﻿ 一类HCV模型的随机稳定性和分岔分析

# 一类HCV模型的随机稳定性和分岔分析Stochastic Dynamics Analysis of HCV Virus Model

Abstract: Gauss white noise was added into the nonlinear HCV model, nonlinear differential equations were established, and the original equation was reduced to a two-dimensional Ito ̂ differential equation by applying the theorem of stochastic central manifold and the correlation theorem of stochastic average method. Then, the maximum Lyapunov index and singular boundary theory are applied to analyze the local stochastic stability and global stochastic stability of the stochastic system respectively, and the stability conditions are obtained. The dynamic bifurcation and phenomenological bifurcation behavior of systems are studied by using stochastic average method for quasi-integrable Hamiltonian systems. Finally, the numerical simulation shows that the infection rate of HCV epidemic will change obviously under the influence of noise.

1. 引言

2. 模型介绍

$\left\{\begin{array}{l}\frac{\text{d}T}{\text{d}t}=\lambda -{\beta }_{1}TV-{\beta }_{2}TI-{d}_{1}T+aI\\ \frac{\text{d}I}{\text{d}t}={\beta }_{1}TV+{\beta }_{2}TI-{d}_{2}T-aI\\ \frac{\text{d}V}{\text{d}t}=kI-{d}_{3}V-\rho VZ\\ \frac{\text{d}Z}{\text{d}t}=cVZ-{d}_{4}V\end{array}$ (1)

${E}_{1}\left(0,\frac{\lambda }{{d}_{1}},0,0\right)$${E}_{2}\left({I}^{\ast },{T}^{\ast },{V}^{\ast },{Z}^{\ast }\right)$，其中：

${I}^{\ast }=\frac{k\lambda {\beta }_{1}+\lambda {\beta }_{2}{d}_{3}-{d}_{1}{d}_{2}{d}_{3}-a{d}_{1}{d}_{3}}{{d}_{2}\left(k{\beta }_{1}+{\beta }_{2}{d}_{3}\right)}$, ${T}^{\ast }=\frac{{d}_{3}\left(a+{d}_{2}\right)}{k{\beta }_{1}+{\beta }_{2}{d}_{3}}$, ${V}^{\ast }=\frac{k\left(k\lambda {\beta }_{1}+\lambda {\beta }_{2}{d}_{3}-{d}_{1}{d}_{2}{d}_{3}-a{d}_{1}{d}_{3}\right)}{{d}_{2}{d}_{3}\left(k{\beta }_{1}+{\beta }_{2}{d}_{3}\right)}$

$\left\{\begin{array}{l}\stackrel{˙}{T}=\lambda -{\beta }_{1}TV-{\beta }_{2}TI-{d}_{1}T+aI+\delta \left(T-{T}^{\ast }\right)\xi \left(t\right)\\ \stackrel{˙}{I}={\beta }_{1}TV+{\beta }_{2}TI-{d}_{2}I-aI+\delta \left(I-{I}^{\ast }\right)\xi \left(t\right)\\ \stackrel{˙}{V}=kI-{d}_{3}V-\rho VZ+\delta \left(V-{V}^{\ast }\right)\xi \left(t\right)\\ \stackrel{˙}{Z}=cVZ-{d}_{4}V+\delta \left(Z-{Z}^{\ast }\right)\xi \left(t\right)\end{array}$ (2)

$\left\{\begin{array}{l}\stackrel{˙}{X}=AX-BY-{\beta }_{1}{T}^{\ast }U-{\beta }_{1}XU-{\beta }_{2}XY+\delta X\xi \left(t\right)\\ \stackrel{˙}{Y}=CX+DY+{\beta }_{1}{T}^{\ast }U+{\beta }_{1}XU+{\beta }_{2}XY+\delta Y\xi \left(t\right)\\ \stackrel{˙}{U}=EY-FU-\rho UY+\delta U\xi \left(t\right)\\ \stackrel{˙}{W}=HW+c{Z}^{\ast }U+cWU+\delta W\xi \left(t\right)\end{array}$ (3)

$A=-\left({\beta }_{1}{V}^{\ast }+{\beta }_{2}{I}^{\ast }+{d}_{1}\right)$, $B={\beta }_{2}{T}^{\ast }-a$, $C={\beta }_{1}{V}^{\ast }+{\beta }_{2}{I}^{\ast }$, $D={\beta }_{2}{T}^{\ast }-{d}_{2}-a$, $E=k-\rho {V}^{\ast }$, $F={d}_{3}+\rho {I}^{\ast }$, $H={d}_{4}-c{V}^{\ast }$

3. 随机HCV模型的初步处理

$f=\left(\lambda +H\right)\left({\lambda }^{3}+a{\lambda }^{2}+b\lambda +c\right)=0$ (4)

$a=F-D-A$, $b=AD-E{\beta }_{1}{T}^{\ast }-AF-BC-DF$, $c=AE{\beta }_{1}{T}^{\ast }+CE{\beta }_{1}{T}^{\ast }+ADF-BCF$

${W}_{loc}^{c}\left(O\right)=\left\{\left({x}_{1},{x}_{2},{x}_{3},{x}_{4}\right)|{x}_{1}=h\left({x}_{2},{x}_{3}\right),{x}_{4}=g\left({x}_{2},{x}_{3}\right),|{x}_{2}|+|{x}_{3}|\le 1\right\}$,

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{2}=-\sqrt{b}{x}_{3}+\varphi \left({x}_{2},{x}_{3}\right)+\delta \left({l}_{12}{x}_{2}+{l}_{13}{x}_{3}\right)\xi \left(t\right)\\ {\stackrel{˙}{x}}_{3}=\sqrt{b}{x}_{2}+\phi \left({x}_{2},{x}_{3}\right)+\delta \left({m}_{12}{x}_{2}+{m}_{13}{x}_{3}\right)\xi \left(t\right)\end{array}$ (5)

$\begin{array}{c}\varphi \left({x}_{2},{x}_{3}\right)=\left({l}_{2}{h}_{1}+{l}_{8}{g}_{1}\right){x}_{2}^{3}+\left({l}_{2}{h}_{2}+{l}_{3}{h}_{1}+{l}_{8}{g}_{2}+{l}_{9}{g}_{1}\right){x}_{2}^{2}{x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({l}_{2}{h}_{4}+{l}_{3}{h}_{2}+{l}_{8}{g}_{4}+{l}_{9}{g}_{2}\right){x}_{2}{x}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{l}_{4}{x}_{2}^{2}+{l}_{5}{x}_{2}{x}_{3}+{l}_{6}{x}_{3}^{2}+\left({l}_{3}{h}_{4}+{l}_{9}{g}_{4}\right){x}_{3}^{3}\end{array}$

$\begin{array}{c}\phi \left({x}_{2},{x}_{3}\right)=\left({m}_{2}{h}_{1}+{m}_{8}{g}_{1}\right){x}_{2}^{3}+\left({m}_{2}{h}_{2}+{m}_{3}{h}_{1}+{m}_{8}{g}_{2}+{m}_{9}{g}_{1}\right){x}_{2}^{2}{x}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({m}_{2}{h}_{4}+{m}_{3}{h}_{2}+{m}_{8}{g}_{4}+{m}_{9}{g}_{2}\right){x}_{2}{x}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{m}_{4}{x}_{2}^{2}+{m}_{5}{x}_{2}{x}_{3}+{m}_{6}{x}_{3}^{2}+\left({m}_{3}{h}_{4}+{m}_{9}{g}_{4}\right){x}_{3}^{3}\end{array}$

$\left\{\begin{array}{l}\text{d}r=\left[\left({\mu }_{1}+\frac{1}{16}{\mu }_{2}\right)r+\frac{1}{8}{\mu }_{3}{r}^{3}\right]\text{d}t+{\left(\frac{1}{8}{\mu }_{4}{r}^{2}\right)}^{\frac{1}{2}}\text{d}{W}_{r}\left(t\right)\\ \text{d}\theta =\left[{\mu }_{5}+\frac{1}{8}{\mu }_{6}{r}^{2}\right]\text{d}t+{\left(\frac{1}{8}{\mu }_{2}\right)}^{\frac{1}{2}}\text{d}{W}_{\theta }\left(t\right)\end{array}$ (6)

${W}_{r}\left(t\right)$${W}_{\theta }\left(t\right)$ 是独立的标准Wiener过程，并有以下标记：

${\mu }_{1}=0$

${\mu }_{2}={\delta }^{2}{l}_{12}^{2}+{\delta }^{2}{m}_{12}^{2}+{\delta }^{2}{l}_{13}^{2}+3{\delta }^{2}{m}_{13}^{2}$

${\mu }_{3}=3\left({l}_{2}{h}_{1}+{l}_{8}{g}_{1}\right)+\left({l}_{2}{h}_{4}+{l}_{3}{h}_{2}+{l}_{8}{g}_{4}+{l}_{9}{g}_{2}\right)+\left({m}_{2}{h}_{2}+{m}_{3}{h}_{2}+{m}_{8}{g}_{2}+{m}_{9}{g}_{1}\right)+\left({m}_{3}{h}_{4}+{m}_{9}{g}_{4}\right)$

${\mu }_{4}=3{\delta }^{2}{l}_{12}^{2}+{\delta }^{2}{m}_{12}^{2}+{\delta }^{2}{l}_{13}^{2}+{\delta }^{2}{m}_{13}^{2}$

${\mu }_{5}=4\sqrt{b}+{\delta }^{2}\left({l}_{12}{l}_{13}-{m}_{12}{m}_{13}\right)$

${\mu }_{6}=3\left({m}_{2}{h}_{1}+{m}_{8}{g}_{1}\right)-\left({m}_{2}{h}_{4}+{m}_{3}{h}_{2}+{m}_{8}{g}_{4}+{m}_{9}{g}_{2}\right)-\left({l}_{2}{h}_{2}+{l}_{3}{h}_{2}+{l}_{8}{g}_{2}+{l}_{9}{g}_{1}\right)-\left({l}_{4}{h}_{3}+{m}_{9}{g}_{4}\right)$

$\text{d}r=\left[\frac{1}{16}{\mu }_{2}r+\frac{1}{8}{\mu }_{3}{r}^{3}\right]\text{d}t+{\left(\frac{1}{8}{\mu }_{4}{r}^{2}\right)}^{\frac{1}{2}}\text{d}W\left(t\right)$ (7)

4. 随机稳定性分析

4.1. 局部随机稳定性

$\text{d}r=\frac{1}{16}{\mu }_{2}r\text{d}t+{\left(\frac{1}{8}{\mu }_{4}{r}^{2}\right)}^{\frac{1}{2}}\text{d}W\left(t\right)$ (8)

$r\left(t\right)=r\left(0\right)\mathrm{exp}\left({\int }_{0}^{t}\left[m\left(0\right)-\frac{{\delta }^{2}\left(0\right)}{2}\right]\text{d}s+{\int }_{0}^{t}\delta \left(0\right)\text{d}W\left(s\right)\right)$

$\lambda =\underset{t\to +\infty }{\mathrm{lim}}\frac{1}{t}\mathrm{ln}\left(‖Z\left(t,{z}_{0}\right)‖\right)=\underset{t\to +\infty }{\mathrm{lim}}\frac{1}{t}\mathrm{ln}{\left(r\left(t\right)\right)}^{\frac{1}{2}}=\frac{m\left(0\right)-\frac{{\delta }^{2}\left(0\right)}{2}}{2}=\frac{{\mu }_{2}-{\mu }_{4}}{32}$

4.2. 全局随机稳定性

$\text{d}r=\frac{1}{16}{\mu }_{2}r\text{d}t+{\left(\frac{1}{8}{\mu }_{4}{r}^{2}\right)}^{\frac{1}{2}}\text{d}W\left(t\right)$ (9)

$r=0$ 时，其为系统的第一类奇异边界；当 $r=+\infty$ 时， ${m}_{r}=+\infty$, $r=+\infty$ 为系统的第二类奇异边界。

, , ,

，则是排斥自然边界；若，则是吸引自然边界；若，则是严格自然边界。

, ,

，则是吸引自然边界；若，则是排斥自然边界；若，则是严格自然边界。

(10)

, ,

，则是排斥自然边界；若，则是吸引自然边界；若，则是严格自然边界。

, , ,

，即是吸引自然边界；若，即是排斥自然边界；若，即是严格自然边界。

5. 随机Hopf分岔

5.1. 动态分岔

(1) 当时，系统为：

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

，于是可得，当时不动点的不变测度是稳定的，当时非平凡状态不变测度

(19)

(2) 当时，令，且则系统(7)变为：

(20)

(21)

(22)

(23)

(24)

5.2. 维象分岔

(25)

(26)

(27)

(28)

6. 数值模拟

(a)的平稳概率密度图 (b)的联合概率密度图

Figure 1. The stationary probability density graph and the corresponding joint probability density graph when

(a)的平稳概率密度图 (b)的联合概率密度图

Figure 2. The stationary probability density graph and the corresponding joint probability density graph when

(a)的平稳概率密度图 (b)的联合概率密度图

Figure 3. The stationary probability density graph and the corresponding joint probability density graph when

7. 结论

NOTES

*第一作者。

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