.2)

由文献[2]知, 当基本再生数 ${\cal R}_0<1$ 时, 系统(1.2)的唯一的无病平衡态是全局渐近稳定的; 当 ${\cal R}_0>1$ 时, 系统(1.2)至少存在一个正平衡态并且系统是一致持续生存的.由于空间异质的影响, 研究系统(1.2)的正平衡态的全局吸引性是非常困难的.注意到, 文献[2]利用波动引理的方法讨论了一类非局部时滞的捕食-食饵反应扩散模型的正平衡点的全局吸引性.随后, 很多学者利用该方法讨论了具有非局部时滞的不同类型传染病模型的正平衡点的全局吸引性, 见文献[3-6].

假定系统(1.2)中的参数都是常数.由 $G(\tau, x, y)$ 是与 $d_{E}\Delta-(\mu+\delta_{E})$ 和Neumann边值条件相关的格林函数及Fourier变换可知

$G(\tau, x, y) = {\rm e}^{-(\mu+\delta_{E})\tau}\left(\frac{1}{\sqrt{4\pi d_E\tau}}\right)^{n}\exp{\left(-\frac{|x-y|^{2}}{4d_E\tau}\right)}.$

定义

$G(d_E\tau, x, y) = \left(\frac{1}{\sqrt{4\pi d_E\tau}}\right)^{n}\exp{\left(-\frac{|x-y|^{2}}{4d_E\tau}\right)},$

故

$G(\tau, x, y) = {\rm e}^{-(\mu+\delta_{E})\tau}G(d_E\tau, x, y).$

由

$\frac{\partial G(\tau, x, y)}{\partial \tau} = d_{E}\Delta G(\tau, x, y)-(\mu+\delta_{E})G(\tau, x, y),$

得

$\frac{\partial G(d_E\tau, x, y)}{\partial \tau} = d_{E}\Delta G(d_E\tau, x, y),$

即 $G(d_E\tau, x, y)$ 是与 $d_{E}\triangle$ 和Neumann边值条件相关的格林函数且 $\int_{\Omega}G(d_{E}\tau, x, y){\rm d}y = 1$ .因此, 系统(1.2)中的参数都是常数时可化为如下系统:

$\begin{eqnarray} \left\{ \begin{array}{ll} S_{t} = d_{S}\Delta S+\lambda-\beta_1SI-(\mu+\alpha)S, \ t>0, \ x\in\Omega, \\ V_{t} = d_{V}\Delta V+\alpha S-\beta_2VI-(\mu+\gamma)V, \ t>0, \ x\in\Omega, \\ { } I_{t} = d_{I}\Delta I+{\rm e}^{-(\mu+d_{E})\tau}\int_{\Omega}G(d_{E}\tau, x, y)[\beta_1S(t-\tau, y)+\beta_2V(t-\tau, y)]I(t-\tau, y){\rm d}y \\ \qquad-(\mu+\delta_{I})I, \ t>0, \ x\in\Omega, \\ { } \frac{\partial S}{\partial{\bf \nu}} = \frac{\partial V}{\partial{\bf \nu}} = \frac{\partial I}{\partial{\bf \nu}} = 0, \ t>0, \ x\in\partial\Omega. \end{array} \right. \end{eqnarray}$

(1.3)

目前, 有关系统(1.3)的正平衡点的全局吸引性方面的结果还没有.最近, 文献[7]通过构造合适的Lyapunov泛函讨论一类非局部时滞的蓝舌病反应扩散模型的正平衡点的全局吸引性.受文献[7]的启发, 本文通过构造Lyapunov泛函研究系统(1.3)的正平衡点的全局吸引性.关于Lyapunov泛函的构造方法, 见文献[8-10].

本文结构安排如下:第二节给出一些预备知识.第三节讨论当基本再生数 ${\cal R}_0>1$ 时系统(1.3)的正平衡点的全局吸引性.最后, 给出简要的结论.

2 预备知识

对 $\tau\geq0$ , 定义 $C_{\tau}: = C([-\tau, 0], {\Bbb X})$ , 范数为 $||\psi|| = \max\limits_{\theta\in[-\tau, 0]}||\psi(\theta)||_{{\Bbb X}},$ 其中 ${\Bbb X}: = C(\bar{\Omega}, {{\Bbb R}} ^{3})$ 是具有范数 $||\psi(\cdot)||_{{\Bbb X}}$ 的Banach空间.进一步, 定义 ${\Bbb X}^{+}: = C(\bar{\Omega}, {{\Bbb R}} _{+}^{3})$ 和 $C_{\tau}^{+}: = C([-\tau, 0], {\Bbb X}^{+})$ .

由文献[1]的定理6.1有

定理2.1 对 $\forall\ \psi\in C_{\tau}^{+}$ , 系统(1.3)在 $[0, \infty)$ 上存在唯一的解 ${\bf u}(t, \cdot, \psi)$ , 且解半流 $\Psi(t) = {\bf u}_{t}(\cdot): C_{\tau}^{+}\rightarrow C_{\tau}^{+}$ , $t\geq0$ 在 $C_{\tau}^{+}$ 上有一个全局紧吸引子.

易知, 系统(1.3)总是存在一个无病平衡点 $E_0 = (S^0, V^0, 0)$ , 其中

$S^0 = \frac{\lambda}{\mu+\alpha}, \quad V^0 = \frac{\alpha\lambda}{(\mu+\alpha)(\mu+\gamma)}.$

而且系统(1.3)的基本再生数

${\cal R}_0 = {\rm e}^{-(\mu+\delta_{E})\tau}\frac{\beta_1S^0+\beta_2V^0}{\mu+\delta_{I}}.$

由文献[11]知, 当 ${\cal R}_0>1$ 时, 系统(1.3)有唯一的正平衡点 $E^* = (S^*, V^*, I^*)$ , 其中

$S^* = \frac{\lambda}{\beta_1I^*+\mu+\alpha}, \quad V^* = \frac{\alpha S^*}{\beta_2I^*+\mu+\gamma},$

$a(I^*)^{2}+bI^*+c = 0,$

这里

$a = \beta_1\beta_2, \quad b = \beta_1(\mu+\gamma)+\beta_2(\mu+\alpha)-{\rm e}^{-(\mu+\delta_{E})\tau}\frac{\beta_1\beta_2\lambda}{\mu+\delta_{I}},$

$c = (\mu+\alpha)(\mu+\gamma)(1-{\cal R}_0).$

再由文献[1]的推论6.7有

定理2.2 (ⅰ)若 ${\cal R}_0<1$ , 则无病平衡点 $E_{0}$ 在 $C_{\tau}^{+}$ 内是全局吸引的.

(ⅱ) 若 ${\cal R}_0>1$ , 则存在一个常数 $\varrho>0$ 使得对 $\forall\ \psi\in C_{\tau}^{+}$ 满足 $\psi_3(0, \cdot)\not\equiv0$ , 关于 $x\in \overline{\Omega}$ 一致成立

$\liminf \limits_{t\rightarrow +\infty}S(t, x)\geq\varrho, \ \; \; \liminf \limits_{t\rightarrow +\infty}V(t, x)\geq\varrho, \ \; \; \liminf \limits_{t\rightarrow +\infty}I(t, x)\geq\varrho.$

3 全局吸引性

下面给出本文的主要结果.

定理3.1 若 ${\cal R}_0>1$ , 则对 $\forall\ \psi\in C_{\tau}^{+}$ 满足 $\psi_3(0, \cdot)\not\equiv0$ , 关于 $x\in \overline{\Omega}$ 一致成立

$\lim\limits_{t\rightarrow \infty}{\bf u}(t, x;\psi) = {\bf u}^{*},$

其中 ${\bf u}^{*} = (S^*, V^*, I^*)$ .

证定义 $\omega(\psi)$ 是半流 $\Psi(t)$ 的轨道 $\gamma^{+}(\psi)$ 的 $\omega$ -极限集.定义集合

${\Bbb H}: = \left\{\psi\in C_{\tau}^{+}:\psi_{i}(0, x)>0, \ \forall x\in\bar{\Omega}, \ i = 1, 2, 3\right\}.$

由定理2.1和2.2(ⅱ)可知, $\omega(\psi)\subset{\Bbb H}$ .为方便讨论, 记 ${\cal X} = {\cal X}(t, x), \ \forall {\cal X}\in\{S, V, I\}.$

构造Lyapunov泛函 ${\Bbb V}:{\Bbb H}\rightarrow {{\Bbb R}}$ 如下:

$\begin{eqnarray*} {\Bbb V}(t) = \int_{\Omega}({\Bbb V}_1(t, x)+{\Bbb V}_2(t, x)){\rm d}x, \end{eqnarray*}$

其中

$\begin{eqnarray*} {\Bbb V}_1(t, x)& = & S^* g\left(\frac{S}{S^*}\right)+V^* g\left(\frac{V}{V^*}\right) +{\rm e}^{(\mu+\delta_{E})\tau}I^* g\left(\frac{I}{I^*}\right), \\ {\Bbb V}_2(t, x)& = & \beta_1S^* I^*\int_{-\tau}^{0}\int_{\Omega}G(d_{E}(-\theta), x, y)g\left(\frac{S(t+\theta, y)I(t+\theta, y)}{S^* I^*}\right){\rm d}y{\rm d}\theta \\ &&+\beta_2V^* I^*\int_{-\tau}^{0}\int_{\Omega}G(d_{E}(-\theta), x, y)g\left(\frac{V(t+\theta, y)I(t+\theta, y)}{V^* I^*}\right){\rm d}y{\rm d}\theta, \end{eqnarray*}$

$g(x) = x-1-\ln x$ , $x>0$ .显然, $x>0$ 时 $g(x)\geq 0$ 且 $\min\limits_{x>0}g(x) = g(1)$ .

注意到

$\lambda = \beta_1S^* I^*+(\mu+\alpha)S^*, \ \alpha S^* = \beta_2V^* I^*+(\mu+\gamma)V^*$

和

${\rm e}^{-(\mu+d_{E})\tau}(\beta_1S^*+\beta_2V^*) = \mu+\delta_{I}.$

通过计算得

$\begin{eqnarray*} \frac{\partial{\Bbb V}_1(t, x)}{\partial t} & = &\left(1-\frac{S^*}{S}\right)(d_{S}\triangle S+(\mu+\alpha)S^*-(\mu+\alpha)S+\beta_1S^* I^*-\beta_1SI)\\ && +\left(1-\frac{V^*}{V}\right)(d_{V}\triangle V+\alpha(S-S^*)+(\mu+\gamma)(V^*-V)+\beta_2V^* I^*-\beta_2VI)\\ & &+{\rm e}^{(\mu+\delta_{E})\tau}\left(1-\frac{I^*}{I}\right)\left(d_{I}\Delta I-{\rm e}^{-(\mu+\delta_{E})\tau} (\beta_1S^*+\beta_2V^*)I\right.\\ &&\left.+{\rm e}^{-(\mu+\delta_{E})\tau}\int_{\Omega}G(d_{E}\tau, x, y)[\beta_1S (t-\tau, y)+\beta_2V(t-\tau, y)]I(t-\tau, y){\rm d}y\right)\\ & = & A-\mu S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{S}{S^*}\right)\right)+\alpha S^*\left(1-\frac{S^*}{S} -\frac{SV^*}{S^* V}+\frac{V^*}{V}\right)\\ && -(\mu+\gamma)V^*\left(g\left(\frac{V^*}{V}\right)+g\left(\frac{V}{V^*}\right)\right)\\ && +\beta_1S^* I^*\left(2-\frac{SI}{S^*I^*}-\frac{S^*}{S}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{S(t-\tau, y)I(t-\tau, y)} {S^* I}{\rm d}y\right)\\ &&+\beta_2V^* I^*\left(2-\frac{VI}{V^*I^*}-\frac{V^*}{V}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{V(t-\tau, y)I(t-\tau, y)} {V^* I}{\rm d}y\right)\\ &&+\int_{\Omega}G(d_{E}\tau, x, y)[\beta_1S(t-\tau, y)+\beta_2V(t-\tau, y)]I(t-\tau, y){\rm d}y, \end{eqnarray*}$

其中

$A = \frac{d_{S}(S-S^*)}{S}\triangle S+\frac{d_{V}(V-V^*)}{V}\triangle V+{\rm e}^{(\mu+\delta_{E})\tau}\frac{d_{I}(I-I^*)}{I}\triangle I.$

$\begin{eqnarray*} \frac{\partial{\Bbb V}_2(t, x)}{\partial t} & = &\beta_1S^* I^*\left[g\left(\frac{SI}{S^* I^*}\right)-\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I^*}\right){\rm d}y\right]\\ &&+\beta_2V^* I^*\left[g\left(\frac{VI}{V^* I^*}\right)-\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I^*}\right){\rm d}y\right]\\ & = &\beta_1S^* I^*\left[\frac{SI}{S^* I^*}-\ln\frac{SI}{S^* I^*}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I^*}{\rm d}y\right.\\ &&\left.+\int_{\Omega}G(d_{E}\tau, x, y)\ln\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I^*}{\rm d}y\right]\\ &&+\beta_2V^* I^*\left[\frac{VI}{V^* I^*}-\ln\frac{VI}{V^* I^*}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I^*}{\rm d}y\right.\\ &&\left.+\int_{\Omega}G(d_{E}\tau, x, y)\ln\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I^*}{\rm d}y\right]. \end{eqnarray*}$

进一步, 有

$\begin{eqnarray*} &&\frac{\partial({\Bbb V}_1(t, x)+{\Bbb V}_2(t, x))}{\partial t}\\ & = & A-\mu S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{S}{S^*}\right)\right)-(\mu+\gamma)V^* g\left(\frac{V^*}{V}\right)-(\mu+\gamma)V^* g\left(\frac{V}{V^*}\right)\\ &&+\alpha S^*\left(g\left(\frac{V^*}{V}\right)-g\left(\frac{S^*}{S}\right)-g\left(\frac{SV^*}{S^* V}\right)\right)\\ &&+\beta_1S^* I^*\left(2-\frac{S^*}{S}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I}{\rm d}y\right.\\ &&\left.-\ln\frac{SI}{S^* I^*}+\int_{\Omega}G(d_{E}\tau, x, y)\ln\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I^*}{\rm d}y\right)\\ &&+\beta_2V^* I^*\left(2-\frac{V^*}{V}-\int_{\Omega}G(d_{E}\tau, x, y)\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I}{\rm d}y\right.\\ &&\left.-\ln\frac{VI}{V^* I^*}+\int_{\Omega}G(d_{E}\tau, x, y)\ln\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I^*}{\rm d}y\right)\\ & = & A-\mu S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{S}{S^*}\right)\right)-(\mu+\gamma)V^* g\left(\frac{V}{V^*}\right)\\ &&-\alpha S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{SV^*}{S^* V}\right)\right)\\ &&-\beta_1S^* I^*\left(g\left(\frac{S^*}{S}\right)+\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I}\right){\rm d}y\right)\\ &&-\beta_2V^* I^*\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{V(t-\tau, y)I(t-\tau, y)}{V^* I}\right){\rm d}y. \end{eqnarray*}$

由于 $\int_{\Omega}\Delta \kappa {\rm d}x = 0$ 和 $\int_{\Omega}\frac{\Delta \kappa}{\kappa}{\rm d}x = \int_{\Omega}\frac{\|\nabla\kappa\|^2}{\kappa^2}{\rm d}x\geq 0$ , 故

$\begin{eqnarray} \frac{{\rm d}{\Bbb V}(t)}{{\rm d}t}& = &\int_{\Omega}\frac{\partial({\Bbb V}_1(t, x)+{\Bbb V}_2(t, x))}{\partial t}{\rm d}x {}\\ &\leq&-\int_{\Omega}\left[\mu S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{S}{S^*}\right)\right) +\alpha S^*\left(g\left(\frac{S^*}{S}\right)+g\left(\frac{SV^*}{S^*V}\right)\right)\right.{}\\ &&\left.+(\mu+\gamma)V^* g\left(\frac{V}{V^*}\right)+\beta_1S^* I^*g\left(\frac{S^*}{S}\right)\right]{{\rm d}x}{}\\ &&-\beta_1S^* I^*\int_{\Omega}\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{S(t-\tau, y)I(t-\tau, y)}{S^* I}\right){\rm d}y{\rm d}x{}\\ &&-\beta_2V^* I^*\int_{\Omega}\int_{\Omega}G(d_{E}\tau, x, y)g\left(\frac{V(t-\tau, y)I(t-\tau, y)} {V^* I}\right){\rm d}y{\rm d}x: = {\cal U}_{\psi}(t). \end{eqnarray}$

(3.1)

对 $t\geq0$ , ${\Bbb V}({\bf u}_{t}(\psi))$ 是非增和有下界的.因此, 存在一个常数 ${\cal G}>0$ 使得 $\lim\limits_{t\rightarrow \infty}{\Bbb V}({\bf u}_{t}(\psi)) = {\cal G}.$ 进一步, 存在一个序列 $l_{n}\rightarrow \infty$ 使得在 ${\Bbb H}$ 内有 $\lim\limits_{n\rightarrow \infty}{\bf u}_{l_n}(\psi) = \varphi$ .这意味着对 $\forall\varphi\in\omega(\psi)$ 有 ${\Bbb V}(\varphi) = {\cal G}$ .故当 $t\geq0$ 时, 由 ${\bf u}_t(\varphi)\in\omega(\psi)$ 知 ${\Bbb V}({\bf u}_t(\varphi)) = {\cal G}$ .故有 $\frac{{\rm d}{\Bbb V}({\bf u}_t(\varphi))}{{\rm d}t} = 0$ .根据不等式(3.1)得

$0 = \frac{{\rm d}{\Bbb V}({\bf u}_t(\varphi))}{{\rm d}t}\leq{\cal U}_{\varphi}(t)\leq0.$

因此, $t\geq0$ 时有 ${\cal U}_{\varphi}(t) = 0$ .由系统(1.3)知, 对 $\forall\ t\geq\tau$ 有 ${\bf u}_t(\varphi) = {\bf u}^*$ .故 $t\geq\tau$ 时, 由于 $\varphi\in\omega(\psi)$ 是任意的, 故得 ${\bf u}_t(\omega(\psi)) = {\bf u}^*$ .利用 $\omega$ -极限集的不变性, 得 $\omega(\psi) = {\bf u}_{\tau}(\omega(\psi)) = {\bf u}^*$ , 即 $\lim\limits_{t\rightarrow \infty}{\bf u}(t, x;\psi) = {\bf u}^{*}$ .

4 结论

当不考虑扩散和空间异质时, 系统(1.3)即为如下的时滞系统:

$\begin{eqnarray} \left\{ \begin{array}{ll} { } \frac{{\rm d}S(t)}{{\rm d}t} = \lambda-\beta_1SI-(\mu+\alpha)S, \\ { } \frac{{\rm d}V(t)}{{\rm d}t} = \alpha S-\beta_2VI-(\mu+\gamma)V, \\ { } \frac{{\rm d}I(t)}{{\rm d}t} = {\rm e}^{-(\mu+d_{E})\tau}[\beta_1S(t-\tau)+\beta_2V(t-\tau)]I(t-\tau)-(\mu+\delta_{I})I. \end{array} \right. \end{eqnarray}$

(4.1)

有关系统(4.1)的正平衡点的全局吸引性的结果可参见文献[12-14].当时滞 $\tau = 0$ 时, 参见文献[15].

当不考虑固定潜伏期时, 系统(1.3)即为如下的反应扩散系统:

$\begin{eqnarray} \left\{ \begin{array}{ll} { } S_{t} = d_{S}\Delta S+\lambda-\beta_1SI-(\mu+\alpha)S, \ t>0, \ x\in\Omega, \\ { } V_{t} = d_{V}\Delta V+\alpha S-\beta_2VI-(\mu+\gamma)V, \ t>0, \ x\in\Omega, \\ { } I_{t} = d_{I}\Delta I+\beta_1SI+\beta_2VI-(\mu+\delta_{I})I, \ t>0, \ x\in\Omega, \\ { } \frac{\partial S}{\partial{\bf \nu}} = \frac{\partial V}{\partial{\bf \nu}} = \frac{\partial I}{\partial{\bf \nu}} = 0, \ t>0, \ x\in\partial\Omega. \end{array} \right. \end{eqnarray}$

(4.2)

系统(4.2)的正平衡点的全局吸引性的结果见文献[16].

当 $\beta_2(x) = 0$ , 系统(1.2)已被文献[17]讨论过.进一步, 文献[18]在文献[17]的基础上考虑时间异质并推广了文献[17]中的结果.注意到文献[17, 18]中并未讨论正平衡态的全局吸引性.对于空间异质的反应扩散传染病模型, 研究其正平衡态的全局吸引性是非常困难的.当 ${\cal R}_0>1$ 时, 本文利用Lyapunov泛函的方法讨论了系统(1.3)的正平衡点的全局吸引性.

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原文顺序

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Analysis of a reaction-diffusion SVIR model with a fixed latent period and non-local infections

2020

... 最近, 文献[1]考虑了如下一类非局部时滞的SVIR反应扩散模型: ...

... 由文献[1]的定理6.1有 ...

... 再由文献[1]的推论6.7有 ...

A non-local delayed and diffusive predator-prey model

2001

... 由文献[2]知, 当基本再生数

${\cal R}_0<1$

时, 系统(1.2)的唯一的无病平衡态是全局渐近稳定的; 当

${\cal R}_0>1$

时, 系统(1.2)至少存在一个正平衡态并且系统是一致持续生存的.由于空间异质的影响, 研究系统(1.2)的正平衡态的全局吸引性是非常困难的.注意到, 文献[2]利用波动引理的方法讨论了一类非局部时滞的捕食-食饵反应扩散模型的正平衡点的全局吸引性.随后, 很多学者利用该方法讨论了具有非局部时滞的不同类型传染病模型的正平衡点的全局吸引性, 见文献[3-6]. ...

... 时, 系统(1.2)至少存在一个正平衡态并且系统是一致持续生存的.由于空间异质的影响, 研究系统(1.2)的正平衡态的全局吸引性是非常困难的.注意到, 文献[2]利用波动引理的方法讨论了一类非局部时滞的捕食-食饵反应扩散模型的正平衡点的全局吸引性.随后, 很多学者利用该方法讨论了具有非局部时滞的不同类型传染病模型的正平衡点的全局吸引性, 见文献[3-6]. ...

A reaction-diffusion malaria model with incubation period in the vector population

2011

... 由文献[2]知, 当基本再生数

${\cal R}_0<1$

时, 系统(1.2)的唯一的无病平衡态是全局渐近稳定的; 当

${\cal R}_0>1$

A vector-bias malaria model with incubation period and diffusion

2012

A diffusive dengue disease model with nonlocal delayed transmission

2015

A time-periodic and reaction-diffusion Dengue fever model with extrinsic incubation period and crowding effects

2020

... 由文献[2]知, 当基本再生数

${\cal R}_0<1$

时, 系统(1.2)的唯一的无病平衡态是全局渐近稳定的; 当

${\cal R}_0>1$

Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease

2021

... 目前, 有关系统(1.3)的正平衡点的全局吸引性方面的结果还没有.最近, 文献[7]通过构造合适的Lyapunov泛函讨论一类非局部时滞的蓝舌病反应扩散模型的正平衡点的全局吸引性.受文献[7]的启发, 本文通过构造Lyapunov泛函研究系统(1.3)的正平衡点的全局吸引性.关于Lyapunov泛函的构造方法, 见文献[8-10]. ...

... ]通过构造合适的Lyapunov泛函讨论一类非局部时滞的蓝舌病反应扩散模型的正平衡点的全局吸引性.受文献[7]的启发, 本文通过构造Lyapunov泛函研究系统(1.3)的正平衡点的全局吸引性.关于Lyapunov泛函的构造方法, 见文献[8-10]. ...

Global stability of a diffusive virus dynamics model with general incidence function and time delay

2015

Impacts of the cell-free and cell-to-cell infection modes on viral dynamics

2018

Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions

2015

Wave propagation of a diffusive epidemic model with latency and vaccination

2021

... 由文献[11]知, 当

${\cal R}_0>1$

时, 系统(1.3)有唯一的正平衡点

$E^* = (S^*, V^*, I^*)$

, 其中 ...

Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence

2016

... 有关系统(4.1)的正平衡点的全局吸引性的结果可参见文献[12-14].当时滞

$\tau = 0$

时, 参见文献[15]. ...

Global stability of a delayed epidemic model with latent period and vaccination strategy

2012

一类具有接种和潜伏期的传染病模型及动力学分析

2019

... 有关系统(4.1)的正平衡点的全局吸引性的结果可参见文献[12-14].当时滞

$\tau = 0$

时, 参见文献[15]. ...

一类具有接种和潜伏期的传染病模型及动力学分析

2019

... 有关系统(4.1)的正平衡点的全局吸引性的结果可参见文献[12-14].当时滞

$\tau = 0$

时, 参见文献[15]. ...

SVIR epidemic models with vaccination strategies

2008

... 有关系统(4.1)的正平衡点的全局吸引性的结果可参见文献[12-14].当时滞

$\tau = 0$

时, 参见文献[15]. ...

Stability and traveling waves of a vaccination model with nonlinear incidence

2018

... 系统(4.2)的正平衡点的全局吸引性的结果见文献[16]. ...

Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

2012

... 当

$\beta_2(x) = 0$

, 系统(1.2)已被文献[17]讨论过.进一步, 文献[18]在文献[17]的基础上考虑时间异质并推广了文献[17]中的结果.注意到文献[17, 18]中并未讨论正平衡态的全局吸引性.对于空间异质的反应扩散传染病模型, 研究其正平衡态的全局吸引性是非常困难的.当

${\cal R}_0>1$