生态环境的不断破坏和国际交流的日趋频繁是影响传染病的暴发和流行的重要因素. 近年来, 新冠肺炎和非典型肺炎等新型传染病不断暴发以及肝炎、肺结核等原有疾病的反复发作给人类的健康和社会经济的发展带来了严重威胁. 运用数学模型研究传染病的传播规律和流行趋势对疾病的预防和控制具有重要的理论和实践意义.

疫苗接种是控制和消除传染病最有效的方法之一, 为了寻求疾病的最佳控制策略, 越来越多的学者致力于具有预防接种的数学模型的动力学研究. 早期的传染病动力学中关于预防接种的工作主要是建立确定性模型, 并围绕平衡点的存在性、稳定性及多种分歧现象的存在性等问题进行研究, 如文献[1-8] 等.

然而, 疾病传播过程中往往受到复杂环境中不确定因素的影响. 具有不同形式噪声干扰的传染病模型近年来受到普遍关注, 如文献[9-12]等. 赵亚男等人^[9]在文献[3]的基础上对感染系数进行随机扰动并提出了一类SIVS模型, 研究发现当噪声较大时, 不论基本再生数是否小于1, 感染人群的数量都将指数衰减为0. 刘群和蒋达清^[10]采取了同样的扰动方法建立了一类接种后不完全免疫的SIVS模型, 给出了疾病灭绝的条件, 且发现较大的噪声能够有效地抑制疾病的流行. 在文献[11-12]中, 赵亚男和蒋达清假设随机扰动项与易感者类($ S $)、感染者类($ I $)和接种者类($ V $)成比例建立了一类自治SIVS随机模型, 研究了由确定型模型的基本再生数决定的遍历性等性质, 并且分别给出了决定疾病灭绝及持久的阈值条件.

以上论文中随机干扰项均为系统状态的线性函数, 在文献[13]中张伟伟等人使用白噪声模拟环境波动, 提出了一类具有非线性扰动的随机非自治SIRI模型, 给出了解的随机有界性、随机持久性和疾病持续存在的解析结果, 且对于具有马尔科夫转换的系统得到了遍历平稳分布和正常返存在的充分条件.

事实上, 影响疾病传播的环境噪声不仅有上述的白噪声(一种功率谱密度为常数的随机过程, 通常用布朗运动广义均方导数表示), 还有一种强度大但数量少的有色噪声, 又称为电报信号噪声^[14-15]. 在现实生活中, 很多疾病的传播受季节、气候、政治人文因素的改变影响较大, 如Covid-19, SARS, 流感等. 这一类因素的功率谱密度函数不为常数, 不能单纯地用布朗运动来描述其随机性, 因此一些学者在传染病建模过程中运用Markov切换描述其随机性, 如文献[16-17]等. 本文拟运用Markov切换描述有色噪声, 建立具有非线性随机扰动及预防接种策略的传染病模型, 并通过李雅普诺夫函数、Chebyshev's不等式、Has'minskii理论、强大数定理等方法研究模型的持久性、灭绝性和遍历性等性质.

胡俊娜^[18]等人提出了转换机制(环境状态之间的转换通常是无记忆的, 下一次切换停留时间服从指数分布)下具有非线性发病率的随机SIVS传染病模型

(2.1) $ \begin{equation} \left\{ \begin{array}{rl} {\rm d}S = &[(1-q_{r(t)})A_{r(t)}-\beta_{r(t)}f(S(t)g(I(t))-(\mu_{r(t)}+p_{r(t)})S(t)+\gamma_{r(t)}I(t)\\ &+\varepsilon_{r(t)}V(t)]{\rm d}t+\sigma_{1r(t)}S(t){\rm d}B_1(t), \\ {\rm d}I = &[\beta_{r(t)}f(S(t)g(I(t))-(\mu_{r(t)}+\gamma_{r(t)}+\alpha_{r(t)})I(t)]{\rm d}t+\sigma_{2r(t)}I(t){\rm d}B_2(t), \\ {\rm d}V = &[q_{r(t)}A_{r(t)}+p_{r(t)}S(t)-(\mu_{r(t)}+\varepsilon_{r(t)})V(t)]{\rm d}t+\sigma_{3r(t)}V(t){\rm d}B_3(t), \end{array} \right. \end{equation} $

其中$ S(t), I(t), V(t) $分别为$ t $时刻易感人群, 感染人群及接种疫苗后获得免疫人群的数量, $ q_{r(t)} $为新生儿的接种率, $ A_{r(t)} $为人口总输入率, $ \beta_{r(t)} $为疾病传播率, $ \mu_{r(t)} $为自然死亡率, $ p_{r(t)} $为易感者的接种免疫率, $ \alpha_{r(t)} $为因疾病造成的死亡率, $ \gamma_{r(t)} $为患者的康复率, $ \varepsilon_{r(t)} $为康复者的免疫失去率, $ r(t) $为完备概率空间$ (\Omega, {\cal F}, \{{\cal F}_t\}_{t\geq0}, {\Bbb P}) $上的右连续马尔科夫链, 它的状态空间为$ \hbar = \{1, 2, 3\cdots , N\} $.定义马尔科夫链的生成元$ \Gamma = (\gamma_{ij})_{N\times N} $为

$ {\Bbb P}\{r(t+\triangle) = j|r(t) = i\} = \left\{ \begin{array}{rl} &\gamma_{ij}\triangle+\circ(\triangle), i \neq j, \\ &1+\gamma_{ij}\triangle+\circ(\triangle), i = j, \\ \end{array} \right. $

其中$ \triangle>0, \gamma_{ij}\geq0(i \neq j) $是从$ i $到$ j $的转移概率, $ \gamma_{ii} = -\sum\limits_{i\neq j} \gamma_{ij}. $作者通过对函数$ f $和$ g $做一般性的假设, 讨论了模型(1.1)解的遍历平稳分布性质及灭绝性和平均持久性.

不难验证, 当$ g(I(t)) = I^{m} $时, $ \frac{g(I)}{I} = I^{m-1} $, 若$ m>1, $则$ \frac{g(I)}{I} $关于$ I $单调非减, 作者未对此情况进行讨论. 基于此, 本文拟考虑具有转换机制的非线性发病率$ \beta(r(t))S I^{m}(m>1) $及非线性随机扰动项

$ S(\sigma_{11}(r(t))+\sigma_{12}(r(t))S){\rm d}B_1(t), I(\sigma_{21}(r(t))+\sigma_{22}(r(t))I){\rm d}B_2(t), $

$ V(\sigma_{31}(r(t))+\sigma_{32}(r(t))V){\rm d}B_3(t). $

建立如下SIVS模型

(2.2) $ \begin{equation} \left\{ \begin{array}{rl} {\rm d}S = &[(1-q(r(t)))A(r(t))-\beta(r(t))S I^{m}-(\mu(r(t))+p(r(t)))S\\ &+\gamma(r(t))I+\varepsilon(r(t))V]{\rm d}t+S(\sigma_{11}(r(t))+\sigma_{12}(r(t))S){\rm d}B_1(t), \\ {\rm d}I = &[\beta(r(t))SI^{m}-(\mu(r(t))+\gamma(r(t))+\alpha(r(t)))I]{\rm d}t\\ &+I(\sigma_{21}(r(t))+\sigma_{22}(r(t))I){\rm d}B_2(t), \\ {\rm d}V = &[q(r(t))A(r(t))+p(r(t))S-(\mu(r(t))+\varepsilon(r(t)))V]{\rm d}t\\ &+V(\sigma_{31}(r(t))+\sigma_{32}(r(t))V){\rm d}B_3(t), \end{array} \right. \end{equation} $

其中$ B_i(t)\ (i = 1, 2, 3) $为标准布朗运动, $ \sigma_{ij}(i = 1, 2, 3, j = 1, 2) $为白噪声强度, 其余参数与模型(2.1)中的意义相同, 且假设布朗运动$ B_i(t) $独立于马尔科夫链$ r(t), $进一步假设$ r(t) $不可约, 这表明马尔科夫链$ r(t) $有唯一的平稳分布$ \pi = \{\pi_1, \pi_2, \cdots , \pi_N\}. $平稳分布在$ \sum\limits_{h = 1}^{N} \pi_h = 1 $和$ \pi_h>0 $的情况下由$ \pi \Gamma $确定, 其中$ h\in\hbar. $本文假设$ i\neq j, \gamma_{ij}>0. $

为了讨论方便, 假设$ A(r(t)), \beta(r(t)), \mu(r(t)), \gamma(r(t)) $为正常数, 其余参数为非负常数, 且$ q(r(t))<1, $记$ \mathbb{R} _+^{n} = \{(x_1, x_2, \cdots , x_n)\in \mathbb{R} ^{n}:x_i>0, i = 1, 2, \cdots , n\}. $对于任意的向量$ g = \{g(1), g(2), \cdots g(N)\}, $定义$ \hat{g} = \min\limits_{k\in\hbar}\{g(k)\} $和$ \check{g} = \max\limits_{k\in\hbar}\{g(k)\}. $对$ [0, +\infty] $上的可积函数$ f, $定义

$ \langle f\rangle_{t} = \frac{1}{t}\int_0^tf(s){\rm d}s, f^{u} = \sup\limits_{t\in[0, \infty]}f(t), f^{l} = \inf\limits_{t\in[0, \infty]}f(t). $

除了各种环境噪声外, 在现实中, 环境的周期变化也是影响传染病传播的一个重要因素, 基于系统(2.2), 忽略有色噪声, 我们建立如下具有周期系数的非自治传染病模型

(2.3) $ \begin{equation} \left\{ \begin{array}{rl} {\rm d}S = &[(1-q(t))A(t)-\beta(t)SI^{m}-(\mu(t)+p(t))S+\gamma(t)I+\varepsilon(t)V]{\rm d}t\\ &+S(\sigma_{11}(t)+\sigma_{12}(t)S){\rm d}B_1(t), \\ {\rm d}I = &[\beta(t)SI^{m}-(\mu(t)+\gamma(t)+\alpha(t))I]{\rm d}t+I(\sigma_{21}(t)+\sigma_{22}(t)I){\rm d}B_2(t), \\ {\rm d}V = &[q(t)A(t)+p(t)S-(\mu(t)+\varepsilon(t))V]{\rm d}t+V(\sigma_{31}(t)+\sigma_{32}(t)V){\rm d}B_3(t), \\ \end{array} \right. \end{equation} $

其中所有系数均为正$ T $ -周期连续函数.

为了讨论模型(2.2)-(2.3)的性质, 我们引入如下结论. 首先考虑随机微分方程

(2.4) $ \begin{equation} \left\{ \begin{array}{rl} {\rm d}x(t) = &f(x(t), t){\rm d}t+g(x(t), t){\rm d}B(t), \\ x(0) = &x_{0}. \end{array} \right. \end{equation} $

引理2.1^[19]  假设系统(2.4)的系数在$ t $上是$ T $ -周期的, 且满足线性增长条件和在每一个区域$ D_l\times[0, \infty), l>0 $满足Lipschitz条件, 进一步假设存在一个函数$ V = V(t, x) $在$ \mathbb{R} _n\times[0, \infty) $上对$ x $是二次连续可微的, 对$ t $是一次连续可微的, 在$ t $上是$ T $ -周期的且满足下列条件

ⅰ) $ \inf\limits_{\|x\|>l}V(t, x)\rightarrow \infty , $当$ l\rightarrow \infty, $

ⅱ) $ LV(t, x)\leq-1, \|x\|>l, $

则系统(2.4)存在$ T $ -周期解, 微分算子$ L $由下式定义

$ L = V_t+V_xf(x, t)+\frac{1}{2}{\rm Trace}[g^{T}(t)V_{xx}g(x, t)], $

其中

$ V_t = \frac{\partial V}{\partial t}, V_x = (\frac{\partial V}{\partial x_1}, V_x = \frac{\partial V}{\partial x_2}, \cdots , V_x = \frac{\partial V}{\partial x_N}), V_{xx}(\frac{\partial^{2}V}{\partial x_i\partial x_j})_{n\times n}. $

为了给出状态转换下微分方程的结果, 定义如下方程

(2.5) $ \begin{equation} \left\{ \begin{array}{rl} {\rm d}X(t) = &f(X(t), r(t)){\rm d}t+g(X(t), r(t)){\rm d}B(t), \\ x(0) = &x_{0}, \quad r(0) = r_{0}, \end{array} \right. \end{equation} $

其中$ f:\mathbb{R} ^{n}\times\hbar\rightarrow \mathbb{R} ^{n}, g:\mathbb{R} ^{n}\times\hbar\rightarrow \mathbb{R} ^{n\times l}. $设$ A(x, k) = g(x, k)g^{T}(x, k) = (A_{ij})_{n\times n}, $$ V:\mathbb{R} \times\hbar\rightarrow \mathbb{R} ^{n} $均为二阶连续可微函数. 定义算子

$ L = \sum\limits^{n}_{i = 1}f_i(x, k)\frac{\partial}{\partial x_i}+\frac{1}{2}\sum\limits^{n}_{i, j = 1}A_{ij}\frac{\partial^{2}}{\partial x_ix_j}+\sum\limits_{i\neq k\in\hbar}\gamma_{ki}[V(x, i)-V(x, k)].\\ $

引理2.2^[20]  若满足以下条件

ⅰ) 对任意的$ i\neq j, \gamma_{ij}>0, $

ⅱ)对于任意的$ k\in\hbar, D(x, k) = (d_{ij}(x, k)) $是对称的, 且对于任意的$ x, \xi\in\mathbb{R} ^n $满足$ \sigma|\xi|^{2}\leq\langle D(x, k)\xi, \xi\rangle\leq\sigma^{-1}|\xi|^{2}, $其中常数$ \sigma\in[0, 1], $

ⅲ) 存在闭包非空的开集$ D, $对于任意的$ k\in\hbar, $存在二阶连续可微的非负函数$ V(\cdot, k):D^{c}\rightarrow \mathbb{R} , $当$ \alpha>0 $时, 对任意的$ (x, k)\in D^{c}\times\hbar, $有$ LV(\cdot, k)\leq-\alpha, $ 则系统(2.5)是遍历和正常返的, 也就是说有唯一的平稳分布$ \mu(\cdot, \cdot), $对任意Borel可测函数$ f:\mathbb{R} \times\hbar\rightarrow \mathbb{R} ^{n} $满足$ \sum\limits^{N}_{k = 1}\int_{\mathbb{R} ^{n}}|f(x, k)|\mu(dk, k)<+\infty, $有

$ {\Bbb P}\bigg\{\lim\limits_{t\rightarrow \infty}\frac{1}{t}\int_0^tf(X(s), r(s)){\rm d}s = \sum\limits^{N}_{k = 1}\int_{\mathbb{R} ^{n}}|f(x, k)|\mu(dk, k)\bigg\} = 1. $

定义1^[13] 若对于任意的$ \delta\in(0, 1) $存在正常数$ \varsigma = \varsigma(\delta), $使得对于任意初始值$ X(0) = X_0\in\mathbb{R} _+^{3}, $系统(2.3)的解$ X(t) $具有以下性质

$ \liminf\limits_{t\rightarrow \infty}{\Bbb P}[|X(t)|>\varsigma]<\delta, $

则称$ X(t) $是随机最终有界的.

定义2^[21] 若对于任意的$ \delta\in(0, 1) $存在一对正常数$ \varsigma = \varsigma(\delta), \chi = \chi(\delta), $使得对于任意初始值$ X(0) = X_0\in\mathbb{R} _+^{3}, $系统(2.3)的解$ X(t) $具有以下性质

$ \liminf\limits_{t\rightarrow \infty}{\Bbb P}[|X(t)|\leq\varsigma]\geq1-\delta, \liminf\limits_{t\rightarrow \infty}{\Bbb P}[|X(t)|>\chi]\geq1-\delta, $

则称$ X(t) $是随机持久的.

定义参数

$ \begin{eqnarray*} \lambda_1& = &\mu^{l}-(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u}), \\ C& = &A^{u}+2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u}) +\frac{1}{4A^{l}}(\mu^{l}+\mu^{u}+\alpha^{u}+(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})\\ &&-(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u}))^{2}. \end{eqnarray*} $

定理3.1 若$ \lambda_1>0, $且对于任意的初值$ (S(0), I(0), V(0))\in \mathbb{R} _+^{3}, $系统(2.3)的解随机最终有界且随机持久.

证  定义$ V = N+\frac{1}{N}, N = S+I+V, $对于$ X(t)\in\mathbb{R} _+^{3}, $当$ |X(t)|\rightarrow \infty $时, 则$ V(X(t))\rightarrow \infty. $根据It$ \hat{\rm o} $公式可得

$ \begin{eqnarray*} LV& = &A(t)-\mu(t)N-\alpha(t)I-\frac{1}{N^{2}}(A(t)-\mu(t)N-\alpha(t)I)+\frac{1}{N^{3}} [S^{2}(\sigma_{11}(t)+\sigma_{12}(t)S)^{2}\\ && +I^{2}(\sigma_{21}(t)+\sigma_{22}(t)I)^{2}+V^{2}(\sigma_{31}(t) +\sigma_{32}(t)V)^{2}]\\ &\leq & A^{u}-\mu^{l}N-\frac{A^{l}}{N^{2}}+\frac{\mu^{u}}{N}+\frac{\alpha^{u}(S+I+V)}{N^2}+\frac{1}{N^{3}}\Big[(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})N^2 \\ &&+(2\sigma_{11}^{u}\sigma_{12}^{u}S^2+2\sigma_{21}^{u}\sigma_{22}^{u}I^3+2\sigma_{31}^{u}\sigma_{32}^{u}V^3)N^3+(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})N^4\Big]\\ &\leq & A^{u}-\mu^{l}N-\frac{A^{l}}{N^{2}}+\frac{\mu^{u}}{N}+\frac{\alpha^{u}}{N}+\frac{1}{N^{3}}\Big[(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})N^2 \\ &&+2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u})N^3+(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})N^4\Big]\\ &\leq & A^{u}-\mu^{l}N-\frac{A^{l}}{N^{2}}+\frac{\mu^{u}}{N}+\frac{\alpha^{u}}{N}+\frac{(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})}{N} +2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u})\\ &&+(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})N\\ &\leq&-[\mu^{l}-(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})](N+\frac{1}{N})+A^{u}+2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u})-\frac{A^{l}}{N^{2}}\\ &&+\frac{1}{N}\Big(\mu^{l}+\mu^{u}+\alpha^{u}+(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})-(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})\Big)\\ &\leq&-\lambda_1V+A^{u}+2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u})-A^{l}\bigg(\frac{l}{N^{2}}-\frac{E}{NA^{l}}+\frac{E^2}{4A^{2l}}\bigg)+\frac{E^2}{4A^{l}}\\ &\leq&-\lambda_1V+A^{u}+2(\sigma_{11}^{u}\sigma_{12}^{u}\vee\sigma_{21}^{u}\sigma_{22}^{u}\vee\sigma_{31}^{u}\sigma_{32}^{u})-A^{l}\bigg(\frac{l}{N}-\frac{E}{2A^{l}}\bigg)^2+\frac{E^2}{4A^{l}}\\ &\leq&-\lambda_1V+C, \end{eqnarray*} $

其中$ E = (\mu^{l}+\mu^{u}+\alpha^{u}+(\sigma_{11}^{2u}\vee\sigma_{21}^{2u}\vee\sigma_{31}^{2u})-(\sigma_{12}^{2u}\vee\sigma_{22}^{2u}\vee\sigma_{32}^{2u})). $

意味着

$ \begin{eqnarray*} {\Bbb E}\Big[{\rm e}^{\lambda_1t}V(t)\Big]& = &{\Bbb E}[V(0)]+{\Bbb E}\bigg[\int_0^t{\rm e}^{\lambda_1s}[\lambda_1V(s)+LV(s)]{\rm d}s\bigg]\\ &\leq&{\Bbb E}[V(0)]+C{\Bbb E}\bigg[\int_0^t{\rm e}^{\lambda_1s}{\rm d}s\bigg]\\ &\leq&{\Bbb E}[V(0)]+\frac{C}{\lambda_1}\Big[{\rm e}^{\lambda_1t}-1\Big]. \end{eqnarray*} $

因此

$ {\Bbb E}[V(t)]\leq {\rm e}^{-\lambda_1t}{\Bbb E}[V(0)]+\frac{C}{\lambda_1}\Big[1-{\rm e}^{\lambda_1t}\Big]\leq{\Bbb E}[V(0)]+\frac{C}{\lambda_1} = :H. $

选取足够大的常数$ \varsigma, $使得$ \frac{C}{\lambda_1\varsigma}<1, $运用Chebyshev's不等式得

$ {\Bbb P}\bigg\{N+\frac{1}{N}>\varsigma\bigg\}\leq\frac{1}{\varsigma}{\Bbb E}\bigg[N+\frac{1}{N}\bigg]\leq\frac{1}{\varsigma}\bigg[{\rm e}^{-\lambda_1t}{\Bbb E}[V(0)]+\frac{C}{\lambda_1}\Big[1-{\rm e}^{\lambda_1t}\Big]\bigg]\leq\frac{H}{\varsigma}. $

意味着

$ \limsup\limits_{t\rightarrow \infty}{\Bbb P}\bigg\{N+\frac{1}{N}>\varsigma\bigg\}\leq \frac{1}{\varsigma}\limsup\limits_{t\rightarrow \infty}{\Bbb E}\bigg[N+\frac{1}{N}\bigg]\leq\frac{C}{\lambda_1\varsigma} = :\delta. $

可得

$ \limsup\limits_{t\rightarrow \infty}{\Bbb P}\{N>\varsigma\}\leq\limsup\limits_{t\rightarrow \infty}{\Bbb P}\bigg[N+\frac{1}{N}>\varsigma\bigg]\leq\delta. $

对于$ |X|\leq N, $有

$ \limsup\limits_{t\rightarrow \infty}{\Bbb P}[|X|\leq\varsigma]<\delta. $

类似地

$ \liminf\limits_{t\rightarrow \infty}{\Bbb P}\{N>\varsigma\}\leq\delta, \quad \liminf\limits_{t\rightarrow \infty}{\Bbb P}\bigg\{\frac{1}{N}>\varsigma\bigg\}\leq\delta. $

所以

$ \liminf\limits_{t\rightarrow \infty}{\Bbb P}\{N\leq\varsigma\}\geq1-\delta, \quad \liminf\limits_{t\rightarrow \infty}{\Bbb P}\bigg\{\frac{N}{\sqrt{3}}\geq\frac{1}{\sqrt{3}\varsigma}\bigg\}\geq1-\delta. $

注意到$ N\geq|X|\geq\frac{N}{\sqrt{3}}, $则有

$ \liminf\limits_{t\rightarrow \infty}{\Bbb P}\{|X|\leq\varsigma\}\geq1-\delta, \quad \liminf\limits_{t\rightarrow \infty}{\Bbb P}\bigg\{|X|\geq\frac{1}{\sqrt{3}\varsigma}\bigg\}\geq1-\delta. $

定理3.1证毕.

下面研究系统(2.3)周期解的存在性. 定义参数

$ R_0^{s} = \frac{\Big\langle\sqrt{(1-q(t))A(t)\beta(t)}\Big\rangle_T^2}{\Big\langle\mu(t)+p(t)+\sigma_{11}^{2}(t)+\frac{2\sigma_{12}^{2u}(A(t)+(\gamma(t)+\varepsilon(t))\lambda_2)}{\sigma_{11}^{l}\sigma_{12}^{l}}\Big\rangle_{T} \big\langle\mu(t)+\gamma(t)+\alpha(t)+\frac{1}{2}\sigma_{21}^{2}(t)\big\rangle_{T}}. $

定理3.2 当$ R_0^{s}>1 $时, 系统(2.3)至少存在一个非平凡正$ T $ -周期解.

证  根据定理3.1的证明可知, 对于任意的$ 0<\delta<1, $存在$ \Omega_\delta\in\Omega, $满足$ {\Bbb P}\{\Omega_\delta\}\geq1-\delta, $并且对于所有的$ \omega\in\Omega_\delta $都有$ N<\frac{H}{\delta} = :\lambda_2, $定义函数

$ V = K(V_1+\omega(t))+V_2+V_3, $

其中$ V_1 = -k_1\ln S-k_2\ln I+k_3(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n}, V_2 = -\ln S-\ln V, V_3 = \frac{1}{\theta}(S+I+V)^{\theta}, $

$ k_1 = \frac{\langle(1-q)A\rangle_T}{\Big\langle\mu(t)+p(t)+\sigma_{11}^{2}(t)+\frac{2\sigma_{12}^{2u}(A(t)+(\gamma(t)+\varepsilon(t))\lambda_2)}{\sigma_{11}^{l}\sigma_{12}^{l}}\Big\rangle_T}, $

$ k_2 = \frac{\langle(1-q)A\rangle_T}{\big\langle\mu(t)+\gamma(t)+\alpha(t)+\frac{1}{2}\sigma_{21}^{2}(t)\big\rangle_T}, k_3 = \frac{2k_1\sigma_{12}^{2u}}{n(1-n)\sigma_{12}^{2l}(\sigma_{11}^{l})^{n}}, K, n, \theta $

均为正常数, 且$ n<1, \theta<1. $运用It$ \hat{\rm o} $公式可得

$ \begin{eqnarray*} LV_1& = &-\frac{k_1}{S}\Big((1-q(t))A(t)-\beta(t)S I^{m}-(\mu(t)+p(t))S+\gamma(t)I+\varepsilon(t)V\Big) \\ &&-\frac{k_2}{I}\Big(\beta(t)S I^{m}-(\mu(t)+\gamma(t)+\alpha(t))I\Big)+\frac{k_1}{2}(\sigma_{11}(t)+\sigma_{12}(t)S)^{2} \\ &&+\frac{k_2}{2}(\sigma_{21}(t)+\sigma_{22}(t)I)^{2}+k_3n\sigma_{12}^{l}\Big[A(t)(\sigma_{11}^{l}+ \sigma_{12}^{l}S)^{n-1}-q(t)A(t)(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-1} \\ &&-\beta(t)S I^{m}(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-1}-(\mu(t)+p(t))S(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-1}+\gamma(t)I(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-1} \\ &&+\varepsilon(t)V(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-1}\Big]-\frac{k_3}{2}n(1-n)\sigma_{12}^{2l}S^{2}(\sigma_{11}^{l}+\sigma_{12}^{l}S)^{n-2}(\sigma_{11}(t)+\sigma_{12}(t)S)^{2}\\ &\leq&-\frac{k_1}{S}((1-q(t))A(t)+k_1\Big(\mu(t)+p(t)+\sigma_{11}^{2}(t)+\frac{2\sigma_{12}^{2u}A(t)}{(1-n)\sigma_{11}^{l}\sigma_{12}^{l}}\Big)\\ &&+\frac{2k_1\sigma_{12}^{2u}}{(1-n)\sigma_{11}^{l}\sigma_{12}^{l}}(\gamma(t)I+\varepsilon(t)V)+k_2\Big(\mu(t)+\gamma(t)+\alpha(t)+\frac{1}{2}\sigma_{21}^{2}(t)\Big)\\ &&-k_2\beta(t)S I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2}(t)I^2+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta(t)I^m\\ &\leq&-2\sqrt{k_1k_2(1-q(t))A(t)\beta(t)}+k_2\beta(t)S+k_1\Big(\mu(t)+p(t)+\sigma_{11}^{2}(t)\\ &&+\frac{2\sigma_{12}^{2u}(A(t)+(\gamma(t)+\varepsilon(t))\lambda_2)}{(1-n)\sigma_{11}^{l} \sigma_{12}^{l}}\Big)+k_2\Big(\mu(t)+\gamma(t)+\alpha(t)+\frac{1}{2}\sigma_{21}^{2}(t)\Big) \\ &&-k_2\beta(t)S I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2}(t)I^2+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta(t)I^m\\ & = &:-R_0(n, t)+k_2\beta(t)S-k_2\beta(t)S I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2}(t)I^2+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta(t)I^m, \end{eqnarray*} $

其中

$ \begin{eqnarray*} R_0(n, t)& = &2\sqrt{k_1k_2(1-q(t))A(t)\beta(t)}-k_1\bigg(\mu(t)+p(t)+\sigma_{11}^{2}(t)\\ &&+\frac{2\sigma_{12}^{2u}(A(t)+(\gamma(t)+\varepsilon(t))\lambda_2)}{(1-n)\sigma_{11}^{l}\sigma_{12}^{l}}\bigg)-k_2\Big(\mu(t)+\gamma(t)+\alpha(t)+\frac{1}{2}\sigma_{21}^{2}(t)\Big). \end{eqnarray*} $

从而有

$ \begin{eqnarray*} L(V_1+\omega(t))&\leq&-R_0(n, t)+k_2\beta(t)S-k_2\beta(t)S I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2}(t)I^2\\ &&+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta(t)I^m+\omega'(t). \end{eqnarray*} $

设$ \omega(t) $是$ T $ -周期函数并且满足$ \omega'(t) = R_0(t)-\langle R_0\rangle_T, $选取足够小的$ n $使$ R_0(n, t)>0, $当$ n\rightarrow0 $时, 可得

$ \begin{eqnarray*} L(V_1+\omega(t))&\leq&-\langle R_0\rangle_T+k_2\beta^uS-k_2\beta^lS I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2u}I^2+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta^uI^m\\ & = &-2\langle(1-q(t))A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+k_2\beta^uS-k_2\beta^lS I^{m-1}+\frac{k_2}{2}\sigma_{22}^{2u}I^2\\ &&+k_2\sigma_{21}(t)\sigma_{22}(t)I+k_1\beta^uI^m. \end{eqnarray*} $

类似地

$ \begin{eqnarray*} LV_2& = &-\frac{1}{S}\Big[(1-q(t))A(t)-\beta(t)S I^{m}-(\mu(t)+p(t))S+\gamma(t)I+\varepsilon(t)V\Big] \\ &&-\frac{1}{V}\Big[q(t)A(t)+p(t)S-(\mu(t)+\varepsilon(t))V\Big]+\frac{1}{2}(\sigma_{11}(t )+\sigma_{12}(t)S)^{2}\\ &&+\frac{1}{2}(\sigma_{31}(t)+\sigma_{32}(t)V)^{2}\\ &\leq&-\frac{(1-q)A}{S}+\beta I^m-p\frac{S}{V}+\sigma_{12}^{2u}S^2+\sigma_{32}^{2u}V^2+2\mu^u+p^u+\varepsilon^u+\sigma_{11}^{2u}+\sigma_{31}^{2u}, \\ LV_3& = &(S+I+V)^{\theta-1}(A(t)-\mu(t)(S+I+V)-\alpha(t)I) -\frac{1}{2}(1-\theta)(S+I+V)^{\theta-2}\\ &&\times \Big[ S^2(\sigma_{11}(t)+\sigma_{12}(t)S)^2+I^2(\sigma_{21}(t)+\sigma_{22}(t)I)^2+V^2(\sigma_{31}(t)+\sigma_{32}(t)V)^2\Big]\\ &\leq & (S+I+V)^{\theta-1}(A(t)-\mu(t)(S+I+V)-\alpha(t)I) \\ &&-\frac{1}{2}(1-\theta)(S+I+V)^{\theta-2}\Big(\sigma_{12}^{2}S^4 +\sigma_{22}^{2}I^4+\sigma_{32}^{2}V^4\Big)\\ &\leq&-\frac{1}{54}(1-\theta)\Theta_2(S+I+V)^{\theta+2}-\mu^l(S+I+V)^\theta+\frac{A^u}{S^{1-\theta}}, \end{eqnarray*} $

其中$ \Theta_2 = \sigma_{12}^{2l}\wedge\sigma_{22}^{2l}\wedge\sigma_{32}^{2l}. $基于上述分析, 可得

$ \begin{eqnarray*} LV&\leq&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+Kk_2\beta^u S-Kk_2\beta^lS I^{m-1}+\frac{Kk_2}{2}\sigma_{22}^{2u}I^2\\ &&+Kk_2\sigma_{21}\sigma_{22}I+Kk_1\beta^uI^m-\frac{(1-q)A}{S}+\beta I^m-p\frac{S}{V}+\sigma_{12}^{2u}S^2+\sigma_{32}^{2u}V^2+2\mu^u+p^u\\ &&+\varepsilon^u+\sigma_{11}^{2u}+\sigma_{31}^{2u}-\frac{1}{54}(1-\theta)\Theta_2 (S+I+V)^{\theta+2}-\mu^l(S+I+V)^\theta+\frac{A^u}{S^{1-\theta}}\\ & = &-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+Kk_2\beta^u S-Kk_2\beta^lS I^{m-1}+\frac{Kk_2}{2}\sigma_{22}^{2u}I^2\\ &&+Kk_2\sigma_{21}\sigma_{22}I+Kk_1\beta^uI^m-p\frac{S}{V}-\frac{(1-q)A}{S}+\beta I^m+\sigma_{12}^{2u}S^2+\sigma_{32}^{2u}V^2\\ &&-\frac{1}{54}(1-\theta)\Theta_2(S+I+V)^{\theta+2}-\mu^l(S+I+V)^\theta+\frac{A^u}{S^{1-\theta}}\\ &\leq&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+\frac{Kk_2}{2}\sigma_{22}^{2u}I^2+Kk_2\beta^u \lambda_2+Kk_2\sigma_{21}^u\sigma_{22}^uI\\ &&-Kk_2\beta^lS I^{m-1}+Kk_1\beta^uI^m+F-\frac{(1-q^u)A^l}{S}+\beta^u I^m+\sigma_{12}^{2u}S^2+\sigma_{32}^{2u}V^2\\ &&-\frac{1}{54}(1-\theta)\Theta_2S^{\theta+2}-\frac{1}{54}(1-\theta)\Theta_2 I^{\theta+2}-\frac{1}{54}(1-\theta)\Theta_2V^{\theta+2}-p^l\frac{S}{V}\\ &&-\mu^l(S+I+V)^\theta+\frac{A^u}{S^{1-\theta}}\\ &\leq&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+\frac{Kk_2}{2}\sigma_{22}^{2u}I^2+Kk_2\beta^u \lambda_2+Kk_2\sigma_{21}^u\sigma_{22}^uI\\ &&-Kk_2\beta^lS I^{m-1}+Kk_1\beta^u\lambda_2^{m-1}I+F-\frac{(1-q^u)A^l}{S}+\beta^u\lambda_2^{m-1}I+\sigma_{12}^{2u}S^2+\sigma_{32}^{2u}V^2\\ &&-\frac{1}{54}(1-\theta)\Theta_2S^{\theta+2}-\frac{1}{54}(1-\theta)\Theta_2 I^{\theta+2}-\frac{1}{54}(1-\theta)\Theta_2V^{\theta+2}\\ &&-p^l\frac{S}{V}-\mu^l(S+I+V)^\theta+\frac{A^u}{S^{1-\theta}}\\ & = &:-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &&-\mu^l(S+I+V)^\theta-Kk_2\beta^lS I^{m-1}-p^l\frac{S}{V}, \end{eqnarray*} $

其中

$ f_1(I) = -\frac{1-\theta}{54}\Theta_2I^{\theta+2}+\frac{Kk_2}{2}\sigma_{22}^{2u}I^2+K(k_2\sigma_{21}^u\sigma_{22}^u+k_1\beta^u\lambda_2^{m-1})I+Kk_2\beta^u \lambda_2+\beta^u\lambda_2^{m-1}I, $

$ f_2(S) = -\frac{1-\theta}{54}\Theta_2S^{\theta+2}+\sigma_{12}^{2u}S^2, \quad f_3(V) = -\frac{1-\theta}{54}\Theta_2V^{\theta+2}+\sigma_{32}^{2u}V^2, $

$ g(S) = -\frac{(1-q^u)A^l}{S}+\frac{A^u}{S^{1-\theta}} = \frac{A^u}{S^{1-\theta}}\Big(-\frac{(1-q^u)A^l}{A^uS^{\theta}}+1\Big), F = 2\mu^u+p^u+\varepsilon^u+\sigma_{11}^{2u}+\sigma_{31}^{2u}. $

选取足够大的正常数$ K $使得

$ -2K\langle(1-q)A\rangle_T((R_0^{s})^{\frac{1}{2}}-1)+F+f_2^u+f_3^u+g^u\leq-2. $

定义有界区域

$ D = \Big\{(S, I, V)\in\mathbb{R} _+^{3}, \varepsilon_1\leq S\leq\frac{1}{\varepsilon_1}, \varepsilon_1\leq I\leq\frac{1}{\varepsilon_1}, \varepsilon_2\leq V\leq\frac{1}{\varepsilon_2}\Big\}. $

选择足够小的正常数$ \varepsilon_1 $和$ \varepsilon_2, \varepsilon_2 = \varepsilon_1^2, $并且满足如下不等式

$ \begin{eqnarray*} &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1(\varepsilon_1)+f_2^u+f_3^u\leq-1, \\ &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1^u+f_2^u+f_3^u-\frac{(1-q^u)A^l}{\varepsilon_1}+\frac{A^u}{\varepsilon_1^{1-\theta}}\leq-1, \\ &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1^u+f_2^u+f_3^u-p^l\frac{1}{\varepsilon_1}\leq-1, \\ &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1\Big(\frac{1}{\varepsilon_1}\Big)+f_2^u+f_3^u\leq-1, \\ &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1^u+f_2\Big(\frac{1}{\varepsilon_1}\Big)+f_3^u+g^u\leq-1, \\ &&-2K\langle(1-q)A\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1^u+f_2^u+f_3\Big(\frac{1}{\varepsilon_1}\Big)+g^u\leq-1. \end{eqnarray*} $

为了讨论方便, 将$ \mathbb{R} _+^{3}\setminus D $分成以下6个区域

$ D_1 = \{(S, I, V)\in\mathbb{R} _+^{3}:0<I<\varepsilon_1\}, D_2 = \{(S, I, V)\in\mathbb{R} _+^{3}:0<S<\varepsilon_1\}, $

$ D_3 = \{(S, I, V)\in\mathbb{R} _+^{3}:\varepsilon_1<S, \varepsilon_1<I, 0<V<\varepsilon_2\}, D_4 = \bigg\{(S, I, V)\in\mathbb{R} _+^{3}:\frac{1}{\varepsilon_1}<I\bigg\}, $

$ D_5 = \bigg\{(S, I, V)\in\mathbb{R} _+^{3}:\frac{1}{\varepsilon_1}<S\bigg\}, D_6 = \bigg\{(S, I, V)\in\mathbb{R} _+^{3}:\frac{1}{\varepsilon_2}<V\bigg\}. $

显然, $ D^c = \bigcup\limits_{i = 1}^6D_i. $下面将分6种情况在$ D^c $证明$ LV\leq-1. $

1) 当$ (S, I, V)\in[0, \infty)\times D_1, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1(\varepsilon_1)+f_2^u+f_3^u. \end{eqnarray*} $

2) 当$ (S, I, V)\in[0, \infty)\times D_2, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1^u+f_2^u+f_3^u-\frac{(1-q^u)A^l}{\varepsilon_1}+\frac{A^u}{\varepsilon_1^{1-\theta}}. \end{eqnarray*} $

3) 当$ (S, I, V)\in[0, \infty)\times D_3, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1^u+f_2^u+f_3^u-p^l\frac{1}{\varepsilon_1}. \end{eqnarray*} $

4) 当$ (S, I, V)\in[0, \infty)\times D_4, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+g^u+f_1(\frac{1}{\varepsilon_1})+f_2^u+f_3^u. \end{eqnarray*} $

5) 当$ (S, I, V)\in[0, \infty)\times D_5, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1^u+f_2(\frac{1}{\varepsilon_1})+f_3^u+g^u. \end{eqnarray*} $

6) 当$ (S, I, V)\in[0, \infty)\times D_6, $有

$ \begin{eqnarray*} LV&\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1(I)+f_2(S)+f_3(V)+g(S)\\ &\leq&-2K\big\langle(1-q)A\big\rangle_T\big((R_0^{s})^{\frac{1}{2}}-1\big)+F+f_1^u+f_2^u+f_3(\frac{1}{\varepsilon_1})+g^u. \end{eqnarray*} $

综上, 对任意的$ (S, I, V)\in[0, \infty)\times D^c $, 有$ LV\leq-1. $

定理4.1  当$ R_1^\star = \frac{\hat{\beta}(1-\check{q})\hat{A}}{\lambda_2(\check{\mu}+\check{p})(\check{\mu}+\check{\gamma}+\check{\alpha}+\frac{1}{2}\check{\sigma}_{21}^{2}+\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^2)}>1, k\in\hbar $时, 系统(2.2)的染病者在平均意义下具有持久性.

证  根据定理3.1可知, 对于任意的$ 0<\delta<1, $存在$ \Omega_\delta\in\Omega, $满足$ {\Bbb P}\{\Omega_\delta\}\geq1-\delta, $并且对于所有的$ \omega\in\Omega_\delta $都有$ N<\frac{H}{\delta} = :\lambda_2. $利用It$ \hat{\rm o} $公式可得

$ \begin{eqnarray*} {\rm d}(S+I)&\geq&(1-\check{q})\hat{A}-(\check{\mu}+\check{p})S-(\check{\mu}+\check{\alpha})I+S(\sigma_{11}(k)+\sigma_{12}(k)S){\rm d}B_1(s)\\ &&+I(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(s). \end{eqnarray*} $

将上式两端从$ 0 $到$ t $积分得

$ \begin{eqnarray*} \phi(t)& = &:\frac{S(t)-S(0)}{t}+\frac{I(t)-I(0)}{t}\\ &\geq&(1-\check{q})\hat{A}-(\check{\mu}+\check{p})\langle S\rangle_t-(\check{\mu}+\check{\alpha})\langle I\rangle_t+\frac{N_1}{t}+\frac{N_2}{t}, \end{eqnarray*} $

其中$ N_1 = \int_0^tS(\sigma_{11}(k)+\sigma_{12}(k)S){\rm d}B_1(s), $$ N_2 = \int_0^tI(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(s). $

由$ \lim\limits_{t\rightarrow \infty}\phi(t) = 0 $可得

$ \langle S\rangle_{t}\geq\frac{1}{\check{\mu}+\check{p}}\bigg[(1-\check{q})\hat{A}-\phi(t)-(\check{\mu}+\check{\alpha})\langle I\rangle_t+\frac{N_1}{t}+\frac{N_2}{t}\bigg], $

运用It$ \hat{\rm o} $公式可得

$ {\rm d}\ln I\geq\bigg[\frac{\hat{\beta}S}{\lambda_2}-(\check{\mu}+\check{\gamma}+\check{\alpha})-\frac{1}{2}\check{\sigma}_{12}^{2} -\check{\sigma}_{21}\check{\sigma}_{22}I-\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}\bigg]{\rm d}t+(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(t). $

将上式从$ 0 $到$ t $积分得

$ \begin{eqnarray*} \frac{\ln I(t)-\ln I(0)}{t}&\geq&\frac{\hat{\beta}}{\lambda_2}\langle S\rangle_{t}-(\check{\mu}+\check{\gamma}+\check{\alpha})-\frac{1}{2}\check{\sigma}_{21}^{2}-\check{\sigma}_{21}\check{\sigma}_{22}\langle I\rangle_t+\frac{N_3}{t}-\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}\\ &\geq&\frac{\hat{\beta}}{(\check{\mu}+\check{p})\lambda_2}\bigg[(1-\check{q})\hat{A}-\phi(t)-(\check{\mu}+\check{\alpha})\langle I\rangle_t+\frac{N_1}{t}+\frac{N_2}{t}\bigg]\\ &&+\frac{N_3}{t}-(\check{\mu}+\check{\gamma}+\check{\alpha})-\frac{1}{2}\check{\sigma}_{21}^{2}-\check{\sigma}_{21}\check{\sigma}_{22}\langle I\rangle_t-\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}, \end{eqnarray*} $

其中$ N_3 = \int_0^t(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(s). $由强大数定理可知

$ \lim\limits_{t\rightarrow \infty}\frac{N_1}{t} = \lim\limits_{t\rightarrow \infty}\frac{N_2}{t} = \lim\limits_{t\rightarrow \infty}\frac{N_3}{t} = 0, \lim\limits_{t\rightarrow \infty}\frac{\ln I(t)-\ln I(0)}{t}\leq0. $

从而得

$ \begin{eqnarray*} \liminf\limits_{t\rightarrow \infty}\langle I\rangle_t&\geq&\frac{1}{\frac{\hat{\beta}(\check{\mu}+\check{\alpha})}{(\check{\mu}+\check{p})\lambda_2}+\check{\sigma}_{21}\check{\sigma}_{22}} \bigg(\frac{\hat{\beta}\lambda_2^{m-1}(1-\check{q})\hat{A}(k)}{(\check{\mu}+\check{p})\lambda_2}-(\check{\mu}+\check{\gamma}+\check{\alpha})-\frac{1}{2}\check{\sigma}_{21}^{2}-\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}\bigg)\\ & = &\frac{1}{\frac{\hat{\beta}(\check{\mu}+\check{\alpha})}{(\check{\mu}+\check{p})\lambda_2}+\check{\sigma}_{21}\check{\sigma}_{22}} \bigg(\check{\mu}+\check{\gamma}+\check{\alpha}+\frac{1}{2}\check{\sigma}_{21}^{2}+\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}\bigg)\\ &&\cdot\bigg(\frac{\hat{\beta}(1-\check{q})\hat{A}}{\lambda_2(\check{\mu}+\check{p})(\check{\mu}+\check{\gamma}+\check{\alpha}+\frac{1}{2}\check{\sigma}_{21}^{2}+\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2})}-1\bigg)\\ & = &:\frac{1}{\frac{\hat{\beta}(\check{\mu}+\check{\alpha})}{(\check{\mu}+\check{p})\lambda_2}+\check{\sigma}_{21}\check{\sigma}_{22}} \bigg(\check{\mu}+\check{\gamma}+\check{\alpha}+\frac{1}{2}\check{\sigma}_{21}^{2}+\frac{1}{2}\check{\sigma}_{22}^{2}\lambda_2^{2}\bigg)(R_1^\star-1)>0. \end{eqnarray*} $

则系统(2.2)的染病者在平均意义下具有持久性.

本节讨论系统(2.2)的遍历平稳分布的存在性. 为此, 定义参数

$ \begin{eqnarray*} R_0^\star& = &\Big\{\Big(\sum\limits_{k = 1}^{N}\pi_k\sqrt{(1-q(k))A(k)\beta(k)}\Big)^2\Big\}/\Big\{\sum\limits_{k = 1}^{N}\pi_k\Big(\mu(k)+p(k)+\sigma_{11}^{2}(k)\\ &&+\frac{2\check{\sigma}_{12}^2(A(k)+(\gamma(k)+\varepsilon(k))\lambda_2)}{\hat{\sigma}_{11}\hat{\sigma}_{12}}\Big) \sum\limits_{k = 1}^{N}\pi_k\Big(\mu(k)+\gamma(k)+\alpha(k)+\frac{1}{2}\sigma_{21}^{2}(k)\Big)\Big\}. \end{eqnarray*} $

定理5.1  当$ R_0^\star>1 $时, 对于任意初值$ (S(0), I(0), V(0), r(0))\in \mathbb{R} _+^{3}\times\hbar, $系统(2.2)的解$ (S(t), I(t), V(t), r(t)) $存在唯一的遍历平稳分布.

证  对$ i\neq j, \gamma_{ij}>0 $成立, 其扰动矩阵为

$ \begin{eqnarray*} A& = &(a_{ij}(S, I, V, K))\\ & = &\left(\begin{array}{ccc} (S(\sigma_{11}(k)+\sigma_{12}(k)S))^2 & 0& 0\\ 0 &(I(\sigma_{21}(k)+\sigma_{22}(k)I))^2 &0\\ 0 &0&(V(\sigma_{31}(k)+\sigma_{32}(k)V))^2 \end{array}\right). \end{eqnarray*} $

为了使引理2.2中的条件ⅲ)成立, 考虑有界开集$ D = [\varepsilon, \frac{1}{\varepsilon}]\times[\varepsilon, \frac{1}{\varepsilon}]\times[\varepsilon, \frac{1}{\varepsilon}], $其中$ \varepsilon>0 $是充分小的正常数. 对任意$ (S, I, V, k)\in D^c(D^c = \mathbb{R} _+^{3}\setminus D)\times\hbar, \xi = (\xi_1, \xi_2, \xi_3)\in\mathbb{R} _+^3, $有

$ \begin{eqnarray*} \sum\limits_{i, j = 1}^{3}a_{ij}(S, I, V, k)\xi_i\xi_j& = &(\sigma_{11}(k)S+\sigma_{12}(k)S^2)^2\xi_1^2+(\sigma_{21}(k)I+\sigma_{22}(k)I^2)^2\xi_2^2\\ &&+(\sigma_{31}(k)V+\sigma_{32}(k)V^2)^2\xi_3^2\\ &\geq & M_1\|\xi\|^2, \end{eqnarray*} $

其中

$ M_{1} = \min\limits_{(S, I, V, k)\in D^c\times\hbar}\Big\{(\sigma_{11}(k)S+\sigma_{12}(k)S^2)^2, (\sigma_{21}(k)I+\sigma_{22}(k)I^2)^2, (\sigma_{31}(k)V+\sigma_{32}(k)V^2)^2\Big\}. $

显然引理2.2中的条件ⅱ)成立. 在$ \mathbb{R} _+^3\times\hbar $上定义函数

$ V = M(V_1+\omega(k))+V_2+V_3, $

其中

$ V_1 = -c_1\ln S-c_2\ln I+c_3(\hat{\sigma}_{11}(k)+\hat{\sigma}_{12}(k)S)^{n}, V_2 = -\ln S-\ln V, V_3 = \frac{1}{\theta}(S+I+V)^{\theta}, $

$ c_1 = \frac{\sum\limits_{k = 1}^{N}\pi_k(1-q(k))A(k)}{\sum\limits_{k = 1}^{N}\pi_k\Big(\mu(k) +p(k)+\sigma_{11}^{2}(k)+\frac{2\check{\sigma}_{12}^2(A(k)+(\gamma(k)+\varepsilon(k))\lambda_2)} {(1-n)\hat{\sigma}_{11}\hat{\sigma}_{12}}\Big)}, $

$ c_2 = \frac{\sum\limits_{k = 1}^{N}\pi_k(1-q(k))A(k)}{\sum\limits_{k = 1}^{N}\pi_k\big(\mu(k)+ \gamma(k)+\alpha(k)+\frac{1}{2}\sigma_{21}^2(k)\big)}, c_3 = \frac{2c_1\check{\sigma}_{12}^2}{n(1-n)\hat{\sigma}_{12}^{2}\hat{\sigma}_{11}^{n}}, $

$ 0<n<1, 0<\theta<1, M $为正常数. 根据It$ \hat{\rm o} $公式得

$ \begin{eqnarray*} L(V_1+\omega(k))&\leq&-\frac{c_1}{S}(1-q(k))A(k)\\ &&+c_1\bigg(\mu(k)+p(k)+\sigma_{11}^2(k)+\frac{2\check{\sigma}_{12}^2(A(k)+(\gamma(k)+ \varepsilon(k))\lambda_2)}{(1-n)\hat{\sigma}_{11}\hat{\sigma}_{12}}\bigg)\\ &&+c_2\Big(\mu(k)+\gamma(k)+\alpha(k)+\frac{1}{2}\sigma_{21}^2(k)\Big)-c_2\beta(k)S I^{m-1}+\frac{c_2}{2}\sigma_{22}^2(k)I^2\\ &&+c_1\beta(k)I^m+c_2\sigma_{21}(k)\sigma_{22}(k)I+\sum\limits_{l\in\hbar}\gamma_{kl}\omega(l)\\ &\leq&-2\sqrt{c_1c_2(1-q(k))A(k)\beta(k)}+c_2\beta(k) S+c_1\bigg(\mu(k)+p(k)+\sigma_{11}^2(k)\\ &&+\frac{2\check{\sigma}_{12}^2(A(k)+(\gamma(k)+\varepsilon(k))\lambda_2)}{(1-n)\hat{\sigma} _{11}\hat{\sigma}_{12}}\bigg)+c_2\Big(\mu(k)+\gamma(k)+\alpha(k)+\frac{1}{2}\sigma_{21}^2(k)\Big)\\ &&-c_2\beta(k)S I^{m-1}+\frac{c_2}{2}\sigma_{22}^2(k)I^2+c_2\sigma_{21}(k)\sigma_{22}(k)I+c_1\beta(k)I^m+ \sum\limits_{l\in\hbar}\gamma_{kl}\omega(l)\\ & = &:-\bar{R}_0(n, k)+\frac{c_2}{2}\sigma_{22}^2(k)I^2+c_2\beta(k) S+c_2\sigma_{21}(k)\sigma_{22}(k)I-c_2\beta(k)S I^{m-1}\\ &&+c_1\beta(k)I^m+\sum\limits_{l\in\hbar}\gamma_{kl}\omega(l), \end{eqnarray*} $

其中

$ \begin{eqnarray*} \bar{R}_0(n, k)& = &2\sqrt{c_1c_2(1-q(k))A(k)\beta(k)}-c_2\Big(\mu(k)+\gamma(k)+\alpha(k)+\frac{1}{2}\sigma_{21}^2(k)\Big)\\ &&-c_1\bigg(\mu(k)+p(k)+\sigma_{11}^2(k)+\frac{2\check{\sigma}_{12}^2(A(k)+(\gamma(k)+ \varepsilon(k))\lambda_2)}{(1-n)\hat{\sigma}_{11}\hat{\sigma}_{12}}\bigg). \end{eqnarray*} $

选取足够小的$ n, $使得$ \bar{R}_0(n, k)>0, $当$ n\rightarrow0^+ $时, 有

$ \begin{eqnarray*} L(V_1+\omega(k))&\leq&-\bar{R}_0(n, k)+\frac{c_2}{2}\check{\sigma}_{22}^2I^2+c_2\check{\beta} S+c_2\check{\sigma}_{21}\check{\sigma}_{22}I-c_2\hat{\beta}S I^{m-1} +c_1\check{\beta}I^m+\sum\limits_{l\in\hbar}\check{\gamma}_{kl}\omega(l)\\ &\rightarrow &-\bar{R}_0(k)+\frac{c_2}{2}\check{\sigma}_{22}^2I^2+c_2\check{\beta} S+c_2\check{\sigma}_{21}\check{\sigma}_{22}I-c_2\hat{\beta}S I^{m-1}+c_1\check{\beta}I^m+\sum\limits_{l\in\hbar}\check{\gamma}_{kl}\omega(l). \end{eqnarray*} $

为了给出$ \omega(l), $定义向量$ \bar{R}_0 = (\bar{R}_0(1), \bar{R}_0(2), \cdots , \bar{R}_0(N)). $由于生成矩阵$ \Gamma $不可约, 对于任意的$ \bar{R}_0(k), $存在集合$ \omega = (\omega(1), \omega(2), \cdots , \omega(N)), $满足泊松系统^[16]: $ \Gamma\omega-\bar{R}_0(k) = -\sum\limits_{h = 1}^{N}\pi_h\bar{R}_0(h). $相应地可得

$ \sum\limits_{l\in\hbar}\gamma_{kl}\omega(l)-\bar{R}_0(k) = -\sum\limits_{h = 1}^{N}\pi_h\bar{R}_0(h). $

综上可得

$ \begin{eqnarray*} L(V_1+\omega(k))&\leq&-2\sum\limits_{k = 1}^{N}\pi_k(1-\check{q})\hat{A}\Big[(R_0^\star)^{\frac{1}{2}}-1\Big]+c_2\check{\beta}S+\frac{c_2}{2}\check{\sigma}_{22}^2I^2-c_2\hat{\beta}S I^{m-1}\\ &&+c_2\check{\sigma}_{21}\check{\sigma}_{22}I+c_1\check{\beta}I^m. \end{eqnarray*} $

运用类似定理3.2的方法可得$ LV\leq-1. $由引理2.2可知, 系统(2.2)的解具有唯一的遍历分布.

在本节中, 给出随机系统(2.2)的染病者灭绝的条件. 定义参数

$ R_1^s = \frac{\check{\beta}\lambda_2^{m-1}}{(\hat{\mu}+\hat{\gamma}+\hat{\alpha}+\frac{1}{2}\hat{\sigma}_{21}^2)}. $

定理6.1 当$ R_1^s<1 $时, 对于任意的初值$ (S(0), I(0), V(0), r(0))\in \mathbb{R} _+^{3}\times\hbar, $系统(2.2) 的染病者是灭绝的.

证  根据It$ \hat{\rm o} $公式有

$ \begin{eqnarray*} {\rm d}[\ln I(t)]& = &\bigg[\frac{\beta(k)S I^m-(\mu(k)+\gamma(k)+\alpha(k))I}{I}-\frac{1}{2}(\sigma_{21}(k)+\sigma_{22}(k)I)^2\bigg]{\rm d}t\\ &&+(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(t)\\ &\leq&\bigg[\check{\beta}S I^{m-1}-(\hat{\mu}+\hat{\gamma}+\hat{\alpha})-\frac{1}{2}(\sigma_{21}^2(k)+\sigma_{22}^2(k)I^2)\bigg]{\rm d}t\\ &&+(\sigma_{21}(k)+\sigma_{22}(k)I){\rm d}B_2(t). \end{eqnarray*} $

两边从$ 0 $到$ t $积分, 可得

$ \begin{eqnarray*} \ln I&\leq&\check{\beta}\int_0^tS\lambda_2^{m-1}{\rm d}s-\int_0^t(\hat{\mu}+\hat{\gamma}+\hat{\alpha}){\rm d}s -\frac{1}{2}\int_0^t(\sigma_{21}^2(k)+\sigma_{22}^2(k)I^2){\rm d}s\\ &&+\int_0^t\sigma_{21}(k){\rm d}B_2(s)+\int_0^t\sigma_{22}(k)I{\rm d}B_2(s)+\ln(I(0))\\ & = &:\check{\beta}\int_0^tS\lambda_2^{m-1}{\rm d}s-\int_0^t(\hat{\mu}+\hat{\gamma}+\hat{\alpha}){\rm d}s -\frac{1}{2}\int_0^t(\sigma_{21}^2(k)+\sigma_{22}^2(k)I^2){\rm d}s\\ &&+M_1(t)+M_2(t)+\ln(I(0)), \end{eqnarray*} $

其中$ M_1(t) = \int_0^t\sigma_{22}(k)I{\rm d}B_2(s), $$ M_2(t) = \int_0^t\sigma_{21}(k){\rm d}B_2(s). $

通过指数鞅不等式, 对于任何正常数$ T, \vartheta, \upsilon, $有

$ {\Bbb P}\Big\{\sup\limits_{0<t\leq T}[M_1(t)-\frac{\vartheta}{2}\langle M_1, M_1\rangle_t]>\upsilon\Big\}\leq {\rm e}^{-\vartheta\upsilon}, $

其中$ \langle M_1, M_1\rangle_t = \int_0^t\sigma_{22}^2(k)I^2{\rm d}s, $令$ T = Q, \vartheta = 1, \upsilon = 2\ln Q. $可得

$ {\Bbb P}\Big\{\sup\limits_{0<t\leq T}[M_1(t)-\frac{1}{2}\langle M_1, M_1\rangle_t]>2\ln Q\Big\}\leq\frac{1}{Q^2}. $

因此对于几乎所有的$ \omega\in\Omega $存在随机整数$ Q_0\in Q_0(\omega), $使得对于$ Q\geq Q_0+M $ (其中$ M $为足够大的正常数), 有

$ \sup\limits_{0<t\leq T}\bigg[M_1-\frac{1}{2}\langle M_1, M_1\rangle_t\bigg]\leq2\ln Q. $

对于所有的$ 0\leq t\leq Q, Q\geq Q_0, $ a.s.有

$ M_1(t)\leq2\ln Q+\frac{1}{2}\langle M_1, M_1\rangle_t = 2\ln Q+\frac{1}{2}\int_0^t\sigma_{22}^2I^2{\rm d}s. $

不难看出, 对于$ 0\leq Q-1\leq t\leq Q, $有

$ \begin{eqnarray*} \frac{\ln I}{t}&\leq&\frac{\check{\beta}}{t}\int_0^tS\lambda_2^{m-1}{\rm d}s-\frac{1}{t}\int_0^t\Big(\hat{\mu}+\hat{\gamma} +\hat{\alpha}+\frac{1}{2}\sigma_{21}^2(k)\Big){\rm d}s-\frac{1}{2t}\int_0^t\sigma_{22}^2(k)I^2{\rm d}s\\ &&+\frac{\ln(I(0))}{t}+\frac{2\ln Q}{t}+\frac{1}{2t}\int_0^t\sigma_{22}^2(k)I^2{\rm d}s+\frac{M_2(t)}{t}\\ &\leq&\frac{\check{\beta}}{t}\int_0^tS\lambda_2^{m-1}{\rm d}s-\frac{1}{t}\int_0^t\Big(\hat{\mu}+\hat{\gamma} +\hat{\alpha}+\frac{1}{2}\sigma_{21}^2(k)\Big){\rm d}s+\frac{\ln(I(0))}{t}+\frac{2\ln Q}{t} +\frac{M_2(t)}{t}, \end{eqnarray*} $

根据强大数定理得$ \lim\limits_{t\rightarrow \infty}\frac{M_2(t)}{t} = 0. $因此

$ \lim\limits_{t\rightarrow \infty}\frac{\ln I}{t}\leq\check{\beta}\lambda_2^{m-1}-(\hat{\mu}+\hat{\gamma} +\hat{\alpha}+\frac{1}{2}\hat{\sigma}_{21}^2)<0. $

从而有$ \lim\limits_{t\rightarrow \infty}\ln I(t) = 0. $定理得证.

本节进行数值模拟, 为此我们使用如下离散方程

$ \left\{ \begin{array}{rl} S_{n+1}& = S_n+\Big((1-q_n)A_n-\beta_nS_nI_n^m-(\mu_n+p_n)S_n+\gamma_nI_n+\varepsilon_nV_n\Big)\Delta t\\ & \quad+S_n(\sigma_{11, n}+\sigma_{12, n}S_n)\Delta B_{1, n}, \nonumber\\ I_{n+1}& = I_n+(\beta_nS_nI_n^m-(\mu_n+\gamma_n+\alpha_n)I_n)\Delta t+I_n(\sigma_{21, n}+\sigma_{22, n}I_n)\Delta B_{2, n}, \nonumber\\ V_{n+1}& = V_n+(q_nA_n+p_nS_n-(\mu_n+\varepsilon_n)V_n)\Delta t+V_n(\sigma_{31, n}+\sigma_{32, n}V_n)\Delta B_{3, n}. \nonumber \end{array} \right. $

选取以下参数值

$ \begin{array}{c} A = 1.5+0.5\sin t, \beta = 5.7+0.2\sin t, \mu = 0.2+0.1\sin t, \gamma = 0.4+0.3\sin t, \\ p = 0.6+0.1\sin t, q = 0.2+0.01\sin t, \varepsilon = 0.35+0.02\sin t, \alpha = 0.3+0.2\sin t, \\ \sigma_{11} = 0.5+0.1\sin t, \sigma_{12} = 0.4+0.1\sin t, \sigma_{21} = 0.2+0.01\sin t, \\ \sigma_{22} = 0.1+0.02\sin t, \sigma_{31} = 0.05+0.01\sin t, \sigma_{32} = 0.06+0.01\sin t. \end{array} $

经计算可得$ R_0^s = 4.99>1, $由定理3.2可知系统(2.3)存在非平凡正周期解(见图 1).

在系统(2.2)中, 我们考虑右连续的马尔科夫链$ r(t) $是取值在$ \hbar = \{1, 2\} $上的, $ r(t) $的稳态分布为$ \pi = (\pi_1, \pi_2) = (\frac{5}{7}, \frac{2}{7}). $选取以下参数值.

(1) 当$ r(t) = 1 $时, $ A = 2, \beta = 3, \mu = 0.65, $$ \gamma = 0.2, $$ p = 0.5, q = 0.2, $$ \varepsilon = 0.25, $$ \alpha = 0.45, \sigma_{11} = 0.1, \sigma_{12} = 0.01, \sigma_{21} = 0.05, \sigma_{22} = 0.01, \sigma_{31} = 0.05, \sigma_{32} = 0.01. $

(2) 当$ r(t) = 2 $时, $ A = 2, \beta = 4, \mu = 0.5, \gamma = 0.35, p = 0.7, q = 0.15, \varepsilon = 0.35, $$ \alpha = 0.5, \sigma_{11} = 0.1, \sigma_{12} = 0.01, \sigma_{21} = 0.05, \sigma_{22} = 0.02, \sigma_{31} = 0.05, \sigma_{32} = 0.02. $

经计算可得$ R_0^\star = 1.25>1, $根据定理5.1可知系统(2.2)存在唯一的遍历平稳分布(见图 2).

选取以下参数值.

(1) 当$ r(t) = 1 $时, $ A = 0.8, \beta = 0.2, \mu = 0.9, \gamma = 0.9, p = 0.35, $$ q = 0.75, \varepsilon = 0.9, $$ \alpha = 0.3, \sigma_{11} = 0.7, \sigma_{12} = 0.7, \sigma_{21} = 0.7, \sigma_{22} = 0.7, \sigma_{31} = 0.7, \sigma_{32} = 0.7. $

(2) 当$ r(t) = 2 $时, $ A = 0.81, \beta = 0.21, \mu = 0.89, \gamma = 0.85, p = 0.36, q = 0.65, \varepsilon = 0.9, $$ \alpha = 0.3, \sigma_{11} = 0.7, \sigma_{12} = 0.7, \sigma_{21} = 0.7, \sigma_{22} = 0.7, \sigma_{31} = 0.7, \sigma_{32} = 0.7. $

经计算可得$ R_1^s = 0.78<1, $故根据定理6.1可知系统的感染者趋于灭绝(见图 3).

本文考虑环境之间的随机切换建立了一类具有非线性随机扰动非自治SIVS模型. 首先忽略有色噪声的影响, 定义了模型的阈值, 证明了随机正周期解的存在性, 并进行数值模拟验证了理论结果. 表明易感人群, 感染人群及接种疫苗后获得免疫的人群的数量由于季节的变化、天气的转变等因素的影响呈周期性变化.

在有色噪声的影响下, 我们证明了模型的平均持久性、遍历性和绝灭性等. 这些数学结论的生物意义如下.

由定理4.1知当$ R_1^\star>1 $时系统(2.2)在平均意义下是持久的, 表明传染病在种群内将长期存在;

由定理5.1知当$ R_0^\star>1 $时系统(2.2)的随机过程$ (S(t), I(t), V(t), r(t)) $是遍历的且具有唯一的平稳分布, 表明虽然易感者、感染者及接种者的数量随着时间的推移而波动, 但它们的平均值收敛于固定值;

由定理6.1知当$ R_1^s<1 $时系统(2.2)是绝灭的, 表明感染者的数量随着时间的推移最终趋于0, 疾病消失.