从过去的霍乱、鼠疫、黑死病、天花等到如今的新冠, 传染病一直是困扰人类的难题之一. 传染病严重威胁人类健康和生命, $ 2019 $年末爆发的新冠肺炎快速地在全球蔓延开来, 不仅危害了人类健康, 也对各国经济造成了巨大影响.

对传染病的众多研究中, 发病机制、传播规律和防治措施等都有重要的现实意义, 其中疫苗接种是防治领域内比较有效的控制手段, 通过接种产生抗体, 从而抵抗疾病的感染^[1-4]. 但是疫苗接种通常采用静脉注射, 这是一个脉冲瞬时过程, 利用脉冲微分方程理论来制定有效的策略, 特别是免疫接种策略, 这样更符合实际^[5].

近年来, 许多生物学数学家对脉冲接种传染病模型产生了浓厚的兴趣. 汪金燕^[6]研究了一类具固定时刻脉冲式疫苗接种策略且带污染环境间接传播的动力学模型, 为手足口病的疫苗策略提供了理论依据. 王来全^[7]考虑了对易感人群进行脉冲接种具有标准发生率的传染病模型, 分析其对传染病预防的效果. 还有部分数学家研究了具有时滞、线性或非线性、垂直传染等因素的脉冲接种模型, 以此来探讨传染病的控制和疾病的消亡等问题^[8-12]. 然而大多数学者只考虑到了注射疫苗瞬间的效应, 没有考虑到人与人的体质不同, 有的人注射疫苗马上起效, 可以看作是一个脉冲的过程, 本文把它称作脉冲接种; 而有的人注射疫苗后需要一小段时间才能产生效果, 这一段时间是非瞬时的, 本文把这段时间称为接种效应区间. 希望通过该模型为现实生活中的疫苗接种提供可靠的决策支持.

本文的主要结构如下, 第二节介绍了模型的背景概念, 第三节给出了相关重要引理, 第四节给出了模型持久的条件, 最后一部分对文章主要结论进行了讨论.

根据上面的讨论, 我们建立了具瞬时脉冲接种与非瞬时脉冲接种效应的一类新的$ SIR $传染病模型.

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _1} - {\beta _1}S(t)I(t) - {d_1}S(t), \\ \frac{{\mathrm{d}I(t)}}{{dt}} = {\beta _1}S(t)I(t) - ({r_1}+{d_1})I(t), \\ \frac{{\mathrm{d}R(t)}}{{dt}} = {r_1}I(t) - {d_1}R(t), \end{array} \right\}t \in (n\tau , (n + l)\tau ], \\ \left. \begin{array}{llc} \Delta S(t) = - {\mu}S(t), \\ \Delta I(t) = 0, \\ \Delta R(t) = {\mu}S(t), \end{array} \right\}t = (n + l)\tau , \\ \left. \begin{array}{llc} \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _2} - {\beta _2}S(t)I(t) - ({d_2}+{u_1})S(t) , \\ \frac{{\mathrm{d}I(t)}}{{dt}} = {\beta _2}S(t)I(t) - ({r_2}+{d_2})I(t), \\ \frac{{\mathrm{d}R(t)}}{{dt}} = {r_2}I(t) - ({d_2}-{\mu_1})R(t), \end{array} \right\}t \in ((n + l)\tau , (n + 1)\tau ], \\ \end{array} \right. \end{equation} $

(2.1)

其中$ S(t) $, $ I(t) $和$ R(t) $分别代表$ t $时刻易感者类、染病者类和恢复者类的人数; $ {\lambda _1} > 0 $代表在$ (n\tau , (n + l)\tau ] $上的出生系数; $ {\beta _1} > 0 $代表在$ (n\tau , (n + l)\tau ] $上的传染系数, 即$ S $和$ I $接触后被传染的概率; $ {d_1} > 0 $在$ (n\tau , (n + l)\tau ] $上的自然死亡率; $ {r_1} > 0 $代表在$ (n\tau , (n + l)\tau ] $上$ I $的恢复率; $ 0 < {\mu} < 1 $代表在$ t = (n + l)\tau $时刻的脉冲接种率. $ {\lambda _2} > 0 $代表在$ ((n+l)\tau , (n + 1)\tau ] $上的出生系数; $ {\beta _2} > 0 $代表在$ ((n+l)\tau , (n + 1)\tau ] $上的传染系数; $ {d_2} > 0 $代表在$ ((n+l)\tau , (n + 1)\tau ] $上的自然死亡率; $ {r_2} > 0 $代表在$ ((n+l)\tau , (n + 1)\tau ] $上$ I $的恢复率; $ {\mu_1} > 0 $代表在$ ((n+l)\tau , (n + 1)\tau ] $上的连续接种率; $ (1-l)\tau $是非瞬时脉冲接种效应区间长度$ (0<l<1) $; $ {\tau}>0 $表示脉冲接种周期.

由于系统$ (2.1) $中有关$ R(t) $的方程是独立的, 因此我们分析仅与$ S(t) $和$ I(t) $有关的方程, 得到如下模型

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _1} - {\beta _1}S(t)I(t) - {d_1}S(t), \\ \frac{{\mathrm{d}I(t)}}{{dt}} = {\beta _1}S(t)I(t) - ({r_1}+{d_1})I(t), \end{array} \right\}t \in (n\tau , (n + l)\tau ], \\ \left. \begin{array}{llc} \Delta S(t) = - {\mu}S(t), \\ \Delta I(t) = 0, \end{array} \right\}t = (n + l)\tau , \\ \left. \begin{array}{llc} \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _2} - {\beta _2}S(t)I(t) - ({d_2}+{\mu_1})S(t) , \\ \frac{{\mathrm{d}I(t)}}{{dt}} = {\beta _2}S(t)I(t) - ({r_2}+{d_2})I(t), \end{array} \right\}t \in ((n + l)\tau , (n + 1)\tau ].\\ \end{array} \right. \end{equation} $

(2.2)

系统$ (2.1) $的解是是一个连续的分段函数$ X:{R_ + } \to R_ + ^3 $, 其解用$ X(t) = {(S(t), I(t), R(t))^\mathrm{T}} $表示, $ X(t) $在$ (n\tau , (n + l)\tau ] $和$ ((n + l), (n + 1)\tau ] $上是连续的, 且$ X(n\tau^{+})=\lim\limits_{t\rightarrow n\tau^{+}}X(t) $和$ X\left((n+l)\tau^{+}\right)=\lim\limits_{t\rightarrow (n+l)\tau^{+}}X(t) $存在. 由文献[13] 易知, $ X(t) $的全局存在性和唯一性由系统$ (2.1) $右边函数的光滑性保证.

由文献[14](147页) 我们容易得到如下引理.

引理3.1   假设$ X(t) $是系统$ (2.1) $的解且$ {X(0^+)}\ge 0 $, 那么当$ t\ge 0 $时, $ X(t)\ge 0 $.

引理3.2   函数$ m\in PC[R^+, R] $满足不等式

$ \begin{equation} \left\{ \begin{array}{lll} m(t)\leq p(t)m(t)+q(t), \\ t\geq t_{0}, t\neq t_{k}, k=1, 2, ..., \\ m(t_{k}^{+}\leq d_{k}m(t_{k})+b_{k}, t=t_{k}, \end{array} \right. \end{equation} $

(3.1)

其中$ p, q\in PC[R^+, R] $且$ {d_k}\ge 0 $, $ {b_k} $是常数. 那么

$ m(t)\leq m(t_{0})\Pi_{t_{0}<t_{k}<t}d_{k}{\rm{exp}}( \int_{t_{0}}^{t}p(s)ds)+\sum\limits_{t_{0}<t_{k}<t}(\Pi_{t_{k}<t_{j}<t}d_{j}{\rm{exp}}(\int_{t_{0}}^{t}p(s)ds))b_{k} \\ + \int_{t_{0}}^{t}\Pi_{t_{0}<t_{k}<t}d_{k}{\rm{exp}}(\int_{s}^{t}p(\sigma)d\sigma)q(s)ds, t\geq t_{0}. $

接下来, 我们证明系统$ (2.1) $的解都是最终一致有界的.

引理3.3   对于系统$ (2.1) $的解$ (S(t), I(t), R(t)) $, 当$ t $足够大时, 存在一个常数$ M>0 $, 使得$ S(t)\le M $, $ I(t)\le M $, $ R(t)\le M $.

证明: 定义$ V(t)=S(t)+I(t)+R(t) $且$ d=\rm{min}\{d_{1}, d_{2}, d_{2}-\mu_{1}\} $, 当$ t\in (n\tau , (n + l)\tau ] $时, 有$ D^{+}V(t)+dV(t)={\lambda _1}-({d_1}-d)S(t)-({d_1}-d)I(t)-({d_1}-d)R(t)\leq {\lambda _1} $.

当$ t\in ((n+l)\tau , (n + 1)\tau ] $时, 有$ D^{+}V(t)+dV(t)={\lambda _2}-({d_2}+{\mu_1}-d)S(t)-({d_2}-d)I(t)-({d_2}-{\mu_1}-d)R(t)\leq {\lambda _2} $.

取$ \lambda=\rm{max}\{{\lambda _1}, {\lambda _2}\} $, 当$ t=(n + l)\tau $时, $ V((n+l)\tau^{+})=S((n + l)\tau)+I((n + l)\tau)+R((n + l)\tau)=V((n + l)\tau) $.

当$ t=(n + 1)\tau $时, $ V((n+1)\tau^{+})=V((n + 1)\tau) $. 根据引理$ 3.2 $可知, 对于$ t\in (n\tau , (n + l)\tau ] $和$ t\in ((n+l)\tau , (n + 1)\tau ] $, 有

$ \begin{align} V(t)&\leq V(0){\rm{exp}}(-dt)+\int_{o}^{t}\lambda {\rm{exp}}(-d(t-s))ds\\ &=V(0){\rm{exp}}(-dt)+\frac{\lambda}{d}(1-{\rm{exp}}(-dt))\\ &\quad \rightarrow \frac{\lambda}{d}, \, \, t\rightarrow \infty \end{align} $

所以$ V(t) $是最终一致有界的. 因此由$ V(t) $的定义可知, 存在一个常数$ M>0 $, 当$ t $足够大时, 使得$ S(t)\le M $, $ I(t)\le M $, $ R(t)\le M $. 证毕.

当$ I(t)=0 $时, 系统$ (2.2) $的子系统如下:

$ \begin{equation} \left\{ \begin{array}{lll} \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _1} - {d_1}S(t), t \in (n\tau , (n + l)\tau ], \\ \Delta S(t) = - {\mu}S(t), t = (n + l)\tau , \\ \frac{{\mathrm{d}S(t)}}{{dt}} = {\lambda _2} - ({d_2}+{\mu_1})S(t) , t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

(3.2)

引理3.4   系统$ (3.2) $有全局渐近稳定的周期解

$ \begin{equation} \begin{array}{l} \widetilde {S(t)} = \left\{ \begin{array}{l} \frac{\lambda_1}{d_1}-(\frac{\lambda_1}{d_1}-S^{*})e^{-{d_1}(t-n\tau)}, t \in (n\tau , (n + l)\tau ], \\ \frac{\lambda_2}{{d_2}+{\mu_1}}-(\frac{\lambda_2}{{d_2}+{\mu_1}}-S^{**})e^{-({d_2}+{\mu_1})(t-(n+l)\tau)}, t \in ((n + l)\tau , (n + 1)\tau ], \end{array} \right. \end{array} \end{equation} $

(3.3)

其中$ S^{*}=\frac{\frac{\lambda_2}{{d_2}+{\mu_1}}+((1-{\mu})\frac{\lambda_1}{d_1}-\frac{\lambda_2}{{d_2}+{\mu_1}})A+(1-(1-{\mu})\frac{\lambda_1}{d_1})B}{1+{\mu}B} $, $ S^{**}=(1-{\mu})(\frac{\lambda_1}{d_1}-(\frac{\lambda_1}{d_1}-S^{*})e^{-{d_1}l\tau}) $, $ A=e^{-({d_2}+{\mu_1})(1-l)\tau}<1 $, $ B=e^{-(({d_2}+{\mu_1})(1-l)+{d_1}l)\tau}<1 $.

证明: 分别考虑系统$ (3.2) $的第一个和第三个方程, 我们得到了在脉冲点之间的解析解,

$ \begin{equation} \left\{ \begin{array}{lll} \ S(t)=\frac{\lambda_1}{d_1}-(\frac{\lambda_1}{d_1}-S(n\tau^{+}))e^{-{d_1}(t-n\tau)}, t \in (n\tau , (n + l)\tau ], \\ \ S(t)=\frac{\lambda_2}{{d_2}+{\mu_1}}-(\frac{\lambda_2}{{d_2}+{\mu_1}}-S((n+l)\tau^{+}))e^{-({d_2}+{\mu_1})(t-(n+l)\tau)}, t \in ((n + l)\tau , (n + 1)\tau ].\nonumber \end{array} \right. \label{a} \end{equation} $

其频闪映射为

$ \begin{equation} S((n+1)\tau^{+})=\frac{\lambda_2}{{d_2}+{\mu_1}}-(\frac{\lambda_2}{{d_2}+{\mu_1}}-(1-{\mu})(\frac{\lambda_1}{d_1}-(\frac{\lambda_1}{d_1}-S(n\tau^{+}))e^{-{d_1}l\tau}))e^{-({d_2}+{\mu_1})(1-l)\tau}. \end{equation} $

(3.4)

由此我们得到$ (3.4) $的唯一不动点

$ \begin{equation} \left\{ \begin{array}{lll} S^{*}=\frac{\frac{\lambda_2}{{d_2}+{\mu_1}}+((1-{\mu})\frac{\lambda_1}{d_1}-\frac{\lambda_2}{{d_2}+{\mu_1}})A+(1-(1-{\mu})\frac{\lambda_1}{d_1})B}{1+{\mu}B}, \\ S^{**}=(1-{\mu})(\frac{\lambda_1}{d_1}-(\frac{\lambda_1}{d_1}-S^{*})e^{-{d_1}l\tau}), \end{array} \right. \end{equation} $

其中$ \qquad A=e^{-({d_2}+{\mu_1})(1-l)\tau}<1 $, $ \qquad B=e^{-(({d_2}+{\mu_1})(1-l)+{d_1}l)\tau}<1 $.

令$ S^{n+1}=S((n+1)\tau^{+}) $, $ S^{n}=S(n\tau^{+}) $, 则$ (3.4) $转换为$ S^{n+1}=f(S^{n}) $. 对$ f(S^{n}) $求导我们有, $ \frac{dS^{n+1}}{dS^{n}}\bigg|_{S^{n} = S^{*}}=-{\mu}B<1 $, 其中$ B=e^{-(({d_2}+{\mu_1})(1-l)+{d_1}l)\tau}<1 $.

所以$ (3.4) $的唯一不动点是全局渐近稳定的, 由文献[15] 可知系统$ (3.2) $有全局渐近稳定的周期解$ \widetilde {S(t)} $.

引理3.5   系统$ (2.2) $的周期解$ ((\widetilde {S(t)}, 0)) $是全局渐近稳定的, 其中$ \widetilde {S(t)} $, $ S^{*} $和$ S^{**} $如$ (3.3) $所示.

定理4.1   如果

$ \begin{align} \frac{\beta_1}{d_1}&({\lambda_1}l\tau + \frac{{\lambda_1}-{d_1}S^{*}}{d_1}(e^{-{d_1}l\tau }-1))-({r_1}+{d_1})l\tau + \frac{\beta_2}{{d_2}+{\mu_1}}({\lambda_2}(1-l)\tau \\ &+\frac{{\lambda_2}-({d_2}+{\mu_1})S^{**}}{{d_2}+{\mu_1}}(e^{-({d_2}+{\mu_1})(1-l)\tau}-1))-({r_2}+{d_2})(1-l)\tau < 0, \end{align} $

(4.1)

成立, 则系统$ (2.2) $的无病周期解$ (\widetilde {S(t)}, 0) $是全局渐近稳定的, 其中$ S^{*} $和$ S^{**} $如$ (3.3) $所示.

证明: 首先证明$ (\widetilde {S(t)}, 0) $的局部稳定性, 定义$ {S_1}(t)=S(t)-\widetilde {S(t)} $, $ {I_1}(t)=I(t) $, 则$ (2.2) $对应的线性系统可写为

$ \begin{equation} \left( \begin{array}{*{20}{c}} \frac{d{S_1}(t)}{dt}\\ \frac{d{I_1}(t)}{dt} \end{array} \right ) = \left( \begin{array}{*{20}{c}} {-{d_1}}&{-{\beta_1}{\widetilde {S(t)}}}\\ {0}&{{\beta_1}{\widetilde {S(t)}}-({r_1}+{d_1})} \end{array} \right ) \left( \begin{array}{*{20}{c}} {S_1}(t)\\ {I_1}(t) \end{array} \right ), t \in (n\tau , (n + l)\tau ].\nonumber \end{equation} $

得到基解矩阵为

$ \begin{equation} {\Phi_1}(t) = \left( {\begin{array}{*{20}{c}} {{\rm{exp}}(-{d_1}t)}&{*_1}\\ {0}&{{\rm{exp}}( \int_{0}^{t}{\beta_1}\widetilde {S(s)}-({r_1}+{d_1})ds)} \end{array}} \right).\nonumber \end{equation} $

系统$ (2.2) $的第三个和第四个方程对应的线性化矩阵为

$ \begin{equation} \left( \begin{array}{*{20}{c}} {S_1}((n+l)\tau^{+})\\ {I_1}((n+l)\tau^{+}) \end{array} \right ) = \left( \begin{array}{*{20}{c}} {1-\mu}&{0}\\ {0}&{1} \end{array} \right ) \left( \begin{array}{*{20}{c}} {S_1}((n+l)\tau)\\ {I_1}((n+l)\tau) \end{array} \right ).\nonumber \end{equation} $

同理我们有

$ \begin{equation} \left( \begin{array}{*{20}{c}} \frac{d{S_1}(t)}{dt}\\ \frac{d{I_1}(t)}{dt} \end{array} \right ) = \left( \begin{array}{*{20}{c}} {-({d_2}+{\mu_1})}&{-{\beta_2}{\widetilde {S(t)}}}\\ {0}&{{\beta_2}{\widetilde {S(t)}}-({r_2}+{d_2})} \end{array} \right ) \left( \begin{array}{*{20}{c}} {S_1}(t)\\ {I_1}(t) \end{array} \right ), t \in ((n + l)\tau , (n + 1)\tau ], \nonumber \end{equation} $

$ \begin{equation} {\Phi_2}(t) = \left( {\begin{array}{*{20}{c}} {{\rm{exp}}(-({d_2}+{\mu_1})t)}&{*_2}\\ {0}&{{\rm{exp}}( \int_{0}^{t}{\beta_2}\widetilde {S(s)}-({r_2}+{d_2})ds)} \end{array}} \right).\nonumber \end{equation} $

$ \begin{equation} \left( \begin{array}{*{20}{c}} {S_1}((n+1)\tau^{+})\\ {I_1}((n+1)\tau^{+}) \end{array} \right ) = \left( \begin{array}{*{20}{c}} {1}&{0}\\ {0}&{1} \end{array} \right ) \left( \begin{array}{*{20}{c}} {S_1}((n+1)\tau)\\ {I_1}((n+1)\tau) \end{array} \right ).\nonumber \end{equation} $

因为在接下来的运算中不需要$ {*_i}(i=1, 2) $, 所以不计算它的精确值. 周期解$ (\widetilde {S(t)}, 0) $的稳定性由

$ \begin{equation} {M} = \left( {\begin{array}{*{20}{c}} {1-\mu}&{0}\\ {0}&{1} \end{array}} \right) \left( \begin{array}{*{20}{c}} {1}&{0}\\ {0}&{1} \end{array} \right ){\Phi_1}(l\tau){\Phi_2}(\tau), \nonumber \end{equation} $

的特征值$ {\lambda_1}=(1-\mu)e^{-({d_1}l+{d_2}+{\mu_1})\tau}<1 $, $ {\lambda_2}={k_1}{k_2} $决定. 其中$ {k_1}=e^{\int_{0}^{l\tau}{\beta_1}\widetilde {S(s)}-({r_1}+{d_1})ds} $, $ {k_2}=e^{\int_{0}^{\tau}{\beta_2}\widetilde {S(s)}-({r_2}+{d_2})ds} $.

根据条件$ (4.1) $和Floquet定理^[13] $ (即{\rho_i}={\rm{exp}}( \int_{0}^{\tau}{\beta_i}\widetilde {S(s)}-({r_i}+{d_i})ds)<1) $, 得到$ {\lambda_2}<1 $成立, 系统$ (2.2) $的无病周期解$ (\widetilde {S(t)}, 0) $局部稳定.

接下来将证明全局吸引性, 令一个$ \varepsilon >0 $, 使得$ {\rho_i} = {\rm{exp}}( \int_{0}^{\tau}({\beta_i}\widetilde {S(s)}+\varepsilon)-({r_i}+{d_i})ds)<1(i=1, 2) $. 由系统$ (2.2) $的第一和第五个方程可知

$ \begin{equation} \left\{ \begin{array}{lll} \frac{dS(t)}{dt}\leq {\lambda_1}-{d_1}S(t), t \in (n\tau , (n + l)\tau ], \\ \frac{dS(t)}{dt}\leq {\lambda_2}-({d_2}+{\mu_1})S(t), t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

考虑如下脉冲微分方程

$ \begin{equation} \left\{ \begin{array}{lll} \frac{{\mathrm{d}{S_2}(t)}}{{dt}} = {\lambda _1} - {d_1}{S_2}(t), t \in (n\tau , (n + l)\tau ], \\ \Delta {S_2}(t) = - {\mu}{S_2}(t), t = (n + l)\tau , \\ \frac{{\mathrm{d}{S_2}(t)}}{{dt}} = {\lambda _2} - ({d_2}+{\mu_1}){S_2}(t) , t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

(4.2)

依据脉冲微分方程比较定理$ ( $参考文献[13] 中的定理$ 3.1.1) $及引理$ 3.5 $, 有$ S(t)\leq {S_2}(t) $, 且当$ t \rightarrow \infty $时, $ {S_2}(t)\rightarrow \widetilde {S(t)} $. 即当所有$ t $充分大时,

$ \begin{equation} S(t)\leq {S_2}(t)\leq\widetilde {S(t)}+\varepsilon, \end{equation} $

(4.3)

为了方便, 假设$ (4.3) $对所有的$ t\ge 0 $均成立. 由$ (2.2) $和$ (4.3) $可以得到

$ \begin{equation} \left\{ \begin{array}{lll} \frac{dI(t)}{dt}\leq (({\beta_1}(\widetilde {S(t)}+\varepsilon)-({r_1}+{d_1})))I(t) , t \in (n\tau , (n + l)\tau ], \\ \frac{dI(t)}{dt}\leq (({\beta_2}(\widetilde {S(t)}+\varepsilon)-({r_2}+{d_2})))I(t) , t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

(4.4)

因此

$ \begin{align} I((n+1)\tau)\leq& I((n+l)\tau^{+}){\rm{exp}}(\int_{n\tau}^{(n+1)\tau}({\beta_2}(\widetilde {S(s)}+\varepsilon)-({r_2}+{d_2}))ds)\\ \leq& I(n\tau^{+}){\rm{exp}}(\int_{n\tau}^{(n+l)\tau}({\beta_1}(\widetilde {S(s)}+\varepsilon)-({r_1}+{d_1}))ds)\\ &{\rm{exp}}(\int_{(n+l)\tau}^{(n+1)\tau}({\beta_2}(\widetilde {S(s)}+\varepsilon)-({r_2}+{d_2}))ds)\\ =&I(n\tau^{+}){\rho_1}{\rho_2}. \end{align} $

所以$ I(n\tau)\leq I(0^{+})({\rho_1}{\rho_2})^{n} $且当$ n\rightarrow \infty $, $ I(n\tau)\rightarrow 0 $. 因此当$ t\rightarrow \infty $, $ I(t)\rightarrow 0 $.

接下来, 将证明当$ t\rightarrow \infty $, $ \widetilde {S(t)}\rightarrow S(t) $. 对于$ {\varepsilon_1}>0 $, 一定存在一个$ {t_0}>0 $, 使得对任意$ t\ge{t_0} $有$ 0<I(t)<{\varepsilon_1} $. 为了不失一般性, 假设对于所有的$ t\ge0 $有$ 0<I(t)<{\varepsilon_1} $. 根据系统$ (2.2) $可以得到

$ \begin{equation} \left\{ \begin{array}{lll} {\lambda_1}-({\beta_1}{\varepsilon_1}+{d_1})S(t)\leq \frac{dS(t)}{dt}\leq {\lambda_1}-{d_1}{S_2}(t), t \in (n\tau , (n + l)\tau ], \\ {\lambda_2}-({\beta_2}{\varepsilon_1}+({d_2}+{\mu_1}))S(t)\leq \frac{dS(t)}{dt}\leq {\lambda_2}-({d_2}+{\mu_1}){S_2}(t) , t \in ((n + l)\tau , (n + 1)\tau ], \end{array} \right. \end{equation} $

(4.5)

且当$ t \rightarrow \infty $时, 有$ {S_3}(t)\leq S(t)\leq {S_2}(t) $, $ {S_3}(t)\rightarrow \widetilde {{S_3}(t)} $和$ {S_2}(t)\rightarrow \widetilde {S(t)} $. $ {S_3}(t) $和$ {S_2}(t) $分别是系统$ (4.2) $和$ (4.6) $的解,

$ \begin{equation} \left\{ \begin{array}{lll} \frac{{\mathrm{d}{S_3}(t)}}{{dt}} = {\lambda _1} - ({\beta_1}{\varepsilon_1}+{d_1}){S_3}(t), t \in (n\tau , (n + l)\tau ], \\ \Delta {S_3}(t) = - {\mu}{S_3}(t), t = (n + l)\tau , \\ \frac{{\mathrm{d}{S_3}(t)}}{{dt}} = {\lambda _2} - ({\beta_2}{\varepsilon_1}+({d_2}+{\mu_1})){S_3}(t) , t \in ((n + l)\tau , (n + 1)\tau ], \end{array} \right. \end{equation} $

(4.6)

其中

$ \begin{equation} \begin{array}{l} \widetilde {{S_3}(t)} = \left\{ \begin{array}{l} \frac{1}{{d_1}+{\beta_1}{\varepsilon_1}}({\lambda _1}-({\lambda _1}-({d_1}+{\beta_1}{\varepsilon_1}){S_3}^{*})e^{-({d_1}+{\beta_1}{\varepsilon_1})(t-n\tau)}), t \in (n\tau , (n + l)\tau ], \\ \frac{1}{{d_2}+{\mu_1}+{\beta_2}{\varepsilon_1}}({\lambda _2}-({\lambda _2}-({d_2}+{\mu_1}+{\beta_2}{\varepsilon_1}){S_3}^{**})e^{-({d_2}+{\mu_1}+{\beta_2}{\varepsilon_1})(t-(n+l)\tau)}), \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad t \in ((n + l)\tau , (n + 1)\tau ], \end{array} \right. \end{array} \end{equation} $

(4.7)

这里的$ {S_3}^{*} $和$ {S_3}^{**} $定义如下:

$ \begin{equation} \left\{ \begin{array}{lll} {S_3}^{*}=\frac{\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}{\varepsilon_1}}+((1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}{\varepsilon_1}} -\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}{\varepsilon_1}}){A_1}+(1-(1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}{\varepsilon_1}}){B_1}}{1+{\mu}{B_1}}>0, \\ {S_3}^{**}=(1-{\mu})(\frac{\lambda_1}{{d_1}+{\beta_1}{\varepsilon_1}}-(\frac{\lambda_1}{{d_1}+{\beta_1}{\varepsilon_1}}-{S_3}^{*})e^{-({{d_1}+{\beta_1} {\varepsilon_1}})l\tau}), \end{array} \right. \end{equation} $

(4.8)

其中$ {A_1}=e^{-({d_2}+{\mu_1}+{\beta_2}{\varepsilon_1})(1-l)\tau}<1 $, $ {B_1}=e^{-(({d_2}+{\mu_1}+{\beta_2}{\varepsilon_1})(1-l)+{d_1}l)\tau}<1 $.

对于任意的$ {\varepsilon_2}>0 $, 存在一个$ {t_1}>0 $, 当$ t>{t_1} $时, 使得$ t $足够大时, 有$ \widetilde{{S_3}(t)}-{\varepsilon_2}\leq S(t)\leq \widetilde{S(t)}+{\varepsilon_2} $, 即$ t \rightarrow \infty $时, $ S(t)\rightarrow \widetilde {S(t)} $. 证毕.

接下来证明系统$ (2.2) $的持久性:

定义4.1   如果存在常数$ m $, $ M>0 $(与初值无关)和一个有限的时间$ {T_0} $, 使得对所有的$ t\ge {T_0} $, 解$ (S(t), I(t)) $的所有初值$ S(0^{+})>0 $, $ I(0^{+})>0 $, $ m\leq S(t)\leq M $, $ m\leq I(t)\leq M $成立, 则系统$ (2.2) $是持久的.

定理4.2   如果

$ \begin{align} \frac{\beta_1}{d_1}&({\lambda_1}l\tau + \frac{{\lambda_1}-{d_1}S^{*}}{d_1}(e^{-{d_1}l\tau }-1))-({r_1}+{d_1})l\tau + \frac{\beta_2}{{d_2}+{\mu_1}}({\lambda_2}(1-l)\tau \\ &+\frac{{\lambda_2}-({d_2}+{\mu_1})S^{**}}{{d_2}+{\mu_1}}(e^{-({d_2}+{\mu_1})(1-l)\tau}-1))-({r_2}+{d_2})(1-l)\tau > 0, \end{align} $

(4.9)

成立, 则系统$ (2.2) $是持久的, 其中$ S^{*} $和$ S^{**} $如$ (3.3) $所示.

证明: 假设$ (S(t), I(t)) $是系统$ (2.2) $关于初值$ S(0)>0 $, $ I(0)>0 $的一个解. 通过引理$ 3.3 $, 证明了存在一个常数$ M>0 $, 当$ t $足够大时, $ S(t)\leq M $, $ I(t)\leq M $. 由系统$ (2.2) $, 当$ t $足够大时, 有

$ \begin{equation} \left\{ \begin{array}{lll} I(t)> I(0^{+})e^{-({r_1}+{d_1})} , t \in (0 , l\tau ], \\ I(t)>I(l\tau^{+})e^{-({r_2}+{d_2})} , t \in ( l\tau , \tau ], \end{array} \right. \end{equation} $

因此只需要找到$ {m_1}>0 $和$ {\varepsilon_3} $, 当$ t $足够大时, 使得$ I(t)\ge {m_1} $. 否则, 我们可以选择$ {m_2}>0 $足够小, 证明当$ t\ge0 $时, $ I(t)\ge {m_2} $不成立. 根据条件$ (4.9) $得到

$ \sigma = \frac{\beta_1}{{d_1}+{\beta_1}{m_2}}({\lambda_1}l\tau + \frac{{\lambda_1}-({{d_1}+{\beta_1}{m_2}}){S_4}^{*}}{d_1}(e^{-{{d_1}+{\beta_1}{m_2}}l\tau }-1))-({\beta_1}{\varepsilon_3}+{r_1}+{d_1}+{\beta_1}{m_2})l\tau + \frac{\beta_2}{{d_2}+{\mu_1}+{\beta_2}{m_2}}({\lambda_2}(1-l)\tau + \frac{{\lambda_2}-({d_2}+{\mu_1}+{\beta_2}{m_2}){S_4}^{**}}{{d_2}+{\mu_1}+{\beta_2}{m_2}}(e^{-({d_2}+{\mu_1} +{\beta_2}{m_2})(1-l)\tau}-1))-({\beta_2}{\varepsilon_3}+{r_2}+\qquad{d_2}+{\beta_2}{m_2})(1-l)\tau > 0 $, 其中$ {S_4}^{*} $和$ {S_4}^{**} $如下所示

$ \begin{equation} \left\{ \begin{array}{lll} {S_4}^{*}=\frac{\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}{m_2}}+((1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}{m_2}} -\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}{m_2}}){A_2}+(1-(1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}{m_2}}){B_2}}{1+{\mu}{B_2}}>0, \\ {S_4}^{**}=(1-{\mu})(\frac{\lambda_1}{{d_1}+{\beta_1}{m_2}}-(\frac{\lambda_1}{{d_1}+{\beta_1}{m_2}}-{S_4}^{*})e^{-({{d_1}+{\beta_1} {m_2}})l\tau}), \end{array} \right. \end{equation} $

(4.10)

其中$ \qquad {A_2}=e^{-({d_2}+{\mu_1}+{\beta_2}{m_2})(1-l)\tau}<1 $, $ \qquad {B_2}=e^{-(({d_2}+{\mu_1}+{\beta_2}{m_2})(1-l)+{d_1}l)\tau}<1 $.

因此, 存在一个$ {T_1}>0 $和$ {\varepsilon_3}>0 $, 使得$ S(t)\ge {S_4}(t)\ge \widetilde {{S_4}(t)}-{\varepsilon_3} $. 对于$ t\ge {T_1} $,

$ \begin{equation} \left\{ \begin{array}{lll} \frac{dI(t)}{dt}\ge (({\beta_1}(\widetilde {{S_4}(t)}+{\varepsilon_3})-({r_1}+{d_1})))I(t) , t \in (n\tau , (n + l)\tau ], \\ \frac{dI(t)}{dt}\ge (({\beta_2}(\widetilde {{S_4}(t)}+{\varepsilon_3})-({r_2}+{d_2})))I(t) , t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

(4.11)

令$ {N_1}\in N $且$ {N_1}\tau > {T_1} $, 将系统$ (4.11) $在区间$ (n\tau , (n + 1)\tau ) $上积分, 当$ n\ge {N_1} $有

$ \begin{align*} I((n+1)\tau)\ge& I(n\tau^{+}){\rm{exp}}((\int_{n\tau}^{(n+l)\tau}({\beta_1}(\widetilde {{S_4}(s)}+{\varepsilon_3})-({r_1}+{d_1}))ds)\\ &+(\int_{(n+l)\tau}^{(n+1)\tau}({\beta_2}(\widetilde {{S_4}(s)}+ {\varepsilon_3})-({r_2}+{d_2}))ds))\nonumber\\ =&I(n\tau^{+})e^{\sigma}.\nonumber \end{align*} $

所以, 当$ k\rightarrow \infty $, $ I(({N_1}+k)\tau)\ge I({N_1}\tau^{+})e^{k\sigma}\rightarrow \infty $, 结果和$ I(t) $有界矛盾. 因此, 存在$ {t_1}>0 $, 使得$ I(t)\ge {m_1} $.

接下来有

$ \begin{equation} \left\{ \begin{array}{lll} \frac{{\mathrm{d}S(t)}}{{dt}} > {\lambda _1} - ({\beta_1}M+{d_1})S(t), t \in (n\tau , (n + l)\tau ], \\ \frac{{\mathrm{d}S(t)}}{{dt}} > {\lambda _2} - ({\beta_2}M+{d_2}+{\mu_1})S(t) , t \in ((n + l)\tau , (n + 1)\tau ].\\ \end{array} \right. \end{equation} $

下面是脉冲微分比较方程

$ \begin{equation} \left\{ \begin{array}{lll} \frac{{\mathrm{d}{S_5}(t)}}{{dt}} ={\lambda _1} - ({\beta_1}M+{d_1}){S_5}(t), t \in (n\tau , (n + l)\tau ], \\ \Delta {S_5}(t) = - {\mu}{S_5}(t), t = (n + l)\tau , \\ \frac{{\mathrm{d}{S_5}(t)}}{{dt}} = {\lambda _2} - ({\beta_2}M+{d_2}+{\mu_1}){S_5}(t) , t \in ((n + l)\tau , (n + 1)\tau ]. \end{array} \right. \end{equation} $

(4.12)

与引理$ 3.5 $相似, 有

$ \begin{equation} \begin{array}{l} \widetilde {{S_5}(t)} = \left\{ \begin{array}{l} \frac{1}{{d_1}+{\beta_1}M}({\lambda _1}-({\lambda _1}-({d_1}+{\beta_1}M){S_5}^{*})e^{-({d_1}+{\beta_1}M)(t-n\tau)}), t \in (n\tau , (n + l)\tau ], \\ \frac{1}{{d_2}+{\mu_1}+{\beta_2}M}({\lambda _2}-({\lambda _2}-({d_2}+{\mu_1}+{\beta_2}M){S_5}^{**})\\e^{-({d_2}+{\mu_1}+{\beta_2}M)(t-(n+l)\tau)}), t \in ((n + l)\tau , (n + 1)\tau ], \end{array} \right. \end{array} \end{equation} $

(4.13)

这里的$ {S_5}^{*} $和$ {S_5}^{**} $定义如下:

$ \begin{equation} \left\{ \begin{array}{lll} {S_5}^{*}=\frac{\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}M}+((1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}M} -\frac{\lambda_2}{{d_2}+{\mu_1}+{\beta_2}M}){A_3}+(1-(1-{\mu})\frac{\lambda_1}{{d_1}+{\beta_1}M}){B_3}}{1+{\mu}{B_3}}>0, \\ {S_5}^{**}=(1-{\mu})(\frac{\lambda_1}{{d_1}+{\beta_1}M}-(\frac{\lambda_1}{{d_1}+{\beta_1}M}-{S_5}^{*})e^{-({{d_1}+{\beta_1} M})l\tau}), \end{array} \right. \end{equation} $

(4.14)

其中$ \qquad {A_3}=e^{-({d_2}+{\mu_1}+{\beta_2}M)(1-l)\tau}<1 $, $ \qquad {B_3}=e^{-(({d_2}+{\mu_1}+{\beta_2}M)(1-l)+{d_1}l)\tau}<1 $.

对于任意的$ {\varepsilon_4} $足够小, 有$ {S_5}(t)>\widetilde {{S_5}(t)}-{\varepsilon_4} $. 根据脉冲微分方程比较定理, 有$ S(t)>{S_5}(t)>\widetilde {{S_5}(t)}-{\varepsilon_4}>({S_5}^{*}+ {S_5}^{**})-{\varepsilon_4}={m_5} $. 证毕.

推论4.1   如果

$ \begin{align} \frac{\beta_1}{d_1}&({\lambda_1}l\tau + \frac{{\lambda_1}-{d_1}S^{*}}{d_1}(e^{-{d_1}l\tau }-1))-({r_1}+{d_1})l\tau + \frac{\beta_2}{{d_2}+{\mu_1}}({\lambda_2}(1-l)\tau \\ &+\frac{{\lambda_2}-({d_2}+{\mu_1})S^{**}}{{d_2}+{\mu_1}}(e^{-({d_2}+{\mu_1})(1-l)\tau}-1))-({r_2}+{d_2})(1-l)\tau > 0, \end{align} $

(4.15)

成立, 则系统$ (2.1) $是持久的, 其中$ S^{*} $和$ S^{**} $如$ (3.3) $所示.

本文建立了具脉冲接种与接种效应的$ SIR $传染病模型, 该模型描述了疫苗在人体内起作用的过程, 并证明了该系统的所有解都是一致最终有界的.如果$ (4.1) $成立, 系统$ (2.1) $的无病周期解是全局渐近稳定的. 如果$ (4.9) $成立, 系统$ (2.1) $是持久的. 根据条件$ (4.1) $和$ (4.9) $, 我们推测疫苗接种初次脉冲起效存在阈值$ \mu^{*} $, 当$ \mu > \mu^{*} $时, 疾病灭绝; 当$ \mu < \mu^{*} $时, 疾病持久. 这表明, 疫苗的初次接种率在疾病灭绝中起着重要作用. 我们也推测疫苗的连续接种率存在阈值$ {\mu_1}^{*} $, 当$ {\mu_1} > {\mu_1}^{*} $时, 疾病灭绝; 当$ {\mu_1} < {\mu_1}^{*} $时, 疾病持久. 该结论表明, 连续接种率对系统无病周期解的稳定性起着重要作用. 同时, 我们还可以推测非瞬时脉冲接种起效区间系数存在阈值$ l^{*} $, 当$ l>l^{*} $时, 疾病灭绝; 当$ l<l^{*} $时, 疾病持久. 根据以上讨论我们可以得知, 控制疾病的减少, 可以适当提高初次接种率或连续接种率, 也可以延长非瞬时脉冲接种起效区间长度. 该模型为现实生活中通过疫苗接种来减少疾病传播提供了可靠的决策支持.