﻿ 具分布时滞的SIR传染病模型的脉冲免疫控制

# 具分布时滞的SIR传染病模型的脉冲免疫控制Pulse Vaccination Control of an SIR Epidemic Model with Distributed Delays

Abstract: This paper investigates a class of an SIR epidemic model with pulse vaccination and distributed delays. By using the impulsive comparison theory and analysis technique, we obtain the sufficient conditions on existence and global asymptotic stability of disease-free periodic solution. Further-more, the permanence of the model is also studied.

1. 引言

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)=\mu -\beta {\int }_{0}^{\tau }f\left(u\right)S\left(t-u\right)I\left(t-u\right)\text{d}u-\mu S\left(t\right),t\ne nT\\ {I}^{\prime }\left(t\right)=\beta {\int }_{0}^{\tau }f\left(u\right)S\left(t-u\right)I\left(t-u\right)\text{d}u-\mu I\left(t\right)-\gamma I\left(t\right),t\ne nT\\ {R}^{\prime }\left(t\right)=\gamma I\left(t\right)-\mu R\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\\ I\left({t}^{+}\right)=I\left(t\right),t=nT\\ R\left({t}^{+}\right)=R\left(t\right)+pR\left(t\right),t=nT\end{array}$ (1)

2. 基本准备

$\Omega =\left\{\left(S,I,R\right)\in {R}^{3}|0\le S,I,R\le 1,且S+I+R=1\right\}.$

$R\left(t\right)=1-\left(S\left(t\right)+I\left(t\right)\right)$ 代入系统(1)，我们主要研究下列二维时滞系统：

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)=\mu -\beta {\int }_{0}^{\tau }f\left(u\right)S\left(t-u\right)I\left(t-u\right)\text{d}u-\mu S\left(t\right),t\ne nT\\ {I}^{\prime }\left(t\right)=\beta {\int }_{0}^{\tau }f\left(u\right)S\left(t-u\right)I\left(t-u\right)\text{d}u-\mu I\left(t\right)-\gamma I\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\\ I\left({t}^{+}\right)=I\left(t\right),t=nT\end{array}$ (2)

${x}^{*}\left(t\right)=\frac{a}{b}\left[1-\frac{p{\text{e}}^{-b\left(t-nT\right)}}{1-\left(1-p\right){\text{e}}^{-bT}}\right],nT

${x}^{\prime }\left(t\right)=a{\int }_{0}^{\tau }f\left(s\right)x\left(t-s\right)\text{d}s-bx\left(t\right),$

3. 系统的灭绝性和持久性

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)=\mu -\mu S\left(t\right),t\ne nT,\\ S\left({t}^{\text{+}}\right)=\left(1-p\right)S\left(t\right),t=nT.\end{array}$

${S}^{*}\left(t\right)=1-\frac{p{\text{e}}^{-\mu \left(t-nT\right)}}{1-\left(1-p\right){\text{e}}^{-\mu T}},t\in \left(nT,\left(n+1\right)T\right].$ (3)

${R}_{1}:=\frac{\beta \left(1-{\text{e}}^{-\mu T}\right)}{\left[1-\left(1-p\right){\text{e}}^{-\mu T}\right]\left[\gamma +\mu \right]}<1,$

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)\le \mu -\mu S\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\end{array}$

$\left\{\begin{array}{l}{{S}^{\prime }}_{1}\left(t\right)=\mu -\mu {S}_{1}\left(t\right),t\ne nT\\ {S}_{1}\left({t}^{+}\right)=\left(1-p\right){S}_{1}\left(t\right),t=nT\end{array}$ (4)

${S}^{*}\left(t\right)\le \frac{1-{\text{e}}^{-\mu T}}{1-\left(1-p\right){\text{e}}^{-\mu T}}:=\sigma ,$

${I}^{\prime }\left(t\right)\le \beta \left(\sigma +{\epsilon }_{1}\right){\int }_{0}^{\tau }f\left(u\right)I\left(t-u\right)\text{d}u-\mu I\left(t\right)-\gamma I\left(t\right).$

${{I}^{\prime }}_{1}\left(t\right)=\beta \left(\sigma +{\epsilon }_{1}\right){\int }_{0}^{\tau }f\left(u\right){I}_{1}\left(t-u\right)\text{d}u-\mu {I}_{1}\left(t\right)-\gamma {I}_{1}\left(t\right).$

${\int }_{0}^{\tau }f\left(u\right)\text{d}u=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}^{*}\left(t\right)=1-\frac{p{\text{e}}^{-\mu \left(t-nT\right)}}{1-\left(1-p\right){\text{e}}^{-\mu T}}\le \frac{1-{\text{e}}^{-\mu T}}{1-\left(1-p\right){\text{e}}^{-\mu T}}:=\theta >0.$

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)\ge \mu \left(1-\epsilon \right)+\beta \theta \epsilon +\mu \rho \epsilon -\mu S\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\\ S\left({0}^{+}\right)=S\left(0\right)\end{array}$

$\left\{\begin{array}{l}{{S}^{\prime }}_{2}\left(t\right)=\mu \left(1-\epsilon \right)+\beta \theta \epsilon +\mu \rho \epsilon -\mu {S}_{2}\left(t\right),t\ne nT\\ {S}_{2}\left({t}^{+}\right)=\left(1-p\right){S}_{2}\left(t\right),t=nT\\ {S}_{2}\left({0}^{+}\right)=S\left(0\right)\end{array}$

${S}_{2}\left(t\right)\to \stackrel{˜}{S}\left(t\right)=\frac{\mu \left(1-\epsilon \right)+\beta \theta \epsilon +\mu \rho \epsilon }{\mu }\left[1-\frac{p{\text{e}}^{-\mu \left(t-kT\right)}}{1-\left(1-p\right){\text{e}}^{-\mu T}}\right],nT

${S}^{*}\left(t\right)+{\epsilon }_{1}\le S\left(t\right)\le \stackrel{˜}{S}\left(t\right)-{\epsilon }_{2}$

${\epsilon }_{1},{\epsilon }_{2},\epsilon \to 0,t\to \infty$，可得 $S\left(t\right)\to \stackrel{˜}{S}\left(t\right)\to {S}^{*}\left(t\right)$。从而得无病周期解全局吸引性。证毕。

${R}_{2}=\frac{\beta \left(1-p\right)\left(1-{\text{e}}^{-\mu T}\right)}{\left[1-\left(1-p\right){\text{e}}^{-\mu T}\right]\left[\gamma +\mu \right]}>1.$

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)\ge \mu -\left(\beta +\mu \right)S\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\end{array}$

$\forall {\epsilon }_{\text{0}}>0,\exists {T}_{\text{0}}>0$，使得当 $t>{T}_{0}$ 时，有 $S\left(t\right)\ge {S}_{0}^{\ast }\left(t\right)-\epsilon$，其中 ${S}_{0}^{\ast }\left(t\right)$ 为下述系统

$\left\{\begin{array}{l}{{S}^{\prime }}_{0}\left(t\right)=\mu -\left(\beta +\mu \right){S}_{0}\left(t\right),t\ne nT\\ {S}_{0}^{+}\left(t\right)=\left(1-p\right){S}_{0}\left(t\right),t=nT\end{array}$

${S}_{0}^{\ast }\left(0\right)=\frac{\mu \left(1-p\right)\left(1-{\text{e}}^{-\left(\beta +\mu \right)T}\right)}{\left(\beta +\mu \right)\left[1-\left(1-p\right){\text{e}}^{-\left(\beta +\mu \right)T}\right]}.$

$S\left(t\right)\ge {S}_{0}^{\ast }\left(0\right)-{\epsilon }_{0}=\frac{\mu \left(1-p\right)\left(1-{\text{e}}^{-\left(\beta +\mu \right)T}\right)}{\left(\beta +\mu \right)\left[1-\left(1-p\right){\text{e}}^{-\left(\beta +\mu \right)T}\right]}-{\epsilon }_{0}:={m}_{s}>0.$

$\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}S\left(t\right)\ge {m}_{s}$

$\left\{\begin{array}{l}{S}^{\prime }\left(t\right)\ge \mu -\beta S\left(t\right)\delta -\mu S\left(t\right),t\ne nT\\ S\left({t}^{+}\right)=\left(1-p\right)S\left(t\right),t=nT\end{array}$

$\left\{\begin{array}{l}{{S}^{\prime }}_{3}\left(t\right)\ge \mu -\beta {S}_{3}\left(t\right)\delta -\mu {S}_{3}\left(t\right),t\ne nT\\ {S}_{3}\left({t}^{+}\right)=\left(1-p\right){S}_{3}\left(t\right),t=nT\end{array}$

${S}_{3}^{\ast }\left(0\right)=\frac{\mu \left(1-p\right)\left(1-{\text{e}}^{-\left(\beta \delta +\mu \right)T}\right)}{\left(\beta \delta +\mu \right)\left[1-\left(1-p\right){\text{e}}^{-\left(\beta \delta +\mu \right)T}\right]}.$

$S\left(t\right)\ge {S}_{3}^{\ast }\left(0\right)-\epsilon =\frac{\mu \left(1-p\right)\left(1-{\text{e}}^{-\left(\beta \delta +\mu \right)T}\right)}{\left(\beta \delta +\mu \right)\left[1-\left(1-p\right){\text{e}}^{-\left(\beta \delta +\mu \right)T}\right]}-\epsilon :=\omega >0.$

$W\left(t\right)=I\left(t\right)+\beta {\int }_{0}^{\tau }f\left(u\right){\int }_{t-u}^{t}S\left(\theta \right)I\left(\theta \right)\text{d}\theta \text{d}u\text{ }\text{ }.$

$\begin{array}{c}{W}^{\prime }\left(t\right)={I}^{\prime }\left(t\right)+\beta {\int }_{0}^{\tau }f\left(u\right)\left(S\left(t\right)I\left(t\right)-S\left(t-s\right)I\left(t-s\right)\right)\text{d}u\\ =\beta {\int }_{0}^{\tau }f\left(u\right)S\left(t\right)I\left(t\right)\text{d}u-\left[\gamma +\mu \right]I\left(t\right)\\ =\left[\gamma +\mu \right]\left[\frac{\beta S\left(t\right){\int }_{0}^{\tau }f\left(u\right)\text{d}u}{\gamma +\mu }-1\right]I\left(t\right)\\ =\left[\gamma +\mu \right]\left[\frac{\beta S\left(t\right)}{\gamma +\mu }-1\right]I\left(t\right)\end{array}$ (5)

$S\left(t\right)\ge \omega$ 代入得

${W}^{\prime }\left(t\right)\ge \left[\gamma +\mu \right]\left[\frac{\beta \omega }{\gamma +\mu }-1\right]I\left(t\right).$

$m=\underset{t\in \left[{T}_{1},{T}_{1}+\tau \right]}{\mathrm{min}}I\left(t\right)>0$，则对于所有 $t>{T}_{2}$$I\left(t\right)\ge m$ 成立。否则存在 ${T}_{2}>0$，使得当 $t\in \left[{T}_{1},{T}_{1}+\tau +{T}_{2}\right]$ 时，有 $I\left(t\right)\ge {I}^{l}m$$I\left({T}_{1}+\tau +{T}_{2}\right)=m,{I}^{\prime }\left({T}_{1}+\tau +{T}_{2}\right)\le 0$。由系统(2)有

${I}^{\prime }\left({T}_{1}+\tau +{T}_{2}\right)\ge \left[\gamma +\mu \right]\left[\frac{\beta \omega }{\gamma +\mu }-1\right]{I}^{l}>\text{0}$

${W}^{\prime }\left(t\right)\ge \left[\gamma +\mu \right]\left[\frac{\beta \omega }{\gamma +\mu }-1\right]{I}^{l}>\text{0},$

$I\left({t}_{2k-1}\right)=I\left({t}_{2k}\right)=\delta ,I\left(t\right)<\delta +\epsilon ,t\in \left[{t}_{2k-1},{t}_{2k}\right];$ (7)

$I\left({t}_{2k}\right)=I\left({t}_{2k\text{+}1}\right)=\delta ,I\left(t\right)\ge \delta ,t\in \left[{t}_{2k},{t}_{2k\text{+}1}\right].$ (8)

$I\left(t\right)<\delta +\epsilon ,t\in \left[{t}_{2k-1},{t}_{2k-1}+l\right]\subset \left[{t}_{2k-1},{t}_{2k}\right],$

${I}^{\prime }\left(t\right)\ge -\left[\gamma +\mu \right]I\left(t\right),t\ge 0.$

$I\left(t\right)\ge I\left({t}_{2k-1}\right){\text{e}}^{-\left[\gamma +\mu \right]\left(t-{t}_{2k-1}\right)}\ge \delta {\text{e}}^{-\left[\gamma +\mu \right]L},t\in \left[{t}_{2k-1},{t}_{2k}\right].$

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