# 具有阶段结构和双线性发生率的HIV模型的稳定性分析Stability Analysis of HIV Models with Stage Structure and Bilinear Incidence

DOI: 10.12677/AAM.2019.82019, PDF, HTML, XML, 下载: 510  浏览: 1,688  科研立项经费支持

Abstract: AIDS is one of the most harmful infectious diseases. In this paper, we study a class of HIV trans-mission models with stage structure and bilinear incidence. The spectral radius method is used to calculate the basic regeneration number R0. Furthermore, we prove that the system has a unique disease-free equilibrium E0 when R0<1 while its global asymptotic stability is obtained by the V-function method and the LaSalle invariant principle; and when R0>1, the system adds an en-demic equilibrium E* which is globally asymptotically stable. Numerical simulations are carried out to verify our theoretical results.

1. 引言

Figure 1. The clinical process of HIV infection in adults

2. 模型建立及预备结论

1) 人口增加率为常数k，且均为易感者。

2) HIV携带者与易感者有相同的自然死亡率。

$\left\{\begin{array}{l}\stackrel{˙}{S}=k-dS-S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right),\\ {\stackrel{˙}{I}}_{1}=-d{I}_{1}-{\omega }_{1}{I}_{1}+S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right),\\ {\stackrel{˙}{I}}_{2}=-d{I}_{2}-{\omega }_{2}{I}_{2}+{\omega }_{1}{I}_{1},\\ {\stackrel{˙}{I}}_{3}=-d{I}_{3}-{\omega }_{3}{I}_{3}+{\omega }_{2}{I}_{2},\\ S\left(0\right)={S}^{0},\text{\hspace{0.17em}}{I}_{1}\left(0\right)={I}_{1}^{0},\text{\hspace{0.17em}}{I}_{2}\left(0\right)={I}_{2}^{0},\text{\hspace{0.17em}}{I}_{3}\left(0\right)={I}_{3}^{0},\end{array}$ (1)

${S}^{0}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{1}^{0}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{2}^{0}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}^{0}>0.$ (2)

$\stackrel{˙}{N}=k-dN-{\omega }_{3}{I}_{3}\le k-dN.$

$0\le N\le \frac{k}{d}+N\left(0\right){\text{e}}^{-dt}.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}N\left(t\right)\le \frac{k}{d}.$

3. 模型的无病平衡点和基本再生数

${I}_{1}={I}_{2}={I}_{3}=0$ 可得系统(1)的无病平衡点 ${E}^{0}\left(\frac{k}{d},0,0,0\right)$ 。记 ${S}_{0}=\frac{k}{d}$ 。下面我们用再生矩阵的方法 [8] [9] 求得基本再生数。

$X=\left({I}_{1},{I}_{2},{I}_{3},S\right)$ 则系统(1)变成 $\stackrel{˙}{X}=\Re \left(x\right)-\aleph \left(x\right)$ ，其中：

$\Re \left(x\right)=\left(\begin{array}{l}S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\\ \text{}0\\ \text{}0\\ \text{}0\end{array}\right),\text{}\aleph \left(X\right)=\left(\begin{array}{l}\text{}\left(d+{\omega }_{1}\right){I}_{1}\\ \text{}\left(d+{\omega }_{2}\right){I}_{2}-{\omega }_{1}{I}_{1}\\ \text{}\left(d+{\omega }_{3}\right){I}_{3}-{\omega }_{2}{I}_{2}\\ S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)+dS-k\end{array}\right).$

$\Re \left(x\right),\aleph \left(x\right)$${E}^{0}$ 处的Jacobian矩阵分别为：

$R=\left[\begin{array}{cccc}{S}_{0}{\beta }_{1}& {S}_{0}{\beta }_{2}& {S}_{0}{\beta }_{3}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]，\text{}H=\left[\begin{array}{cccc}d+{\omega }_{1}& 0& 0& 0\\ -{\omega }_{1}& d+{\omega }_{2}& 0& 0\\ 0& -{\omega }_{2}& d+{\omega }_{3}& 0\\ {S}_{0}{\beta }_{1}& {S}_{0}{\beta }_{2}& {S}_{0}{\beta }_{3}& d\end{array}\right].$

${R}_{0}=\rho \left(R{H}^{-1}\right)=\frac{{S}_{0}{\beta }_{1}\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)+{S}_{0}{\beta }_{2}{\omega }_{1}\left(d+{\omega }_{3}\right)+{S}_{0}{\beta }_{3}{\omega }_{1}{\omega }_{2}}{\left(d+{\omega }_{1}\right)\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)}.$

$J\left({E}^{0}\right)=\left(\begin{array}{cccc}-d& -{S}_{0}{\beta }_{1}& -{S}_{0}{\beta }_{2}& -{S}_{0}{\beta }_{3}\\ 0& -d-{\omega }_{1}+{S}_{0}{\beta }_{1}& {S}_{0}{\beta }_{2}& {S}_{0}{\beta }_{3}\\ 0& {\omega }_{1}& -d-{\omega }_{2}& 0\\ 0& 0& {\omega }_{2}& -d-{\omega }_{3}\end{array}\right).$

$A=\left(\begin{array}{ccc}-d-{\omega }_{1}+{S}_{0}{\beta }_{1}& {S}_{0}{\beta }_{2}& {S}_{0}{\beta }_{3}\\ {\omega }_{1}& -d-{\omega }_{2}& 0\\ 0& {\omega }_{2}& -d-{\omega }_{3}\end{array}\right).$

$\begin{array}{c}\mathrm{det}\left(A\right)={S}_{0}{\beta }_{3}{\omega }_{1}{\omega }_{2}-\left(d+{\omega }_{3}\right)\left[d\left(d+{\omega }_{3}\right)-{S}_{0}{\beta }_{1}\left(d+{\omega }_{2}\right)+{\omega }_{1}\left(-{S}_{0}{\beta }_{2}+d+{\omega }_{2}\right)\right]\\ =\left(d+{\omega }_{1}\right)\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)\left[\frac{{S}_{0}{\beta }_{1}\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)+{S}_{0}{\beta }_{1}{\omega }_{1}\left(d+{\omega }_{3}\right)+{S}_{0}{\beta }_{3}{\omega }_{1}{\omega }_{2}}{\left(d+{\omega }_{1}\right)\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)}-1\right]\\ =\left(d+{\omega }_{1}\right)\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)\left({R}_{0}-1\right).\end{array}$

${R}_{0}>1$$\mathrm{det}\left(A\right)>0$ 。此时 $J\left({E}^{0}\right)$ 必有一个正的特征根，因此无病平衡的 ${E}^{0}$ 是不稳定的。下面说明当 ${R}_{0}<1$ 时， ${E}^{0}$ 是局部渐近稳定的。

$Tr\left(A\right)=-3d-{\omega }_{1}-{\omega }_{2}-{\omega }_{3}+{S}_{0}{\beta }_{1}$ .

${A}^{\left[2\right]}=\left(\begin{array}{ccc}-d-{\omega }_{1}+{S}_{0}{\beta }_{1}-d-{\omega }_{2}& 0& -{S}_{0}{\beta }_{3}\\ {\omega }_{2}& -d-{\omega }_{1}+{S}_{0}{\beta }_{1}-d-{\omega }_{3}& 0\\ 0& {\omega }_{1}& -d-{\omega }_{2}-d-{\omega }_{3}\end{array}\right).$

$\begin{array}{c}Tr\left({A}^{\left[2\right]}\right)={S}_{0}{\omega }_{1}\left[-{\beta }_{3}{\omega }_{2}+{\beta }_{2}\left(2d-{S}_{0}{\beta }_{1}+{\omega }_{1}+{\omega }_{2}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(2d-{S}_{0}{\beta }_{1}+{\omega }_{1}+{\omega }_{2}\right)\left(2d-{S}_{0}{\beta }_{1}+{\omega }_{1}+{\omega }_{3}\right)\left(2d+{\omega }_{2}+{\omega }_{3}\right)\end{array}$

${R}_{0}<1$$Tr\left({A}^{\left[2\right]}\right)<0$ 。因此无病平衡点 ${E}^{0}$ 是局部渐近稳定的 [6] 。

$V\left(S,{I}_{1},{I}_{2},{I}_{3}\right)=\left(S-{S}_{0}-{S}_{0}ln\frac{S}{{S}_{0}}\right)+{a}_{1}{I}_{1}+{a}_{2}{I}_{2}+{a}_{3}{I}_{3}.$

$\begin{array}{c}\stackrel{˙}{V}=\stackrel{˙}{S}\left(1-\frac{{S}_{0}}{S}\right)+{a}_{1}{\stackrel{˙}{I}}_{1}+{a}_{2}{\stackrel{˙}{I}}_{2}+{a}_{3}{\stackrel{˙}{I}}_{3}=\left(1-\frac{{S}_{0}}{S}\right)\left[k-dS-S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{1}\left[-d{I}_{1}-{\omega }_{1}{I}_{1}+S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}\left(-d{I}_{2}-{\omega }_{2}{I}_{2}+{\omega }_{1}{I}_{1}\right)+{a}_{3}\left(-d{I}_{3}-{\omega }_{3}{I}_{3}+{\omega }_{2}{I}_{2}\right)\end{array}$

$\begin{array}{c}=-dS{\left(1-\frac{{S}_{0}}{S}\right)}^{2}-S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)+{S}_{0}\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{1}\left[-d{I}_{1}-{\omega }_{1}{I}_{1}+S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}\left(-d{I}_{2}-{\omega }_{2}{I}_{2}+{\omega }_{1}{I}_{1}\right)+{a}_{3}\left(-d{I}_{3}-{\omega }_{3}{I}_{3}+{\omega }_{2}{I}_{2}\right)\end{array}$

${a}_{1}=1,\text{\hspace{0.17em}}{a}_{2}=\frac{{S}_{0}{\beta }_{2}\left(d+{\omega }_{3}\right)+{\omega }_{2}{S}_{0}{\beta }_{3}}{\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)},\text{\hspace{0.17em}}{a}_{3}=\frac{{S}_{0}{\beta }_{3}}{d+{\omega }_{3}}$ ，则上式右端可变成：

$\begin{array}{c}\stackrel{˙}{V}=-dS{\left(1-\frac{{S}_{0}}{S}\right)}^{2}+\left[{S}_{0}{\beta }_{1}-\left(d+{\omega }_{1}\right)+{a}_{2}{\omega }_{1}\right]{I}_{1}=-dS{\left(1-\frac{{S}_{0}}{S}\right)}^{2}-\left(d+{\omega }_{1}\right)\left[1-\frac{{S}_{0}{\beta }_{1}+{a}_{2}{\omega }_{1}}{d+{\omega }_{1}}\right]{I}_{1}\\ =-dS{\left(1-\frac{{S}_{0}}{S}\right)}^{2}-\left(d+{\omega }_{1}\right)\left(1-{R}_{0}\right){I}_{1}.\end{array}$

4. 地方病平衡点的存在性和渐近稳定性

$\left\{\begin{array}{l}k-dS-S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)=0\\ -d{I}_{1}-{\omega }_{1}{I}_{1}+S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)=0\\ -d{I}_{2}-{\omega }_{2}{I}_{2}+{\omega }_{1}{I}_{1}=0\\ -d{I}_{3}-{\omega }_{3}{I}_{3}+{\omega }_{2}{I}_{2}=0\end{array}$

${S}^{*}=\frac{{S}_{0}}{{R}_{0}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{2}^{*}=\frac{d{S}_{0}{\omega }_{1}\left({R}_{0}-1\right)}{{R}_{0}\left(d+{\omega }_{1}\right)\left(d+{\omega }_{2}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{1}^{*}=\frac{\left(d+{\omega }_{2}\right){I}_{2}^{*}}{{\omega }_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}^{*}=\frac{{\omega }_{2}{I}_{2}^{*}}{d+{\omega }_{3}}.$

$\mathrm{min}\left\{\underset{t\to +\infty }{\mathrm{lim}\mathrm{inf}}S\left(t\right),\underset{t\to +\infty }{\mathrm{lim}\mathrm{inf}}{I}_{1}\left(t\right),\underset{t\to +\infty }{\mathrm{lim}\mathrm{inf}}{I}_{2}\left(t\right),\underset{t\to +\infty }{\mathrm{lim}\mathrm{inf}}{I}_{3}\left(t\right)\right\}>C.$

$V\left(S,{I}_{1},{I}_{2},{I}_{3}\right)=\left(S-{S}^{*}-{S}^{*}ln\frac{S}{{S}^{*}}\right)+{a}_{1}\left({I}_{1}-{I}_{1}^{*}-{I}_{1}^{*}\mathrm{ln}\frac{{I}_{1}}{{I}_{1}^{*}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}\left({I}_{2}-{I}_{2}^{*}-{I}_{2}^{*}\mathrm{ln}\frac{{I}_{2}}{{I}_{2}^{*}}\right)+{a}_{3}\left({I}_{3}-{I}_{3}^{*}-{I}_{3}^{*}\mathrm{ln}\frac{{I}_{3}}{{I}_{3}^{*}}\right).$

$\begin{array}{c}\stackrel{˙}{V}=\left(1-\frac{{S}^{*}}{S}\right)\stackrel{˙}{S}+{a}_{1}\left(1-\frac{{I}_{1}^{*}}{{I}_{1}}\right){\stackrel{˙}{I}}_{1}+{a}_{2}\left(1-\frac{{I}_{2}^{*}}{{I}_{2}}\right){\stackrel{˙}{I}}_{2}+{a}_{3}\left(1-\frac{{I}_{3}^{*}}{{I}_{3}}\right){\stackrel{˙}{I}}_{3}\\ =\left(1-\frac{{S}^{*}}{S}\right)\left[k-dS-S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{1}\left(1-\frac{{I}_{1}^{*}}{{I}_{1}}\right)\left[-d{I}_{1}-{\omega }_{1}{I}_{1}+S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}\left(1-\frac{{I}_{2}^{*}}{{I}_{2}}\right)\left(-d{I}_{2}-{\omega }_{2}{I}_{2}+{\omega }_{1}{I}_{1}\right)+{a}_{3}\left(1-\frac{{I}_{3}^{*}}{{I}_{3}}\right)\left(-d{I}_{3}-{\omega }_{3}{I}_{3}+{\omega }_{2}{I}_{2}\right)\end{array}$ (3)

$\begin{array}{l}=\left(k+d{S}^{*}+{a}_{1}d{I}_{1}^{*}+{a}_{1}{\omega }_{1}{I}_{1}^{*}+{a}_{2}d{I}_{2}^{*}+{a}_{2}{\omega }_{2}{I}_{2}^{*}+{a}_{3}d{I}_{3}^{*}+{a}_{3}{\omega }_{3}{I}_{3}^{*}\right)+\left(-d-{a}_{1}{\beta }_{1}{I}_{1}^{*}\right)S\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left({a}_{1}-1\right)S\left({\beta }_{1}{I}_{1}+{\beta }_{2}{I}_{2}+{\beta }_{3}{I}_{3}\right)+\left({S}^{*}{\beta }_{1}-{a}_{1}d-{a}_{1}{\omega }_{1}+{a}_{2}{\omega }_{1}\right){I}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left({S}^{*}{\beta }_{2}-{a}_{2}d-{a}_{2}{\omega }_{2}+{a}_{3}{\omega }_{2}\right){I}_{2}+\left({S}^{*}{\beta }_{3}-{a}_{3}d-{a}_{3}{\omega }_{3}\right){I}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left(-k\frac{{S}^{*}}{S}-{a}_{1}{\beta }_{2}{I}_{1}^{*}\frac{S{I}_{2}}{{I}_{1}}-{a}_{1}{\beta }_{3}{I}_{1}^{*}\frac{S{I}_{3}}{{I}_{1}}-{a}_{2}{\omega }_{1}{I}_{2}^{*}\frac{{I}_{1}}{{I}_{2}}-{a}_{3}{\omega }_{2}{I}_{3}^{*}\frac{{I}_{2}}{{I}_{3}}\right)\end{array}$

$\begin{array}{l}c=k+d{S}^{*}+{a}_{1}d{I}_{1}^{*}+{a}_{1}{\omega }_{1}{I}_{1}^{*}+{a}_{2}d{I}_{2}^{*}+{a}_{2}{\omega }_{2}{I}_{2}^{*}+{a}_{3}d{I}_{3}^{*}+{a}_{3}{\omega }_{3}{I}_{3}^{*},\\ w=\frac{S}{{S}^{*}},\text{}x=\frac{{I}_{1}}{{I}_{1}^{*}},\text{}y=\frac{{I}_{2}}{{I}_{2}^{*}},\text{}z=\frac{{I}_{3}}{{I}_{3}^{*}}.\end{array}$

${b}_{1}\left(2-w-\frac{1}{w}\right)+{b}_{2}\left(4-\frac{1}{w}-\frac{wz}{x}-\frac{y}{z}-\frac{x}{y}\right)+{b}_{3}\left(3-\frac{1}{w}-\frac{wy}{x}-\frac{x}{y}\right),$

$\begin{array}{ccc}\left\{\begin{array}{l}2{b}_{1}+4{b}_{2}+3{b}_{3}=c\\ {b}_{1}=\left(d+{a}_{1}{I}_{1}^{*}{\beta }_{1}\right){S}^{*}\\ {a}_{1}-1=0\\ {b}_{1}+{b}_{2}+{b}_{3}=k\\ -{a}_{2}\left(d+{\omega }_{2}\right)+{a}_{2}{\omega }_{2}+{S}^{*}{\beta }_{2}=0\\ -{a}_{3}\left(d+{\omega }_{3}\right)+{S}^{*}{\beta }_{3}=0\\ {b}_{3}={S}^{*}{\beta }_{2}{I}_{2}^{*}\\ {b}_{2}={S}^{*}{\beta }_{3}{I}_{3}^{*}={a}_{3}{\omega }_{2}{I}_{2}^{*}\\ {b}_{2}+{b}_{3}={a}_{2}{\omega }_{1}{I}_{1}^{*}\end{array}& ⇔& \left\{\begin{array}{l}{b}_{1}=\left(d+{a}_{1}{I}_{1}^{*}{\beta }_{1}\right){S}^{*}>0\\ {b}_{2}={S}^{*}{\beta }_{3}{I}_{3}^{*}>0\\ {b}_{3}={S}^{*}{\beta }_{2}{I}_{2}^{*}>0\\ {a}_{1}=1\\ {a}_{2}=\frac{{S}^{*}{\beta }_{3}{\omega }_{2}+{S}^{*}{\beta }_{2}\left(d+{\omega }_{3}\right)}{\left(d+{\omega }_{2}\right)\left(d+{\omega }_{3}\right)}\\ {a}_{3}=\frac{{S}^{*}{\beta }_{3}}{d+{\omega }_{3}}\end{array}\end{array}$

$\stackrel{˙}{V}={b}_{1}\left(2-w-\frac{1}{w}\right)+{b}_{2}\left(4-\frac{1}{w}-\frac{wz}{x}-\frac{y}{z}-\frac{x}{y}\right)+{b}_{3}\left(3-\frac{1}{w}-\frac{wy}{x}-\frac{x}{y}\right)\le 0,$

$\stackrel{˙}{V}=0$ 当且仅当 $w=\frac{1}{w}=\frac{wz}{x}=\frac{y}{z}=\frac{x}{y}=\frac{wy}{x}=1$ ，即 $\left\{\stackrel{˙}{V}=0\right\}=\left\{{E}^{*}\right\}$ 。由LaSalle不变原理知地方病平衡点 ${E}^{\text{*}}$$\stackrel{\circ }{\Omega }$ 上是渐近稳定的。

5. 数值模拟

${S}^{0}=10000,\text{}{I}_{1}^{0}=15,\text{}{I}_{2}^{0}=50,\text{}{I}_{3}^{0}=50.$

1) 当 ${R}_{0}<1$ 时的情形

$\begin{array}{l}k=10,\text{\hspace{0.17em}}\text{\hspace{0.17em}}d=0.01,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{1}=\frac{1}{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{2}=\frac{1}{96},\\ {\omega }_{3}=\frac{1}{13},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{1}=2×{10}^{-4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{2}=8×{10}^{-6},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{3}=7.6×{10}^{-5}.\end{array}$ (4)

$S\left(20000\right)=999.9983,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{1}\left(20000\right)=4.5139×{10}^{-6},\text{\hspace{0.17em}}{I}_{2}\left(20000\right)=7.4608×{10}^{-4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}\left(20000\right)=9.1416×{10}^{-5}.$

ii) 当 ${R}_{0}>1$ 时的情形

$\begin{array}{l}k=15,\text{\hspace{0.17em}}\text{\hspace{0.17em}}d=0.01,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{1}=\frac{1}{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{2}=\frac{1}{96},\\ {\omega }_{3}=\frac{1}{13},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{1}=2×{10}^{-4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{2}=8×{10}^{-6},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{3}=7.6×{10}^{-5}.\end{array}$ (5)

Figure 2. Three-dimensional projection of phase portrait of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$, and the parameter values are shown in (4)

Figure 3. Images of $S,{I}_{1},{I}_{2},{I}_{3}$ of the solution of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$ and the parameter values are shown in(4)

Figure 4. Images of $S,{I}_{1},{I}_{2},{I}_{3}$ of the solution of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$ and the parameter values are shown in (4), time of integration: [18000, 20000]

Figure 5. Three-dimensional projection of phase portrait of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$, and the parameter values are shown in (5)

Figure 6. Images of $S,{I}_{1},{I}_{2},{I}_{3}$ of the solution of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$ and the parameter values are shown in (5)

Figure 7. Images of $S,{I}_{1},{I}_{2},{I}_{3}$ of the solution of system (1) with ${S}^{0}=10000$, ${I}_{1}^{0}=15$, ${I}_{2}^{0}=50$, ${I}_{3}^{0}=50$ and the parameter values are shown in (5), time of integration : [19000,20000]

${E}^{*}\left(1109.15,1.29849,190.799,22.8649\right).$

$\begin{array}{l}S\left(20000\right)=1.1092×{10}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{1}\left(20000\right)=1.2983,\\ {I}_{2}\left(20000\right)=190.7990,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}\left(20000\right)=22.8649.\end{array}$

6. 结论

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