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数学物理学报, 2023, 43(5): 1595-1606

具有媒体报道的 SVIR 传染病模型的生存性分析

李丹1, 魏凤英,2,*, 毛学荣3

1福州大学数学与统计学院 福州 350116

2福州大学运筹学与控制论福建省高校重点实验室 福州 350116

3思克莱德大学数学与统计系 格拉斯哥 G1 1XH

Survival Analysis of an SVIR Epidemic Model with Media Coverage

Li Dan1, Wei Fengying,2,*, Mao Xuerong3

1School of Mathematics and Statistics, Fuzhou University, Fuzhou 350116

2Key Laboratory of Operations Research and Control of Universities in Fujian, Fuzhou University, Fuzhou 350116

3Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK

通讯作者: * 魏凤英，Email: weifengying@fzu.edu.cn

收稿日期: 2022-04-22   修回日期: 2022-10-31  

基金资助: 国家自然科学基金-国际(地区)合作与交流项目(61911530398)
福建省科技厅项目(2021L3018)
福建省自然科学基金(2021J01621)
英国皇家学会(WM160014)
英国皇家学会(英国皇家学会沃尔夫森研究优异奖)
英国皇家学会和牛顿基金(NA160317)
英国皇家学会和牛顿基金(皇家学会-牛顿高级奖学金)
工程和物理科学研究委员会(EP/K503174/1)

Received: 2022-04-22   Revised: 2022-10-31  

Fund supported: National Natural Science Foundation of China(61911530398)
Special Projects of the Central Government Guiding Local Science, Technology Development(2021L3018)
Natural Science Foundation of Fujian Province of China(2021J01621)
Royal Society, UK(WM160014)
Royal Society, UK (Royal Society Wolfson Research Merit Award)
Royal Society and the Newton Fund, UK(NA160317)
Royal Society and the Newton Fund, UK (Royal Society-Newton Advanced Fellowship)
EPSRC, the Engineering and Physical Sciences Research Council(EP/K503174/1)

摘要

该文研究了具有Logistic增长和媒体报道的饱和发生率的随机SVIR模型. 为了研究模型的动力学性质, 首先证明了随机模型全局正解的存在唯一性, 其次通过构造合适的李雅普诺夫函数, 探究疾病持久和灭绝的充分性条件. 研究表明: 当Rs0>1时, 疾病长时间持续存在. 当Re0<1时, 疾病在流行一段时间后灭绝. 最后, 通过数值模拟验证了以上结论.

关键词： 传染病模型; 疫苗; 媒体报道; 持久性与灭绝性; 平稳分布

Abstract

We consider the long-term properties of a stochastic SVIR epidemic model with media coverage and the logistic growth in this paper. We firstly derive the fitness of a unique global positive solution. Then we construct appropriate Lyapunov functions and obtain the existence of ergodic stationary distribution when Rs0>1 is valid, and also derive sufficient conditions for persistence in the mean. Moreover, the exponential extinction to the density of the infected is figured out when Re0<1 holds.

Keywords： Epidemic model; Vaccination; Media coverage; Persistence and extinction; Stationary distribution

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本文引用格式

李丹, 魏凤英, 毛学荣. 具有媒体报道的 SVIR 传染病模型的生存性分析[J]. 数学物理学报, 2023, 43(5): 1595-1606

Li Dan, Wei Fengying, Mao Xuerong. Survival Analysis of an SVIR Epidemic Model with Media Coverage[J]. Acta Mathematica Scientia, 2023, 43(5): 1595-1606

1 引言

当疾病在某个地区出现时, 该地的疾病控制中心的首要任务就是竭尽全力防止疾病传播, 重要的预防措施之一就是通过大众传媒向人们普及正确的疾病预防知识. 在传染病传播的早期阶段, 媒体对传染病的报道能够加强公众的防范认识, 避免大众途经感染者停留的地方, 减少感染者的数量. 目前已经建立了一些数学模型研究媒体报道对传染病动力学影响[4,7,10,16,18,33,39,42]. 特别是Cui等[7]、 Caraballo等[4]、 Tchuenche等[32]、 Sun[31]等利用非线性函数f(I)=β−β1Ib+I研究媒体报道的影响, β代表在媒体报道之前, 接种者和感染者之间的接触率, 而β1代表在媒体报道后的最大降低接触率. 众所周知, 媒体的报道并不能完全阻止传染病的传播, 因此, β>β1的假设是合理的. 饱和发病率g(I)=I1+aI反映了感染者的行为变化和拥挤效应, a称为半饱和常数[11]. 当感染者数量较大时, g(I)会趋近于1a[1,3,21,25,26,28,30,34,35,38]. 文献[30]研究具有饱和发生率的传染病模型, 研究表明: 疫苗接种率提高使得感染者的密度降低. 其他发病率函数可以参考相关工作[6,13,14,17,19,24,27,36].

本文假设某个区域的易感者数量变化在一段时间内服从Logistic增长[2,8,15,20,29], 并且接种者随着时间的推移失去疫苗免疫再次回到易感人群, 利用饱和发生率描述感染者的拥挤效应, 因此建立随机SVIR模型

{dS(t)=[γS(1−SK)+θV−ζS−ρSI]dt+σ1SdB1(t),dV(t)=[ζS−θV−(β−β1Ib+I)VI1+aI−μV]dt+σ2VdB2(t),dI(t)=[ρSI+(β−β1Ib+I)VI1+aI−(μ+δ+τ)I]dt+σ3IdB3(t),dR(t)=(τI−μR)dt+σ4RdB4(t).
(1.1)

其中, S(t),V(t),I(t),R(t) 分别表示人群中t时刻易感者、接种者、感染者和恢复者的数量, γ 和K分别代表内禀增长率和环境最大容纳量; μ 表示自然死亡率; δ代表因病死亡率; τ表示感染者的恢复率; ρ表示易感者和感染者的接触率; θ 表示疫苗失效率; ζ 表示易感者接种疫苗率; a,b 都是半饱和常数; B1(t),B2(t),B3(t),B4(t)是相互独立的标准布朗运动; σ1,σ2,σ3,σ4表示白噪声的强度; (Ω,F,{Ft}t⩾是完备的概率空间, 且滤子\{\mathcal{F}_{t}\}_{t\geqslant 0}满足通常情况.

由于恢复者密度的变化不影响易感者、接种者和感染者的密度,所以模型(1.1)变换为

\begin{equation} \begin{cases} \displaystyle \mbox{d}S(t)=\left[\gamma S\Big(1-\frac{S}{K}\Big)+\theta V-\zeta S -\rho SI\right]\mbox{d}t +\sigma_{1}S\mbox{d}B_{1}(t), \\ \displaystyle \mbox{d}V(t)=\left[\zeta S-\theta V- \Big(\beta-\beta_{1}\frac{I}{b+I}\Big)\frac{ VI}{1+a I}-\mu V\right]\mbox{d}t +\sigma_{2}V\mbox{d}B_{2}(t), \\ \displaystyle \mbox{d}I(t)=\left[\rho SI +\Big(\beta-\beta_{1}\frac{I}{b+I}\Big)\frac{ VI}{1+a I}-(\mu+\delta+\tau)I\right]\mbox{d}t +\sigma_{3}I\mbox{d}B_{3}(t). \end{cases} \end{equation}
(1.2)

下面, 我们将证明模型(1.2)全局正解的存在唯一性, 寻找持久和遍历平稳分布存在的充分条件, 以及疾病灭绝的充分条件.

2 解的存在唯一性和持久性

模型(1.2)全局正解的存在性和唯一性是模型主要动力学性质之一. 对任意 t\geqslant0, 模型(1.2)的解记为\textbf{X}(t)=(S(t),V(t),I(t))^{\mbox{T}}. 定义 \mbox{d}\textbf{B}(t)=(\mbox{d}B_{1}(t), \mbox{d}B_{2}(t), \mbox{d}B_{3}(t))^{\mbox{T}}. 设\mathbb{R}^{n}是一个n维欧氏空间, X(t)是\mathbb{R}^{n}中的齐次马尔可夫过程, 则随机微分方程为

\begin{equation*} \mbox{d}X(t)=F(X(t))\mbox{d}t+\sum\limits_{l=1}^{n}g_{l}(X(t))\mbox{d}B_{l}(t), \end{equation*}

令初始值为 X(0) = X_{0}\in \mathbb{R}^{n}, g_{l}=(g^l_1,g^l_2,\cdots,g^l_n)^T 扩散矩阵为 A(X)=(a_{ij}(X))_{n\times n}, 其中 a_{ij}(X)=\sum\limits_{l=1}^{n}g_{l}^{i}(X)g_{l}^{j}(X). 定义随机微分方程的微分算子

\begin{equation*} \mathcal{L}=\sum\limits_{l=1}^{n}F_{i}(X)\frac{\partial}{\partial X_{i}}+ \frac{1}{2}\sum\limits_{l=1}^{n}a_{ij}(X)\frac{\partial^{2}}{\partial X_{i}\partial X_{j}}. \end{equation*}

根据文献[21,34,35,38], 容易得到定理2.1, 证明细节省略.

定理2.1 对于任意初值(S(0), V(0), R(0))^{\mbox{T}}\in \mathbb{R}^3_+, 模型(1.2)存在唯一的正解, 且该解将以概率1留在\mathbb{R}^3_+ .

根据文献[40,41]的结论, 得到引理2.1, 省略证明过程.

引理2.1 模型(1.2)的解\textbf{X}(t)具有以下性质

\begin{equation*} \lim\limits_{t\rightarrow \infty}\frac{1}{t}X_i(t)=0, \quad \lim\limits_{t\rightarrow \infty}\frac{1}{t}\ln X_i(t)\leqslant 0\quad (i=1,2,3) \quad\mbox{a.s.}. \end{equation*}

若 \max\{\sigma_{1}^{2}, \sigma_{2}^{2}, \sigma_{3}^{2}\}<2\mu, 则

\begin{equation*} \limsup\limits_{t\rightarrow \infty}\frac{1}{t}\int_{0}^{t} X_i(s)\mbox{d}B_{i}(s)= 0 \quad (i=1,2,3) \quad \mbox{a.s.}. \end{equation*}

依据文献[21,定理4.2]、文献[37,定理3.1]和文献[5,定理4.1]方法, 给出了疾病的持久的充分条件. 令

\begin{equation} {R}_{0}^{s}=\frac{p_{1}}{p_{2}p_{3}(1+E)}, \quad\, A=\frac{p_{1} \Big[(2\beta-\beta_{1})\Big(1-\frac{\sigma_{1}^{2}}{2\gamma}-\frac{\zeta}{\gamma}\Big)+3\rho p_{2}\Big]}{\gamma p_{2}^2p_{3} \Big(1-\frac{\sigma_{1}^{2}}{2\gamma}-\frac{\zeta}{\gamma}\Big)}, \end{equation}
(2.1)
\begin{equation} \displaystyle p_{1}=(\beta-\beta_{1})\zeta K\Big(1-\frac{\sigma_{1}^{2}}{2\gamma}-\frac{\zeta}{\gamma}\Big)^{3}, \quad p_{2}=\theta+\mu+\frac{\sigma_{2}^{2}}{2}, \quad p_{3}=\mu+\delta+\tau+\frac{\sigma_{3}^{2}}{2}. \end{equation}
(2.2)

定理2.2 若满足条件 {R}_{0}^{s}>1, \ \sigma_1^2<2(\gamma-\zeta), \ \max\{\sigma_{1}^{2}, \sigma_{2}^{2}, \sigma_{3}^{2}\}<2\mu, 则疾病是持久的, 且

\begin{equation*} \mathop{\liminf}\limits_{t\rightarrow\infty}A\langle I\rangle_{t}\geqslant(1+E)({R}_{0}^{s}-1)>0\quad \mbox{a.s.}. \end{equation*}

证 构造如下C^2函数

\begin{align*} V_{1}=\frac{a}{\mu+\delta+\tau}(S+V+I)-c_{1}\ln V-c_{2}\ln I, \quad V_{2}=\frac{2(S+V)}{3\gamma K}-\frac{\ln S}{\gamma}, \end{align*}

其中c_1和c_2是待定正常数, 对V_{1} 应用Itô公式, 得到

\begin{matrix} \mbox{d}V_{1}=\displaystyle \mathcal{L}V_{1}&\mbox{d}t +\frac{aS\sigma_{1}}{\mu+\delta+\tau}\mbox{d}B_{1}(t) +\Big(\frac{aV}{\mu+\delta+\tau}-c_{1}\Big)\sigma_{2}\mbox{d}B_{2}(t) +\Big(\frac{aI}{\mu+\delta+\tau}-c_{2}\Big)\sigma_{3}\mbox{d}B_{3}(t), \\ \displaystyle\mathcal{L}V_{1} & <E-aI -\frac{c_{1}\zeta S}{V}+c_{1}p_2+c_{2}p_3 +\Big(\beta-\frac{\beta_{1}I}{b+I}\Big)\Big(c_{1}\frac{I}{1+a I}- c_{2}\frac{V}{1+a I}\Big) \\ & < -\frac{c_{1}\zeta S}{V}- \frac{c_{2}\big(\beta-\beta_{1}\big)V}{1+a I}-(1+a I)+1+c_{1}p_2+c_{2}p_3 +E+c_{1}(2\beta-\beta_{1}) I \\ &< -3\sqrt[3]{c_{1}c_{2}(\beta-\beta_{1})\zeta S}+c_{1}p_2 +c_{2}p_3+E+c_{1}(2\beta-\beta_{1}) I+1, \end{matrix}
(2.3)

并且

\begin{equation} \displaystyle\, E=\displaystyle\mathop{\max}\limits_{S\in \mathbb R_{+}}\Big \{\frac{a \gamma S}{\mu+\delta+\tau} \Big(1-\frac{S}{K}\Big)-c_{2}\rho S\Big\}=\frac{K [a \gamma-c_{2}\rho(\mu+\delta+\tau)]^{2}} {4a \gamma(\mu+\delta+\tau)}. \end{equation}
(2.4)

对V_{2}应用Itô公式, 得到

\mbox{d}V_{2}=\mathcal{L}V_{2}\mbox{d}t +\Big(\frac{2S}{3\gamma K}-\frac{1}{\gamma}\Big)\sigma_{1}\mbox{d}B_{1}(t)+\frac{2V}{3\gamma K}\sigma_{2}\mbox{d}B_{2}(t),

其中

\begin{equation} \mathcal{L}V_{2} <\frac{2S}{3K}\Big(1-\frac{S}{K}\Big)+\frac{S}{K}-1 +\frac{\zeta}{\gamma}+\frac{\rho I}{\gamma}+\frac{\sigma_{1}^{2}}{2\gamma} <\sqrt [3] {\frac{S}{K}}+\frac{\rho I}{\gamma}-\Big(1-\frac{\sigma_{1}^{2}}{2\gamma}-\frac{\zeta}{\gamma}\Big). \end{equation}
(2.5)

定义V_{3}=V_{1}+3\sqrt [3] {c_{1}c_{2}(\beta-\beta_{1})\zeta K}V_{2}, 先对V_{3}应用Itô公式, 然后结合(2.3)和(2.5)式, 得到

\begin{matrix} \mathcal{L}V_{3} &<1+E+c_{1} p_2+c_{2}p_3+ \frac{3\rho I}{\gamma}\sqrt [3]{c_{1}c_{2}(\beta-\beta_{1})\zeta K}\notag\\ &\quad\ -3\sqrt [3] {c_{1}c_{2}(\beta-\beta_{1})\zeta K}\Big(1-\frac{\sigma_{1}^{2}}{2\gamma}-\frac{\zeta}{\gamma}\Big)+c_{1}(2\beta-\beta_{1})I. \end{matrix}
(2.6)

根据(2.2)式, 令

\begin{equation*} \displaystyle c_{1}=\frac{p_{1}}{p_{2}^{2}p_{3}}, \quad c_{2}=\frac{p_{1}}{p_{2}p_{3}^{2}}, \end{equation*}

因此

\begin{equation} \mathcal{L}V_{3}<-(1+E)({R}_{0}^{s}-1)+AI:=-\lambda+AI. \end{equation}
(2.7)

定义一个行向量\textbf{D}_1(t)和列向量 \mbox{d}\textbf{B}(t)=(\mbox{d}B_{1}(t), \mbox{d}B_{2}(t), \mbox{d}B_{3}(t))^{\mbox{T}}, 因此

\begin{equation} \displaystyle\mbox{d}V_{3} = \mathcal{L}V_{3}\mbox{d}t +\textbf{D}_1(t)\mbox{d}\textbf{B}(t), \end{equation}
(2.8)

对等式(2.8)两边同时积分, 再除以t, 结合(2.7)式得

\begin{equation} \frac{1}{t}[V_{3}(t)-V_{3}(0)]< A\langle I\rangle_{t}-\lambda+\Psi_{1}(t), \end{equation}
(2.9)

其中

\begin{matrix} \displaystyle\Psi_{1}(t)&=\frac{1}{t}\int_{0}^{t}\textbf{D}_1(s)\mbox{d}\textbf{B}(s)\notag\\ &=\frac{1}{t}\int_{0}^{t}\Big[\frac{a S(s)}{\mu+\delta+\tau}+3\sqrt[3]{c_{1}c_{2} (\beta-\beta_{1})\zeta K} \Big(\frac{2S(s)}{3\gamma K}-\frac{1}{\gamma}\Big)\Big]\sigma_{1}\mbox{d}B_{1}(s)\notag\\ &\quad+\frac{1}{t}\int_{0}^{t}\Big(\frac{a V(s)}{\mu+\delta+\tau} +\sqrt[3]{c_{1}c_{2}(\beta-\beta_{1}) \zeta K}\frac{2V(s)}{\gamma K}-c_{1}\Big)\sigma_{2}\mbox{d}B_{2}(s)\notag\\ &\quad +\frac{1}{t}\int_{0}^{t}\Big(\frac{a I(s)}{\mu+\delta+\tau}-c_{2}\Big)\sigma_{3}\mbox{d}B_{3}(s). \end{matrix}
(2.10)

根据引理2.1和强大数定理[23] 可知\limsup\limits_{t\rightarrow\infty}\Psi_{1}(t) = 0, 对(2.9)式取下极限得\mathop{\liminf}\limits_{t\rightarrow\infty}A\langle I\rangle_{t}> \lambda>0. 证毕.

3 平稳分布的存在性

根据文献[38,定理3.3], 文献[5,注5.1], 得到定理3.1.

定理3.1 若{R}_{0}^{s}>1, 则模型(1.2)存在平稳分布, 并且具有遍历性.

证 定义有界集为

\begin{equation} \displaystyle D_{\varepsilon}=\Big\{\textbf{X}\in \mathbb R_{+}^{3}, \varepsilon\leqslant S\leqslant \frac{1}{\varepsilon}, \varepsilon^{2}\leqslant V \leqslant \frac{1}{\varepsilon^{2}}, \varepsilon\leqslant I \leqslant \frac{1}{\varepsilon} \Big\}, \end{equation}
(3.1)

这里的\varepsilon 是非常小的正常数, 且满足以下条件

\begin{matrix} \displaystyle &\frac{AM\rho\varepsilon}{\mu+\delta+\tau}\leqslant \min\Big\{m(m+2)(\mu+\delta+\tau), \frac{\gamma(m+3)}{2K}, \frac{m+3}{m+2}\Big\}, \end{matrix}
(3.2)
\begin{matrix} &F+1\leqslant \min\Big\{\frac{\zeta}{\varepsilon}, \frac{\gamma}{4K\varepsilon^{m+3}}, \frac{m(\mu+\delta+\tau)}{2\varepsilon^{m+2}}, \frac{\mu m}{\varepsilon^{2m+4}}\Big\}. \end{matrix}
(3.3)

模型(1.2)扩散矩阵为

\displaystyle\tilde{A}=\mbox{diag}\{\sigma_{1}^{2}S^{2}, \sigma_{2}^{2}V^{2}, \sigma_{3}^{2}I^{2}\}=(a_{ij})_{3\times 3},

设模型(1.2)的扩散项最小值为

\eta=\mathop{\min}\limits_{\textbf{X}\in D_{\varepsilon}}\{\sigma_{1}^{2}S^{2}, \sigma_{2}^{2}V^{2}, \sigma_{3}^{2}I^{2}\}>0,

对任意的 \textbf{X}\in D_{\varepsilon}, \xi=(\xi_{1}, \xi_{2}, \xi_{3})^{\mbox{T}}\in \mathbb R_{+}^{3}, 都有

\begin{equation*} \begin{aligned} \displaystyle \sum\limits_{i, j=1}^{n}a_{ij}\xi_{i}\xi_{j} =(\xi_{1}, \xi_{2}, \xi_{3})\tilde{A}(\xi_{1}, \xi_{2}, \xi_{3})^{{\mbox{T}}} =(\sigma_{1}S)^{2}\xi_{1}^{2}+(\sigma_{2}V)^{2}\xi_{2}^{2}+(\sigma_{3}I)^{2}\xi_{3}^{2} \geqslant\eta\|\xi\|^{2}, \end{aligned} \end{equation*}

这意味着文献[5,注5.1]条件(i)已满足.

构造Lyapunov函数

W=M(V_3+V_4)+V_5+V_6,

其中

\begin{equation*} V_4=\frac{A}{\mu+\delta+\tau}(V+I), \quad V_5=\frac{1}{m+2}(S+I+V)^{m+2}, \quad V_6=-\ln V, \end{equation*}

m>0 是充分小的常数满足

\begin{equation} \displaystyle m<\frac{\mu-0. 5(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2})} {\mu+0. 5(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2})}, \end{equation}
(3.4)

M>0 是充分大的常数满足

\begin{equation} \displaystyle -M\lambda+B+\theta+\mu+\frac{\sigma_{2}^{2}}{2}+\frac{2\beta-\beta_{1}}{a}\leqslant -2. \end{equation}
(3.5)

显然, W(\textbf{X}) 是连续函数, 所以存在最小值W(\bar{\textbf{X}}), 因此可构造非负的 C^{2} 函数

\begin{equation} Q=M(V_{3}+V_{4})+V_{5}+V_{6}-W(\bar{\textbf{X}}). \end{equation}
(3.6)

依次对V_{4}, V_{5}, V_{6} 应用 Itô 公式, 可得

\begin{matrix} \displaystyle\mathcal{L}V_{4} &=\frac{A}{\mu+\delta+\tau}\Big[\zeta S-(\theta+\mu)V +\rho SI-(\mu+\delta+\tau)I\Big]\notag\\ &<\frac{A\zeta S}{\mu+\delta+\tau}+\frac{A\rho SI}{\mu+\delta+\tau}-AI, \end{matrix}
(3.7)
\begin{matrix} \quad\displaystyle\mathcal{L}V_{5}&=(S+V+I)^{m+1} \Big[\gamma S\Big(1-\frac{S}{K}\Big)-\mu V-(\mu+\delta+\tau)I\Big]\notag\\ &\quad+\frac{m+1}{2}(S+V+I)^{m}(\sigma_{1}^{2}S^{2}+\sigma_{2}^{2}V^{2}+\sigma_{3}^{2}I^{2})\notag\\ &<\gamma S(S+V+I)^{m+1}-\frac{\gamma}{K}S^{m+3}-\mu V^{m+2}-(\mu+\delta+\tau)I^{m+2}\notag\\ &\quad+\frac{m+1}{2}(S+V+I)^{m+2}(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2}), \end{matrix}
(3.8)
\begin{matrix} \mathcal{L}V_{6}&=\displaystyle -\frac{\zeta S}{V}+\Big(\beta-\frac{\beta_{1}I}{b+I}\Big) \frac{I}{1+aI}+p_2 \leqslant-\frac{\zeta S}{V}+\frac{2\beta-\beta_{1}}{a}+p_2, \end{matrix}
(3.9)

其中

\begin{align*} \displaystyle B=\mathop{\max}\limits_{\textbf{X}\in \mathbb R_{+}^{3}} &\Big\{-\frac{\gamma}{2K}S^{m+3}-\mu(1-m) V^{m+2}-(\mu+\delta+\tau)(1-m)I^{m+2}\\ &\quad+\gamma S(S+V+I)^{m+1}+\frac{AM\zeta S}{\mu+\delta+\tau} +\frac{m+1}{2}(S+V+I)^{m+2}(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2})\Big\}, \end{align*}

结合(2.7)式和(3.7)-(3.9)式, 我们得到

\begin{matrix} \displaystyle\mathcal{L}Q &< \frac{MA\rho SI} {\mu+\delta+\tau}-M\lambda -\frac{\gamma}{2K}S^{m+3}-\mu m V^{m+2}\notag\\ &\quad\ -m(\mu+\delta+\tau)I^{m+2}+B-\frac{\zeta S}{V}+\frac{2\beta-\beta_{1}}{a}+p_2. \end{matrix}
(3.10)

将\mathbb R_{+}^{3}\backslash D_{\varepsilon} 分成六个子区间, 证明 \mathcal{L}Q 在这六个区间内满足\mathcal{L}Q<-1, 其中\mathbb R_{+}^{3}\backslash D_{\varepsilon}=D_{1}\cup D_{2}\cup D_{3}\cup D_{4}\cup D_{5}\cup D_{6}.

\begin{gathered} D_{1} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},0＜S＜\varepsilon\right\}, D_{2} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},0＜I＜\varepsilon\right\}, \\ D_{3} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},S>\varepsilon,I>\varepsilon,0＜V＜\varepsilon^{2}\right\},D_{4} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},S>\frac1\varepsilon\right\}, \\ D_{5} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},I>\frac1\varepsilon\right\}, D_{6} =\left\{\mathbf{X}\in\mathbb{R}_{+}^{3},V>\frac1{\varepsilon^{2}}\right\}. \end{gathered}
(3.11)

情形 1 若\textbf{X}\in D_{1}, 则

\begin{equation*} SI\leqslant\varepsilon I\leqslant \varepsilon\frac{m+1+I^{m+2}}{m+2}, \end{equation*}

根据(3.2), (3.5), (3.10)式, 可得

\begin{equation*} \mathcal{L}Q<-2+\frac{AM\rho \varepsilon(m+1)}{(\mu+\delta+\tau)(m+2)} +\Big[\frac{AM\rho\varepsilon}{(\mu+\delta+\tau)(m+2)} -m(\mu+\delta+\tau)\Big]I^{m+2} \leqslant-1. \end{equation*}

情形 2 若\textbf{X}\in D_{2}, 则

\begin{equation*} SI\leqslant\varepsilon S\leqslant \varepsilon\frac{m+2+S^{m+3}}{m+3}, \end{equation*}

根据(3.2), (3.5), (3.10)式, 可得

\begin{equation*} \mathcal{L}Q\displaystyle<-2+\frac{AM \rho\varepsilon(m+2)} {(\mu+\delta+\tau)(m+3)}+\Big(\frac{AM \rho\varepsilon} {(\mu+\delta+\tau)(m+3)}-\frac{\gamma}{2K}\Big)S^{m+3}\leqslant-2+1=-1. \end{equation*}

情形 3 若\textbf{X}\in D_{3}, 则通过(3.3), (3.10)式可得

\begin{equation*} \mathcal{L}Q\displaystyle< -\frac{\zeta S}{V} +F< -\frac{\zeta}{\varepsilon}+F\leqslant-1, \end{equation*}

其中

\begin{equation*} F=\displaystyle\mathop{\sup}\limits_{\textbf{X}\in \mathbb R_{+}^{3}}\left\{\frac{AM\rho SI}{\mu+\delta+\tau} -\frac{\gamma}{4K}S^{m+3}-\frac{m}{2}(\mu+\delta+\tau)I^{m+2}+B+p_2+\frac{2\beta-\beta_{1}}{a}\right\}. \end{equation*}

情形 4 若 \textbf{X}\in D_{4}, 则通过(3.3), (3.10)式可得

\begin{equation*} \mathcal{L}Q<-\frac{\gamma}{4K}S^{m+3}+F <-\frac{\gamma}{4K\varepsilon^{m+3}}+F \leqslant-1. \end{equation*}

情形 5 若\textbf{X}\in D_{5}, 则通过(3.3), (3.10)式可得

\begin{equation*} \mathcal{L}Q<-\frac{m}{2}(\mu+\delta+\tau)I^{m+2}+F <-\frac{(\mu+\delta+\tau)m}{2 \varepsilon^{m+2}}+F \leqslant-1. \end{equation*}

情形 6 若\textbf{X}\in D_{6}, 则通过(3.3), (3.10)式可得

\begin{equation*} \mathcal{L}Q<-\mu m V^{m+2}+F <-\frac{\mu m}{\varepsilon^{2m+4}}+F \leqslant-1. \end{equation*}

所以文献[5,注5.1]条件(ii)已满足; 因此模型(1.2)存在遍历的平稳分布. 证毕.

4 疾病的灭绝性

定理4.1 若满足条件

\begin{equation} \displaystyle \displaystyle {R}_{0}^{e}=\frac{\gamma K(\rho\mu+\rho\theta+2\beta\zeta-\beta_{1}\zeta)}{4\big(\mu+\delta+\tau+0. 5\sigma_{3}^{2}\big)\zeta\mu}<1, \quad \displaystyle \max\{\sigma_{1}^{2}, \sigma_{2}^{2}, \sigma_{3}^{2}\}<2\mu, \end{equation}
(4.1)

则疾病将指数灭绝.

证 对模型(1.2) 的第一个方程两边同时从0到t 积分, 然后再除以t, 最后进行适当地放缩

\begin{equation} \displaystyle \frac{1}{t}[S(t)-S(0)]<\frac{\gamma K}{4}+\theta\langle V\rangle_{t}-\zeta\langle S\rangle_{t}+\frac{\sigma_{1}}{t}\int_{0}^{t}S(s)\mbox{d}B_{1}(s), \end{equation}
(4.2)

根据引理2.1, 可知

\begin{equation*} \limsup\limits_{t\rightarrow\infty} \left\{\frac{\sigma_{1}}{t}\int_{0}^{t}S(s)\mbox{d}B_{1}(s)-\frac{1}{t}[S(t)-S(0)]\right\}=0. \end{equation*}

对(4.2)式两边同时取上确界极限

\begin{equation} \mathop{\limsup}\limits_{t\rightarrow\infty}\langle S\rangle_{t} <\frac{\gamma K}{4\zeta}+\frac{\theta}{\zeta} \mathop{\limsup}\limits_{t\rightarrow\infty}\langle V\rangle_{t}. \end{equation}
(4.3)

定义一个行向量 \textbf{D}_2(t)=(\sigma_{1}S, \sigma_{2}V, \sigma_{3}I), 对模型(1.2)三个方程之和先积分再除以t可得

\begin{equation} \displaystyle \mu\langle V\rangle_{t}<\Big\langle\gamma S\Big(1-\frac{S}{K}\Big)\Big\rangle_{t} +\Psi_2(t)<\frac{\gamma K}{4}+\Psi_2(t), \end{equation}
(4.4)

其中

\begin{equation*} \displaystyle\Psi_2(t)=\frac{1}{t}\int_{0}^{t}\textbf{D}_2(s)\mbox{d}\textbf{B}(s) -\frac{1}{t}[S(t)-S(0)]-\frac{1}{t}[V(t)-V(0)]-\frac{1}{t}[I(t)-I(0)], \end{equation*}

由引理2.1可知 \limsup\limits_{t\rightarrow\infty}\Psi_2(t)=0, 对(4.4)式取上确界极限得

\begin{equation} \mathop{\limsup}\limits_{t\rightarrow\infty}\langle V\rangle_{t}<\frac{\gamma K}{4\mu}. \end{equation}
(4.5)

现在对函数\ln I(t) 应用Itô公式, 得到

\begin{equation} \mbox{d}\ln I(t)=\mathcal{L}\ln I(t) \mbox{d}t+\sigma_{3}\mbox{d}B_{3}(t), \end{equation}
(4.6)

其中

\begin{equation*} \mathcal{L}\ln I(t)=\rho S+\Big(\beta-\frac{\beta_{1}I}{b+I}\Big)\frac{VI}{1+aI}-p_{3}. \end{equation*}

对(4.6)式两侧同时0到t 积分, 再除以t, 得

\begin{equation} \displaystyle \frac{1}{t}[\ln I(t)-\ln I(0)] <\rho\langle S\rangle_{t} +(2\beta-\beta_{1})\langle V\rangle_{t}-p_3+\frac{\sigma_{3}B_{3}(t)}{t}, \end{equation}
(4.7)

根据强大数定理[23] 可得 \lim\limits_{t\rightarrow\infty}\frac{\sigma_{3}B_{3}(t)}{t}=0, 对 (4.7) 式两侧取上极限, 则有

\begin{matrix} \mathop{\limsup}\limits_{t\rightarrow\infty}\frac{1}{t}\ln I(t) &<\rho\mathop{\limsup}\limits_{t\rightarrow\infty}\langle S\rangle_{t} +(2\beta-\beta_{1})\mathop{\limsup}\limits_{t\rightarrow\infty}\langle V\rangle_{t} -p_3\notag\\ &<\frac{\rho\gamma K}{4\zeta}\Big(1+\frac{\theta}{\mu}\Big)+(2\beta-\beta_{1})\frac{\gamma K}{4\mu} -p_3\notag\\ &=p_3({R}_{0}^{e}-1)<0. \end{matrix}
(4.8)

这说明 \mathop{\lim}\limits_{t\rightarrow\infty}I(t)=0, 证明完成.

5 数值模拟

Milstein高阶方法[12]和截断的Euler-Maruyama 方法[9,22]都可以应用于模型(1.2)的数值模拟, 但是两种方法的模拟都具有很高的相似性. 因此本文采用随机微分方程的Milstein 高阶方法写出模型(1.2)离散化方程, 细节见文献[21,24,37].

例 5.1 通过数值模拟展现持久性的理论结果. 设模型(1.2)初值为S(0)=1, V(0)=0. 8, I(0)=0.8, 其他参数取值为\gamma=0. 5, K=5. 5, \beta=0. 9, \beta_{1}=0. 1, \tau=0. 1, \delta=0. 08, \zeta=0. 15, \theta=0. 1, \rho=0. 085, \mu=0. 15, \sigma_{1}=0. 05, \sigma_{2}=0. 05, \sigma_{3}=0. 05, a=0. 85, b=2. 这组参数值满足定理2.2的条件 R_{0}^{s}\approx2. 6024>1, 0. 15=\mu>0. 5(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2})=0. 00125, 0. 7=2(\gamma-\zeta)>\sigma_1^2=0. 0025.

如图1所示, 易感者、接种者和感染者密度在长时间内平均持久; 如图2所示, 当\beta_{1}变化时, 易感者和接种者密度增加, 感染者的密度减少, 但是易感者、接种者和感染者密度仍然是平均持久的; 从图3和图4注意到, 当\zeta和\theta 分别增加时, 易感者和接种者的密度具有相反的性质. 研究表明: 在传染病传播的早期阶段, 媒体报道的干预有利于疾病控制.

图1

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图1   易感者、 接种者、感染者的持久性


图2

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图2   当 \beta_{1} 增加时, 易感者、感染者、接种者的密度变化


图3

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图3   当\zeta增加时, 易感者和接种者密度变化


图4

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图4   当\theta增加时, 易感者、接种者密度变化


例 5.2 疾病的灭绝性在此例子中讨论, 设模型(1.2)初值S(0)=1, V(0)=0.8, I(0)=0.8, 以及其他参数取值为 \gamma=0.65, K=1.25, \beta=0.9, \beta_{1}=0.1, \tau=0.65, \delta=0.35, \zeta=0.3, \theta=0.2, \rho=0.8, \mu=0.5, \sigma_{1}=0.05, \sigma_{2}=0.05, \sigma_{3}=0.05, a=0.85, b=0.3. 这些参数值满足定理4.1的条件 R_{0}^{e}\approx0.9628<1,\ 0.5=\mu>0.5(\sigma_{1}^{2}\vee\sigma_{2}^{2}\vee\sigma_{3}^{2})=0.0012.

研究表明: 当\zeta, \delta, \tau, \mu, \sigma_{3} 增加和 \theta 减少时(图5-图7), 感染者密度趋于灭绝的时间缩短; 当\beta_{1}增加时, 媒体报道对疾病的灭绝有一定的影响(图8).

图5

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图5   当\zeta和\theta 增加时, 疾病的灭绝性


图6

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图6   当\tau和\sigma_{3} 增加时, 疾病的灭绝性


图7

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图7   当\delta和\mu分别增加时, 疾病的灭绝性


图8

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图8   当\beta_{1}增加时, 疾病的灭绝性


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