﻿ 寨卡病毒溶瘤治疗脑癌的数学模型动力学分析 Dynamic Analysis of Mathematical Model in Brain Cancer Treatment by Zika Virus Oncolysis

Vol. 08  No. 02 ( 2019 ), Article ID: 28970 , 15 pages
10.12677/AAM.2019.82032

Dynamic Analysis of Mathematical Model in Brain Cancer Treatment by Zika Virus Oncolysis

Tingmei Yang, Jian Liu

School of Mathematics and Information Science, Guangzhou University, Guangzhou Guangdong

Received: Jan. 30th, 2019; accepted: Feb. 14th, 2019; published: Feb. 22nd, 2019

ABSTRACT

Based on the Zika Virus specifically targeting to kill glioma stem cells, and having no effect on normal cells, we build a mathematical model about normal cells and brain tumor cells competing nutrient in culture dish adding Zika Virus Oncolytic therapy by analyzing the existence and stability of the equilibrium, and get the minimum effective dose parameters expression when the Oncolytic virus therapy is the most effective. Finally, we can verify the result by numerical simulation.

Keywords:Zika Virus, Brain Cancer, Oncolytic Therapy, Stability, Drug Dose, Numerical Simulation

1. 引言

2. 基础模型

2017年，王子子建立了一个基于细胞增长的抑制的溶瘤病毒治疗模型 [12] 。该文章在正常细胞和肿瘤细胞竞争的反应扩散模型基础上，结合算子半群理论推导，引入溶瘤病毒项，并得到肿瘤细胞根除平衡点的全局稳定性条件。具体模型如下：

$\left\{\begin{array}{l}{\partial }_{t}{u}_{1}\left(t,x\right)={d}_{1}\Delta {u}_{1}\left(t,x\right)+{u}_{1}\left(t,x\right)\left({a}_{1}-{b}_{1}{u}_{1}\left(t,x\right)-{c}_{1}{u}_{2}\left(t,x\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,x\in \Omega \text{\hspace{0.17em}},\\ {\partial }_{t}{u}_{2}\left(t,x\right)={d}_{2}\Delta {u}_{2}\left(t,x\right)+{u}_{2}\left(t,x\right)\left({a}_{2}-{b}_{2}{u}_{1}\left(t,x\right)-{c}_{2}{u}_{2}\left(t,x\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mu {u}_{2}\left(t,x\right){u}_{3}\left(t,x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,x\in \Omega \text{\hspace{0.17em}},\\ {\partial }_{t}{u}_{3}\left(t,x\right)={d}_{3}\Delta {u}_{3}\left(t,x\right)+B-d{u}_{3}\left(t,x\right)+\gamma \mu {\int }_{\Omega }\Gamma \left(x,y,\tau \right){u}_{2}\left(t-\tau ,y\right){u}_{3}\left(t-\tau ,y\right)\text{d}y,\text{\hspace{0.17em}}t>0,x\in \Omega \text{\hspace{0.17em}},\\ {\partial }_{n}{u}_{1}={\partial }_{n}{u}_{2}={\partial }_{n}{u}_{3}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,x\in \Omega \text{\hspace{0.17em}},\\ {u}_{1}\left(\theta ,x\right)={u}_{1}^{0}\left(\theta ,x\right),{u}_{2}\left(\theta ,x\right)={u}_{2}^{0}\left(\theta ,x\right),{u}_{3}\left(\theta ,x\right)={u}_{3}^{0}\left(\theta ,x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta \in \left[-\tau ,0\right],\text{\hspace{0.17em}}x\in \Omega \text{\hspace{0.17em}}.\end{array}$

$\left\{\begin{array}{l}\frac{dx}{dt}=x\left({a}_{1}-{b}_{1}x-{c}_{1}y\right),\\ \frac{dy}{dt}=y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right)-{r}_{1}yz\text{\hspace{0.17em}},\\ \frac{dz}{dt}=B+{r}_{2}yz-dz\text{\hspace{0.17em}}.\end{array}$ (1)

3. 系统的非负性和有界性分析

$\left\{\begin{array}{l}\frac{dx}{dt}=x\left({a}_{1}-{b}_{1}x-{c}_{1}y\right),\\ \frac{dy}{dt}=y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right).\end{array}$ (2)

$\left(0,0\right),\left(\frac{{a}_{1}}{{b}_{1}},0\right),\left(0,\frac{{a}_{2}}{{b}_{2}}\right),\left(\frac{{a}_{2}{c}_{1}-{a}_{1}{b}_{2}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}},\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right).$

$l\equiv y+x-M,$

$M=\mathrm{max}\left\{\frac{{a}_{1}}{{b}_{1}},\frac{{a}_{2}}{{b}_{2}},\frac{{a}_{2}{c}_{1}-{a}_{1}{b}_{2}+{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right\}.$

$\begin{array}{l}\frac{\text{d}l}{\text{d}t}=\frac{\text{d}y}{\text{d}t}+\frac{\text{d}x}{\text{d}t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right)+x\left({a}_{1}-{b}_{1}x-{c}_{1}y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\left(M-x\right)\left[{a}_{2}-{b}_{2}\left(M-x\right)-{c}_{2}x\right]+x\left[{a}_{1}-{b}_{1}x-{c}_{1}\left(M-x\right)\right]|}_{l=0}.\end{array}$

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right)-{r}_{1}yz\text{\hspace{0.17em}},\\ \frac{\text{d}z}{\text{d}t}=B+{r}_{2}yz-dz\text{\hspace{0.17em}}.\end{array}$

$L=y+\frac{{r}_{1}}{{r}_{2}}z,$

$\begin{array}{c}\frac{\text{d}L}{\text{d}t}=\frac{\text{d}y}{\text{d}t}+\frac{{r}_{1}}{{r}_{2}}\frac{\text{d}z}{\text{d}t}\\ =y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right)-{r}_{1}yz+\frac{{r}_{1}}{{r}_{2}}\left(B+{r}_{2}yz-dz\right)\\ =y\left({a}_{2}-{b}_{2}y-{c}_{2}x\right)+\frac{{r}_{1}B}{{r}_{2}}-\frac{{r}_{1}}{{r}_{2}}dz\\ \le y\left({a}_{2}-mM\right)+\frac{{r}_{1}B}{{r}_{2}}-\frac{{r}_{1}}{{r}_{2}}dz\\ =\frac{{r}_{1}B}{{r}_{2}}-n\left(y+\frac{{r}_{1}}{{r}_{2}}z\right)=\frac{{r}_{1}B}{{r}_{2}}-nL\end{array}$

$\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}\left(y+\frac{{r}_{1}}{{r}_{2}}z\right)\le \frac{1}{n}\frac{{r}_{1}B}{{r}_{2}}.$

$m=\mathrm{min}\left\{{b}_{2},{c}_{2}\right\},n=\mathrm{min}\left\{|{a}_{2}-nm|,d\right\}.$

$\Gamma =\left\{\left(x,y,z\right)|x+y\le M,y+\frac{{r}_{1}}{{r}_{2}}z\le \frac{1}{n}\frac{{r}_{1}B}{{r}_{2}}\right\}.$

4. 平衡点的存在性与稳定性分析

2) 当 $B>\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$ 时，系统(1)除了存在平凡平衡点 ${E}_{0}\left(0,0,\frac{d}{B}\right)$，还存在肿瘤灭绝平衡点 ${E}_{1}\left(\frac{{a}_{1}}{{b}_{1}},0,\frac{B}{d}\right)$

3) 当 $B<\frac{d{a}_{2}}{{r}_{1}}$$\frac{{a}_{1}}{{c}_{1}}<{y}_{2}<\mathrm{min}\left\{\frac{{a}_{2}}{{b}_{2}},\frac{d}{{r}_{2}}\right\}$ 时，除了存在平凡平衡点 ${E}_{0}\left(0,0,\frac{d}{B}\right)$，肿瘤灭绝平衡点 ${E}_{1}\left(\frac{{a}_{1}}{{b}_{1}},0,\frac{B}{d}\right)$，还存在正常细胞灭绝平衡点 ${E}_{2}\left(0,{y}_{2},\frac{{a}_{2}-{b}_{2}{y}_{2}}{{r}_{1}}\right)$，其中

${y}_{2}=\frac{{a}_{2}{r}_{2}+{b}_{2}d-\sqrt{{\left({a}_{2}{r}_{2}+{b}_{2}d\right)}^{2}-4{r}_{2}{b}_{2}\left({a}_{2}d-B{r}_{1}\right)}}{2{b}_{2}{r}_{2}}$

4) 当 $B<\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$${c}_{1}{c}_{2}<{b}_{1}{b}_{2},{a}_{1}{c}_{2}<{a}_{2}{b}_{1},{y}_{3}<\mathrm{min}\left\{\frac{{a}_{1}}{{c}_{1}},\frac{d}{{r}_{2}}\right\}$ 时，除了存在平凡平衡点 ${E}_{0}\left(0,0,\frac{d}{B}\right)$ 和肿瘤灭绝平衡点 ${E}_{1}\left(\frac{{a}_{1}}{{b}_{1}},0,\frac{B}{d}\right)$，还存在正平衡点 ${E}_{3}\left(\frac{{a}_{1}-{c}_{1}{y}_{3}}{{b}_{1}},{y}_{3},\frac{B}{d-{r}_{2}{y}_{3}}\right)$，其中

${y}_{3}=\frac{-{h}_{2}-\sqrt{{h}_{2}^{2}-4{h}_{1}{h}_{3}}}{2{h}_{1}},$

${h}_{1}={b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2},{h}_{2}={c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d,{h}_{3}={a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d.$

3) 当 $y>\frac{{a}_{1}}{{c}_{1}}$ 时，溶瘤病毒治疗最终失败，有 $x=0$ 得到 $\left\{\begin{array}{l}{a}_{2}-{b}_{2}y-{r}_{1}z=0\\ B+{r}_{2}yz-dz=0\end{array}$。要使平衡点 ${z}_{2}$ 存在，由该方程组的第一个式子知， ${a}_{2}-{b}_{2}{y}_{2}>0$ 则必有 ${y}_{2}<\frac{{a}_{2}}{{b}_{2}}$，由方程组的第二个式子知， ${r}_{2}{y}_{2}-d<0$ 则必有 ${y}_{2}<\frac{d}{{r}_{2}}$，故 ${y}_{2}<\mathrm{min}\left\{\frac{{a}_{2}}{{b}_{2}},\frac{d}{{r}_{2}}\right\}$。把 $z=\frac{{a}_{2}-{b}_{2}y}{{r}_{1}}$ 代入方程组得到

${r}_{2}{b}_{2}{y}^{2}-\left({a}_{2}{r}_{2}+{b}_{2}d\right)y+{a}_{2}d-B{r}_{1}=0.$

$f\left(y\right)={r}_{2}{b}_{2}{y}^{2}-\left({a}_{2}{r}_{2}+{b}_{2}d\right)y+{a}_{2}d-B{r}_{1},$

$B<\frac{{a}_{2}d}{{r}_{1}}$，则 ${y}_{1}^{\ast }{y}_{2}^{\ast }=\frac{{a}_{2}d-B{r}_{1}}{{r}_{2}{b}_{2}}$，即进一步假设 $\frac{{a}_{2}}{{b}_{2}}\le \frac{d}{{r}_{2}}$，则函数 $f\left(y\right)$ 的对称轴 $y=\frac{1}{2}\left(\frac{{a}_{2}}{{b}_{2}}+\frac{d}{{r}_{2}}\right)\le \frac{{a}_{2}}{{b}_{2}}$

$f\left(0\right)={a}_{2}d-B{r}_{1}>0,$

$f\left(\frac{{a}_{2}}{{b}_{2}}\right)={r}_{2}{b}_{2}{\left(\frac{{a}_{2}}{{b}_{2}}\right)}^{2}-\left({a}_{2}{r}_{2}+{b}_{2}d\right)\frac{{a}_{2}}{{b}_{2}}+{a}_{2}d-B{r}_{1}=\frac{{r}_{2}{a}_{2}^{2}}{{b}_{2}}-\frac{{a}_{2}^{2}{r}_{2}}{{b}_{2}}-{a}_{2}d+{a}_{2}d-B{r}_{1}=-B{r}_{1}<0.$

$f\left(y\right)$$\left(0,\frac{{a}_{2}}{{b}_{2}}\right)$ 有一实数根。

$f\left(\frac{d}{{r}_{2}}\right)={r}_{2}{b}_{2}{\left(\frac{d}{{r}_{2}}\right)}^{2}-\left({a}_{2}{r}_{2}+{b}_{2}d\right)\frac{d}{{r}_{2}}+{a}_{2}d-B{r}_{1}=\frac{{b}_{2}{d}^{2}}{{r}_{2}}-\frac{{d}^{2}{b}_{2}}{{r}_{2}}-{a}_{2}d+{a}_{2}d-B{r}_{1}=-B{r}_{1}<0.$

$f\left(y\right)$$\left(0,\frac{d}{{r}_{2}}\right)$ 有一实数根。若 $B\ge \frac{{a}_{2}d}{{r}_{1}}$，此时有 $f\left(0\right)={a}_{2}d-B{r}_{1}<0$，且 $f\left(\frac{{a}_{2}}{{b}_{2}}\right)<0,f\left(\frac{d}{{r}_{2}}\right)<0$，故 $f\left(y\right)$$\left(0,\frac{{a}_{2}}{{b}_{2}}\right)$$\left(0,\frac{d}{{r}_{2}}\right)$ 均无解。由以上讨论可知，当满足 $B<\frac{d}{{r}_{1}}{a}_{2}$$\frac{{a}_{1}}{{c}_{1}}<{y}_{2}<\mathrm{min}\left\{\frac{{a}_{2}}{{b}_{2}},\frac{d}{{r}_{2}}\right\}$ 时，系统(1)存在正常细胞灭绝平衡点 ${E}_{2}\left(0,{y}_{2},\frac{{a}_{2}-{b}_{2}{y}_{2}}{{r}_{1}}\right)$，其中

${y}_{2}=\frac{{a}_{2}{r}_{2}+{b}_{2}d-\sqrt{{\left({a}_{2}{r}_{2}+{b}_{2}d\right)}^{2}-4{r}_{2}{b}_{2}\left({a}_{2}d-B{r}_{1}\right)}}{2{b}_{2}{r}_{2}}.$

4) 由 $\left\{\begin{array}{l}{a}_{1}-{b}_{1}x-{c}_{1}y=0\\ {a}_{2}-{b}_{2}y-{c}_{2}x-{r}_{1}z=0\\ B+{r}_{2}yz-dz=0\end{array}$ 可知 $\left\{\begin{array}{l}x=\frac{{a}_{1}-{c}_{1}y}{{b}_{1}}\\ z=\frac{B}{d-{r}_{2}y}\end{array}$，要使平衡点 ${x}_{3},{z}_{3}$ 存在，则必有 ${a}_{1}-{c}_{1}{y}_{3}>0$$d-{r}_{2}{y}_{3}>0$。即 $\left\{\begin{array}{l}{y}_{3}<\frac{{a}_{1}}{{c}_{1}}\\ {y}_{3}<\frac{d}{{r}_{2}}\end{array}$，故 ${y}_{3}<\mathrm{min}\left\{\frac{{a}_{1}}{{c}_{1}},\frac{d}{{r}_{2}}\right\}$。接下来讨论 ${y}_{3}$ 平衡点 ${y}_{3}$ 满足方程

$\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){y}_{3}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right){y}_{3}+{a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d=0$

$\begin{array}{l}{h}_{1}={b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2},{h}_{2}={c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d,\\ {h}_{3}={a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d.\end{array}$

$f\left(y\right)=\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){y}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right)y+\left({a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d\right),$

① 当 ${c}_{1}{c}_{2}<{b}_{1}{b}_{2},{a}_{1}{c}_{2}<{a}_{2}{b}_{1}$$0 时，则 ${h}_{1}>0,{h}_{2}<0,{h}_{3}>0$，进一步假设 $\frac{{a}_{1}}{{c}_{1}}\le \frac{d}{{r}_{2}}$，则函数 $f\left(y\right)$ 的对称轴

$y=\frac{1}{2}\left(\frac{{c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d}{\left({c}_{1}{c}_{2}-{b}_{1}{b}_{2}\right){r}_{2}}\right)=\frac{1}{2}\left(\frac{d}{{r}_{2}}+\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right)\le \frac{{a}_{1}}{{c}_{1}}.$

$\begin{array}{c}f\left(\frac{{a}_{1}}{{c}_{1}}\right)=\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){\left(\frac{{a}_{1}}{{c}_{1}}\right)}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right)\frac{{a}_{1}}{{c}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d\right)\\ ={b}_{1}d\left({a}_{2}-\frac{{a}_{1}{b}_{2}}{{c}_{1}}\right)+{b}_{1}{r}_{2}\frac{{a}_{1}}{{c}_{1}}\left(\frac{{a}_{1}{b}_{2}}{{c}_{1}}-{a}_{2}\right)-{r}_{1}{b}_{1}B\\ ={b}_{1}\left(\frac{{a}_{1}{b}_{2}}{{c}_{1}}-{a}_{2}\right)\left({r}_{2}\frac{{a}_{1}}{{c}_{1}}-d\right)-{r}_{1}{b}_{1}B.\end{array}$

$\frac{{a}_{1}}{{c}_{1}}\le \frac{d}{{r}_{2}}$，得 $\frac{{r}_{2}{a}_{1}}{{c}_{1}}-d<0$，由 $\frac{1}{2}\left(\frac{d}{{r}_{2}}+\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right)\le \frac{{a}_{1}}{{c}_{1}}$$\frac{{a}_{1}}{{c}_{1}}\le \frac{d}{{r}_{2}}$，得 $\frac{{a}_{1}{b}_{2}}{{c}_{1}}-{a}_{2}>0$。所以有 $f\left(\frac{{a}_{1}}{{c}_{1}}\right)<0$

$0，因此 $f\left(0\right)={a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d>0$。故 $f\left(y\right)$$\left(0,\frac{{a}_{1}}{{c}_{1}}\right)$ 有实数根。

$y=\frac{1}{2}\frac{{c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d}{\left({c}_{1}{c}_{2}-{b}_{1}{b}_{2}\right){r}_{2}}=\frac{1}{2}\left(\frac{d}{{r}_{2}}+\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right)\le \frac{d}{{r}_{2}},$

$\begin{array}{c}f\left(\frac{d}{{r}_{2}}\right)=\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){\left(\frac{d}{{r}_{2}}\right)}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right)\frac{d}{{r}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d\right)\\ =-{r}_{1}{b}_{1}B<0.\end{array}$

$0，因此 $f\left(0\right)={a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d>0$。故 $f\left(y\right)$$\left(0,\frac{d}{{r}_{2}}\right)$ 有实数根。

② 当 ${c}_{1}{c}_{2}>{b}_{1}{b}_{2}$$B>\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$ 时， ${h}_{1}<0,{h}_{3}<0$，则

$f\left(0\right)={a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d<0.$

$\frac{{a}_{1}}{{c}_{1}}\le \frac{d}{{r}_{2}}$，则函数 $f\left(y\right)$ 的对称轴

$y=\frac{1}{2}\frac{{c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d}{\left({c}_{1}{c}_{2}-{b}_{1}{b}_{2}\right){r}_{2}}=\frac{1}{2}\left(\frac{d}{{r}_{2}}+\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right)\le \frac{{a}_{1}}{{c}_{1}},$

$\begin{array}{c}f\left(\frac{{a}_{1}}{{c}_{1}}\right)=\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){\left(\frac{{a}_{1}}{{c}_{1}}\right)}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right)\frac{{a}_{1}}{{c}_{1}}+\left({a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d\right)\\ ={b}_{1}d\left({a}_{2}-\frac{{a}_{1}{b}_{2}}{{c}_{1}}\right)+{b}_{1}{r}_{2}\frac{{a}_{1}}{{c}_{1}}\left(\frac{{a}_{1}{b}_{2}}{{c}_{1}}-{a}_{2}\right)-{r}_{1}{b}_{1}B\\ ={b}_{1}\left(\frac{{a}_{1}{b}_{2}}{{c}_{1}}-{a}_{2}\right)\left({r}_{2}\frac{{a}_{1}}{{c}_{1}}-d\right)-{r}_{1}{b}_{1}B<0.\end{array}$

$f\left(y\right)$$\left(0,\frac{{a}_{1}}{{c}_{1}}\right)$ 无实根。

$\frac{{a}_{1}}{{c}_{1}}\ge \frac{d}{{r}_{2}}$，则函数 $f\left(y\right)$ 的对称轴

$y=\frac{1}{2}\frac{{c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d}{\left({c}_{1}{c}_{2}-{b}_{1}{b}_{2}\right){r}_{2}}=\frac{1}{2}\left(\frac{d}{{r}_{2}}+\frac{{a}_{1}{c}_{2}-{a}_{2}{b}_{1}}{{c}_{1}{c}_{2}-{b}_{1}{b}_{2}}\right)\le \frac{d}{{r}_{2}},$

$\begin{array}{c}f\left(\frac{d}{{r}_{2}}\right)=\left({b}_{1}{b}_{2}{r}_{2}-{c}_{1}{c}_{2}{r}_{2}\right){\left(\frac{d}{{r}_{2}}\right)}^{2}+\left({c}_{1}{c}_{2}d+{a}_{1}{c}_{2}{r}_{2}-{a}_{2}{b}_{1}{r}_{2}-{b}_{1}{b}_{2}d\right)\frac{d}{{r}_{2}}+\left({a}_{2}{b}_{1}d-{r}_{1}{b}_{1}B-{a}_{1}{c}_{2}d\right)\\ =-{r}_{1}{b}_{1}B<0.\end{array}$

$f\left(y\right)$$\left(0,\frac{d}{{r}_{2}}\right)$ 无实根。因此，当 $B<\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$${c}_{1}{c}_{2}<{b}_{1}{b}_{2},{a}_{1}{c}_{2}<{a}_{2}{b}_{1},{y}_{3}<\mathrm{min}\left\{\frac{{a}_{1}}{{c}_{1}},\frac{d}{{r}_{2}}\right\}$ 时，系统(1)存在一正平衡点 ${E}_{3}\left(\frac{{a}_{1}-{c}_{1}{y}_{3}}{{b}_{1}},{y}_{3},\frac{B}{d-{r}_{2}{y}_{3}}\right)$，其中 ${y}_{3}=\frac{-{h}_{2}-\sqrt{{h}_{2}^{2}-4{h}_{1}{h}_{3}}}{2{h}_{1}}$

$J\left(E\right)=\left(\begin{array}{ccc}{a}_{1}-2{b}_{1}x-{c}_{1}y& -{c}_{1}x& 0\\ -{c}_{2}y& {a}_{2}-2{b}_{2}y-{c}_{2}x-{r}_{1}z& -{r}_{1}y\\ 0& {r}_{2}z& {r}_{2}y-d\end{array}\right).$

$J\left({E}_{0}\right)=\left(\begin{array}{ccc}{a}_{1}& 0& 0\\ 0& {a}_{2}-\frac{{r}_{1}B}{d}& 0\\ 0& \frac{{r}_{2}B}{d}& -d\end{array}\right).$

$\left(\lambda -{a}_{1}\right)\left(\lambda -{a}_{2}+\frac{{r}_{1}B}{d}\right)\left(\lambda +d\right)=0.$

$J\left({E}_{1}\right)=\left(\begin{array}{ccc}-{a}_{1}& -\frac{{a}_{1}{c}_{1}}{{b}_{1}}& 0\\ 0& {a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}-\frac{{r}_{1}B}{d}& 0\\ 0& \frac{{r}_{2}B}{d}& -d\end{array}\right)\text{\hspace{0.17em}}.$

$\left(\lambda +{a}_{1}\right)\left(\lambda -{a}_{2}+\frac{{a}_{1}{c}_{2}}{{b}_{1}}+\frac{{r}_{1}B}{d}\right)\left(\lambda +d\right)=0.$

${\lambda }_{1}=-{a}_{1},{\lambda }_{2}={a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}-\frac{{r}_{1}B}{d},{\lambda }_{3}=-d.$

$B>\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$ 时，特征根 ${\lambda }_{2}<0$，又由于 ${\lambda }_{1}<0,{\lambda }_{3}<0$，故肿瘤灭绝平衡点 ${E}_{1}\left(\frac{{a}_{1}}{{b}_{1}},0,\frac{B}{d}\right)$ 是局部渐近稳定的。此外，当 $B<\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)$ 时，显然有特征根 ${\lambda }_{2}>0$，从而肿瘤灭绝平衡 ${E}_{1}\left(\frac{{a}_{1}}{{b}_{1}},0,\frac{B}{d}\right)$ 是不稳定的。

$\left\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}=x\left({a}_{1}-{b}_{1}x\right),\\ \frac{\text{d}z}{\text{d}t}=B-dz\text{\hspace{0.17em}}.\end{array}$ (3)

$\frac{\frac{1}{xz}\left({a}_{1}x-{b}_{1}{x}^{2}\right)}{\partial x}+\frac{\frac{1}{xz}\left(B-dz\right)}{\partial z}=-\frac{{b}_{1}}{z}-\frac{B}{x{z}^{2}}<0.$

$J\left({E}_{2}\right)=\left(\begin{array}{ccc}{a}_{1}-{c}_{1}{y}_{2}& 0& 0\\ -{c}_{2}{y}_{2}& -{b}_{2}{y}_{2}& -{r}_{1}{y}_{2}\\ 0& {r}_{2}\frac{{a}_{2}-{b}_{2}{y}_{2}}{{r}_{1}}& {r}_{2}{y}_{2}-d\end{array}\right).$

$\left(\lambda -{a}_{1}+{c}_{1}{y}_{2}\right)\left[\left(\lambda +{b}_{2}{y}_{2}\right)\left(\lambda -{r}_{2}{y}_{2}+d\right)+{r}_{2}{y}_{2}\left({a}_{2}-{b}_{2}{y}_{2}\right)\right]=0.$

${\lambda }^{2}+\left({b}_{2}{y}_{2}+d-{r}_{2}{y}_{2}\right)\lambda +\left({b}_{2}d{y}_{2}+{a}_{2}{r}_{2}{y}_{2}-2{b}_{2}{r}_{2}{y}_{2}^{2}\right)=0.$

${y}_{2}<\frac{d}{{r}_{2}}$${y}_{2}<\frac{{a}_{2}}{{b}_{2}}$ 得到

${\lambda }_{2}+{\lambda }_{3}={r}_{2}{y}_{2}-{b}_{2}{y}_{2}-d<0,$

${\lambda }_{2}{\lambda }_{3}={b}_{2}d{y}_{2}+{a}_{2}{r}_{2}{y}_{2}-2{b}_{2}{r}_{2}{y}_{2}^{2}={y}_{2}\left[{b}_{2}\left(d-{r}_{2}{y}_{2}\right)+{r}_{2}\left({a}_{2}-{b}_{2}{y}_{2}\right)\right]>0.$

${\lambda }_{2}<0,{\lambda }_{3}<0$。因此，当 $B<\frac{d}{{r}_{1}}{a}_{2}$$\frac{{a}_{1}}{{c}_{1}}<{y}_{2}<\mathrm{min}\left\{\frac{{a}_{2}}{{b}_{2}},\frac{d}{{r}_{2}}\right\}$ 时正常细胞灭绝平衡点 ${E}_{2}\left(0,{y}_{2},\frac{{a}_{2}-{b}_{2}{y}_{2}}{{r}_{1}}\right)$ 是局部渐近稳定的。

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=y\left({a}_{2}-{b}_{2}y-{r}_{1}z\right),\\ \frac{\text{d}z}{\text{d}t}=B+{r}_{2}yz-dz\text{\hspace{0.17em}}\text{​}.\end{array}$ (4)

$\frac{\frac{1}{yz}y\left({a}_{2}-{b}_{2}y-{r}_{1}z\right)}{\partial y}+\frac{\frac{1}{yz}\left(B+{r}_{2}yz-dz\right)}{\partial z}=-\frac{{b}_{2}}{z}-\frac{B}{y{z}^{2}}<0.$

$J\left({E}_{3}\right)=\left(\begin{array}{ccc}{c}_{1}{y}_{3}-{a}_{1}& -\frac{{c}_{1}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}& 0\\ -{c}_{2}{y}_{3}& {a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}& -{r}_{1}{y}_{3}\\ 0& \frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}& {r}_{2}{y}_{3}-d\end{array}\right).$

${\lambda }^{3}+{e}_{1}{\lambda }^{2}+{e}_{2}\lambda +{e}_{3}=0.$

$\begin{array}{l}{e}_{1}=-\left[\left({r}_{2}{y}_{3}-d\right)+\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)+\left({c}_{1}{y}_{3}-{a}_{1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left({a}_{1}-{c}_{1}{y}_{3}\right)+\left(d-{r}_{2}{y}_{3}\right)+{y}_{3}\left({b}_{2}-\frac{{c}_{1}{c}_{2}}{{b}_{1}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\frac{{a}_{1}{c}_{2}}{{b}_{1}}-\left({a}_{2}-{b}_{2}{y}_{3}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\right]>0.\end{array}$

$\begin{array}{l}{e}_{2}=\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\left({r}_{2}{y}_{3}-d\right)+\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}{r}_{1}{y}_{3}+\left({c}_{1}{y}_{3}-{a}_{1}\right)\left({r}_{2}{y}_{3}-d\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\left({c}_{1}{y}_{3}-{a}_{1}\right)+{c}_{2}{y}_{3}\frac{{c}_{1}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }=\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\left({r}_{2}{y}_{3}-d+{c}_{1}{y}_{3}-{a}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left({a}_{1}-{c}_{1}{y}_{3}\right)\left(\frac{{c}_{1}{c}_{2}{y}_{3}}{{b}_{1}}-{r}_{2}{y}_{3}+d\right)+\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}{r}_{1}{y}_{3}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\left({a}_{2}-{b}_{2}{y}_{3}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)-\frac{{a}_{1}{c}_{2}}{{b}_{1}}+{y}_{3}\left(\frac{{c}_{1}{c}_{2}}{{b}_{1}}-{b}_{2}\right)\right]\left({r}_{2}{y}_{3}-d+{c}_{1}{y}_{3}-{a}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left({a}_{1}-{c}_{1}{y}_{3}\right)\left(\frac{{c}_{1}{c}_{2}{y}_{3}}{{b}_{1}}-{r}_{2}{y}_{3}+d\right)+\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}{r}_{1}{y}_{3}>0.\end{array}$

$\begin{array}{c}{e}_{3}={c}_{2}{y}_{3}{c}_{1}\frac{{a}_{1}-{c}_{1}{y}_{3}}{{b}_{1}}\left({r}_{2}{y}_{3}-d\right)-\left({c}_{1}{y}_{3}-{a}_{1}\right)\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\left({r}_{2}{y}_{3}-d\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left({c}_{1}{y}_{3}-{a}_{1}\right){r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\\ =\left({a}_{1}-{c}_{1}{y}_{3}\right)\left[{c}_{2}{y}_{3}{c}_{1}\frac{{r}_{2}{y}_{3}-d}{{b}_{1}}+\left({a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)\left({r}_{2}{y}_{3}-d\right)+{r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\right]\\ =\left({a}_{1}-{c}_{1}{y}_{3}\right)\left[\left({r}_{2}{y}_{3}-d\right)\left(\frac{{c}_{2}{y}_{3}{c}_{1}}{{b}_{1}}+{a}_{2}-2{b}_{2}{y}_{3}-\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}\right)+{r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\right]\end{array}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left({a}_{1}-{c}_{1}{y}_{3}\right)\left[\left({r}_{2}{y}_{3}-d\right)\left({a}_{2}-{b}_{2}{y}_{3}-\frac{B{r}_{1}}{d-{r}_{2}{y}_{3}}-{b}_{2}{y}_{3}+\frac{{c}_{1}{c}_{2}{y}_{3}}{{b}_{1}}\right)+{r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\right]\\ =\left({a}_{1}-{c}_{1}{y}_{3}\right)\left[\left({r}_{2}{y}_{3}-d\right)\left(\frac{{c}_{2}\left({a}_{1}-{c}_{1}{y}_{3}\right)}{{b}_{1}}-{b}_{2}{y}_{3}+\frac{{c}_{1}{c}_{2}{y}_{3}}{{b}_{1}}\right)+{r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\right]\\ >\left({a}_{1}-{c}_{1}{y}_{3}\right)\left[\left({r}_{2}{y}_{3}-d\right)\left(\frac{{a}_{1}{c}_{2}}{{b}_{1}}-\frac{{b}_{2}{a}_{1}}{{c}_{1}}\right)+{r}_{1}{y}_{3}\frac{B{r}_{2}}{d-{r}_{2}{y}_{3}}\right]>0.\end{array}$

5. 数值模拟

${a}_{1}=1,{a}_{2}=2,{b}_{1}=0.5,{b}_{2}=1.5,{c}_{1}=1.5,{c}_{2}=0.5,{r}_{1}=0.4,{r}_{2}=0.5,B=0.2,d=0.7$ 时， $B<\frac{d}{{r}_{1}}{a}_{2}=3.5$，最终得到正常细胞灭绝平衡点 ${E}_{2}\left(0,1.0383,1.1061\right)$ 是全局渐近稳定的(如图2所示)。

${a}_{1}=1,{a}_{2}=2,{b}_{1}=0.5,{b}_{2}=1,{c}_{1}=0.4,{c}_{2}=0.5,{r}_{1}=0.4,{r}_{2}=0.5,B=1,d=0.7$ 时， $B<\frac{d}{{r}_{1}}\left({a}_{2}-\frac{{a}_{1}{c}_{2}}{{b}_{1}}\right)=1.75$，最终得到正平衡点 ${E}_{3}\left(1.7032,0.3710,1.9436\right)$ 是局部渐近稳定的(如图3所示)。

Figure 1. The Stability of tumor extinction equilibrium E1

Figure 2. The Stability of normal cell extinction equilibrium E2

Figure 3. The Stability of the positive equilibrium E3

6. 结论

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