﻿ 具有混合切换的随机丙型肝炎病毒感染模型的长期性行为

# 具有混合切换的随机丙型肝炎病毒感染模型的长期性行为Long Time Behavior of Stochastic Hepatitis C Virus Infection Model with Mixed Switching

Abstract: This paper mainly studies the extinction and persistence of a stochastic hepatitis C virus infection model (HCV model) with mixed switching. First, we prove the existence and uniqueness of the so-lution of the stochastic HCV model. Secondly, we prove the existence and uniqueness of the invariant probability measure of the stochastic HCV model by using the theory of Feller property, invariant control set and Krylov Bogoliubov theorem. Finally, by using the strong ergodicity theorem, Borel-Cantelli lemma and iterated logarithm law, the conditions of extinction and persistence of the stochastic HCV model are obtained.

1. 引言

$\left\{\begin{array}{l}\frac{\text{d}{W}_{t}}{\text{d}t}=S+a{W}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)-b{W}_{t}-\left(1-\eta \right)\beta {V}_{t}{W}_{t}+q{I}_{t}\\ \frac{\text{d}{I}_{t}}{\text{d}t}=\nu {I}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)+\left(1-\eta \right)\beta {V}_{t}{W}_{t}-\omega {I}_{t}-q{I}_{t}\\ \frac{\text{d}{V}_{t}}{\text{d}t}=\left(1-ϵ\right)p{I}_{t}-c{V}_{t},\end{array}$ (1.1)

$\left\{\begin{array}{l}\frac{\text{d}{W}_{t}}{\text{d}t}={S}_{{\alpha }_{t}}+{a}_{{\alpha }_{t}}{W}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)-{b}_{{\alpha }_{t}}{W}_{t}-\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}+{q}_{{\alpha }_{t}}{I}_{t},\\ \frac{\text{d}{I}_{t}}{\text{d}t}={\nu }_{{\alpha }_{t}}{I}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)+\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}-{\omega }_{{\alpha }_{t}}{I}_{t}-{q}_{{\alpha }_{t}}{I}_{t},\\ \frac{\text{d}{V}_{t}}{\text{d}t}=\left(1-{ϵ}_{{\alpha }_{t}}\right){p}_{{\alpha }_{t}}{I}_{t}-{c}_{{\alpha }_{t}}{V}_{t},\end{array}$ (1.2)

$ℙ\left({\alpha }_{t+\Delta }=j|{\alpha }_{t}=i\right)=\left\{\begin{array}{ll}{q}_{ij}\Delta +o\left(\Delta \right),\hfill & i\ne j,\hfill \\ 1+{q}_{ii}\Delta +o\left(\Delta \right),\hfill & i=j,\hfill \end{array}$ (1.3)

$ℙ\left({\alpha }_{t+\Delta }=j|{\alpha }_{t}=i,{X}_{t}=x\right)=\left\{\begin{array}{ll}{q}_{ij}\left(x\right)\Delta +o\left(\Delta \right),\hfill & i\ne j,\hfill \\ 1+{q}_{ii}\left(x\right)\Delta +o\left(\Delta \right),\hfill & i=j,\hfill \end{array}$ (1.4)

$\left\{\begin{array}{l}\text{d}{W}_{t}=\left\{{S}_{{\alpha }_{t}}+{a}_{{\alpha }_{t}}{W}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)-{b}_{{\alpha }_{t}}{W}_{t}-\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}+{q}_{{\alpha }_{t}}{I}_{t}\right\}\text{d}t,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left({b}_{{\alpha }_{t}}^{e}-{a}_{{\alpha }_{t}}^{e}\right){W}_{t}\text{d}{B}_{t}^{\left(1\right)}-\frac{{a}_{{\alpha }_{t}}^{e}{I}_{t}}{{W}_{\mathrm{max}}}{W}_{t}\text{d}{B}_{t}^{\left(2\right)}-\left(1-{\eta }_{{\alpha }_{t}}^{e}\right){\beta }_{{\alpha }_{t}}^{e}{V}_{t}{W}_{t}\text{d}{B}_{t}^{\left(3\right)},\\ \text{d}{I}_{t}=\left\{{\nu }_{{\alpha }_{t}}{I}_{t}\left(1-\frac{{W}_{t}+{I}_{t}}{{W}_{\mathrm{max}}}\right)+\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}-{\omega }_{{\alpha }_{t}}{I}_{t}-{q}_{{\alpha }_{t}}{I}_{t}\right\}\text{d}t,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left({\omega }_{{\alpha }_{t}}^{e}+{q}_{{\alpha }_{t}}^{e}-{\nu }_{{\alpha }_{t}}^{e}\right){I}_{t}\text{d}{B}_{t}^{\left(1\right)}-\frac{{\nu }_{{\alpha }_{t}}^{e}{W}_{t}}{{W}_{\mathrm{max}}}{I}_{t}\text{d}{B}_{t}^{\left(2\right)},\\ \text{d}{V}_{t}=\left\{\left(1-{ϵ}_{{\alpha }_{t}}\right){p}_{{\alpha }_{t}}{I}_{t}-{c}_{{\alpha }_{t}}{V}_{t}\right\}\text{d}t-{c}_{{\alpha }_{t}}^{e}{V}_{t}\text{d}{B}_{t}^{\left(1\right)}\end{array}$ (1.5)

2. 具有独立于状态的随机HCV模型的灭绝性与持久性

· ${\mathcal{H}}_{1}$ 对所有的 $i\in \mathcal{M}$$F\left(\cdot ,i\right):ℝ\to {ℝ}_{+}$ 是局部利普希茨连续的，且存在常数 $c>0$，使得 $F\left(x,i\right)\le c\left(1+|x|\right),x\in ℝ$

· ${\mathcal{H}}_{2}$ 连续时间马尔科夫链 ${\left({\alpha }_{t}\right)}_{t\ge 0}$ 是不可约和正递归的，且其具有不变概率测度 $\pi =\left({\pi }_{1},{\pi }_{2},\cdots ,{\pi }_{M}\right)$

${N}_{t}\le {N}_{0}{\text{e}}^{-{\int }_{0}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}+{\int }_{0}^{t}\text{ }{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}.$ (2.1)

Proof. 首先由随机模型(1.2)可以得到

$\begin{array}{l}\text{d}{N}_{t}=\left\{{S}_{{\alpha }_{t}}-\left({b}_{{\alpha }_{t}}-{a}_{{\alpha }_{t}}\right){W}_{t}-\left({\omega }_{{\alpha }_{t}}-{\nu }_{{\alpha }_{t}}\right){I}_{t}-\left({W}_{t}+{I}_{t}\right)\frac{{b}_{{\alpha }_{t}}{W}_{t}+{a}_{{\alpha }_{t}}{I}_{t}}{{W}_{\mathrm{max}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{ }\\ \text{ }\end{array}+\left(1-{ϵ}_{{\alpha }_{t}}\right){p}_{{\alpha }_{t}}{I}_{t}-{c}_{{\alpha }_{t}}{V}_{t}\right\}\text{d}t,\text{ }t>0.\end{array}$ (2.2)

$\text{d}{N}_{t}\le \left\{{S}_{{\alpha }_{t}}-\left({b}_{{\alpha }_{t}}-{a}_{{\alpha }_{t}}\right){W}_{t}-\left({\omega }_{{\alpha }_{t}}-{\nu }_{{\alpha }_{t}}\right){I}_{t}+\left(1-{ϵ}_{{\alpha }_{t}}\right){p}_{{\alpha }_{t}}{I}_{t}-{c}_{{\alpha }_{t}}{V}_{t}\right\}\text{d}t.$ (2.3)

${\Upsilon }_{{\alpha }_{t}}=min\left\{{b}_{{\alpha }_{t}}-{a}_{{\alpha }_{t}},{\omega }_{{\alpha }_{t}}-{\nu }_{{\alpha }_{t}}-{p}_{{\alpha }_{t}}+{ϵ}_{{\alpha }_{t}}{p}_{{\alpha }_{t}},{c}_{{\alpha }_{t}}\right\}$，则有

$\text{d}{N}_{t}\le \left\{{S}_{{\alpha }_{t}}-{\Upsilon }_{{\alpha }_{t}}{N}_{t}\right\}\text{d}t.$

$\text{d}{N}_{t}\le \left\{{S}_{{\alpha }_{{\tau }_{k}}}-{\Upsilon }_{{\alpha }_{{\tau }_{k}}}{N}_{t}\right\}\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{\tau }_{k},{\tau }_{k+1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\in ℕ.$

$\text{d}\left({\text{e}}^{{\Upsilon }_{{\alpha }_{{\tau }_{k}}}t}{N}_{t}\right)$ 应用伊藤链式法，有

$\text{d}\left({\text{e}}^{{\Upsilon }_{{\alpha }_{{\tau }_{k}}}t}{N}_{t}\right)={\Upsilon }_{{\alpha }_{{\tau }_{k}}}{\text{e}}^{{\Upsilon }_{{\alpha }_{{\tau }_{k}}}t}{N}_{t}\text{d}t+{\text{e}}^{{\Upsilon }_{{\alpha }_{{\tau }_{k}}}t}\text{d}{N}_{t}\le {\text{e}}^{{\Upsilon }_{{\alpha }_{{\tau }_{k}}}t}{S}_{{\alpha }_{{\tau }_{k}}}\text{d}t,$

$\begin{array}{c}{N}_{t}\le {\text{e}}^{-{\Upsilon }_{{\alpha }_{{\tau }_{k}}}\left(t-{\tau }_{k}\right)}{N}_{{\tau }_{k}}+{\int }_{{\tau }_{k}}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\Upsilon }_{{\alpha }_{{\tau }_{k}}}\left(t-s\right)}\text{d}s={\text{e}}^{-{\int }_{{\tau }_{k}}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}{N}_{{\tau }_{k}}+{\int }_{{\tau }_{k}}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s\\ \le {\text{e}}^{-{\int }_{{\tau }_{k}}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}\left({\text{e}}^{-{\int }_{{\tau }_{k-1}}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}{N}_{{\tau }_{k-1}}+{\int }_{{\tau }_{k-1}}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s\right)+{\int }_{{\tau }_{k}}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s\\ ={\text{e}}^{-{\int }_{{\tau }_{k-1}}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}{N}_{{\tau }_{k-1}}+{\int }_{{\tau }_{k-1}}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s\le \cdots \le {\text{e}}^{-{\int }_{0}^{t}{\Upsilon }_{{\alpha }_{s}}\text{d}s}{N}_{0}+{\int }_{0}^{t}{S}_{{\alpha }_{s}}{\text{e}}^{-{\int }_{s}^{t}{\Upsilon }_{{\alpha }_{u}}\text{d}u}\text{d}s.\end{array}$ (2.4)

${W}_{t}\le {N}_{t}\le {N}_{0}{\text{e}}^{-\stackrel{^}{\Upsilon }t}+\stackrel{⌣}{S}/\stackrel{^}{\Upsilon }.$ (2.5)

${\chi }_{t}\left(\xi \right)=\frac{1}{t}{\int }_{0}^{t}{M}_{s}\left({w}_{0},{i}_{0},{v}_{0},i;\xi \right)\text{d}s.$

${\chi }_{t}\left({B}_{K}\left(0\right)×\mathcal{M}\right)=\frac{1}{t}{\int }_{0}^{t}{M}_{s}\left({w}_{0},{i}_{0},{v}_{0},i;{B}_{K}\left(0\right)\text{d}s×\mathcal{M}\right)\ge 1-\frac{1}{K}\underset{t\ge 0}{sup}E{N}_{t}\ge 1-\epsilon .$

$\frac{\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}}{{I}_{t}}\le {\Psi }_{t}+{\Phi }_{{\alpha }_{t}}$ (2.6)

${\Delta }_{1}:=\frac{{\sum }_{i\in \mathcal{M}}{\pi }_{i}{\Phi }_{i}}{{\sum }_{i\in \mathcal{M}}{\pi }_{i}\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)}<1.$ (2.7)

$\underset{t\to \infty }{lim}{I}_{t}=0,\text{ }\text{a}\text{.s}.,\text{ }\underset{t\to \infty }{lim}{V}_{t}=0,\text{ }\text{a}\text{.s}.$ (2.8)

$\frac{1}{t}{\int }_{0}^{t}\left\{\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\right\}\text{d}s=\underset{i\in \mathcal{M}}{\sum }\text{ }\text{ }{\pi }_{i}{S}_{i}.$ (2.9)

$\begin{array}{c}\frac{\text{d}}{\text{d}t}\mathrm{ln}{I}_{t}={\nu }_{{\alpha }_{t}}\left(1-\frac{W+I}{{W}_{m}}\right)+\frac{\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}}{{I}_{t}}-{\omega }_{{\alpha }_{t}}-{q}_{{\alpha }_{t}}\\ =-\frac{{\nu }_{{\alpha }_{t}}\left({W}_{t}+{I}_{t}\right)}{{W}_{m}}+\frac{\left(1-\eta \right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}}{{I}_{t}}+{\nu }_{{\alpha }_{t}}-{\omega }_{{\alpha }_{t}}-{q}_{{\alpha }_{t}}\\ \le \frac{\left(1-{\eta }_{{\alpha }_{t}}\right){\beta }_{{\alpha }_{t}}{V}_{t}{W}_{t}}{{I}_{t}}-\left({\omega }_{{\alpha }_{t}}+{q}_{{\alpha }_{t}}-{\nu }_{{\alpha }_{t}}\right)\\ \le {\Psi }_{t}+{\Phi }_{t}-\left({\omega }_{{\alpha }_{t}}+{q}_{{\alpha }_{t}}-{\nu }_{{\alpha }_{t}}\right).\end{array}$ (2.10)

$ln\left(\frac{{I}_{t}}{{I}_{0}}\right)\le {\int }_{0}^{t}\text{ }{\Psi }_{s}\text{d}s+{\int }_{0}^{t}\left({\Phi }_{s}-\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right)\right)\text{d}s.$ (2.11)

$\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}\frac{\mathrm{ln}{I}_{t}}{t}\le \underset{i\in \mathcal{M}}{\sum }\left\{{\Phi }_{i}-\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)\right\}{\pi }_{i},\text{ }\text{a}\text{.s}.$

$\text{d}{V}_{t}\le \left\{\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{I}_{t}-\stackrel{^}{v}{V}_{t}\right\}\text{d}t,$

$\text{d}\left({\text{e}}^{ct}{V}_{t}\right)$ 应用伊藤链式法，有

${V}_{t}\le {V}_{0}{\text{e}}^{-\stackrel{^}{c}t}+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{0}^{t}\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}{I}_{s}\text{d}s$ (2.12)

${I}_{t}\left(\omega \right)\le \frac{\stackrel{^}{c}\epsilon }{3\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}},\text{ }t\ge T,\text{ }\omega \in {\Omega }_{0}.$

$\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{T}^{t}\text{ }\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}{I}_{s}\left(\omega \right)\text{d}s\le \frac{\epsilon }{3},\text{ }\omega \in {\Omega }_{0},\text{ }t\ge T.$

$t\ge T\vee \left(\frac{1}{\stackrel{^}{c}}ln\frac{3{V}_{0}}{\epsilon }\right)\vee \left(T+\frac{1}{\stackrel{^}{c}}ln\frac{3\left(1-\stackrel{^}{ϵ}\right)\left({N}_{0}+\frac{\stackrel{⌣}{S}}{\stackrel{^}{\gamma }}\right)}{\stackrel{^}{c}\epsilon }\right),$

$\begin{array}{c}{V}_{t}\left(\omega \right)\le {V}_{0}{\text{e}}^{-\stackrel{^}{c}t}+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{0}^{T}\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}{I}_{s}\left(\omega \right)\text{d}s+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{T}^{t}\text{ }\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}{I}_{s}\left(\omega \right)\text{d}s\\ \le \frac{\epsilon }{3}+{V}_{0}{\text{e}}^{-\stackrel{^}{c}t}+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{0}^{T}\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}{I}_{s}\left(\omega \right)\text{d}s\\ \le \frac{\epsilon }{3}+{V}_{0}{\text{e}}^{-\stackrel{^}{c}t}+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}\left({N}_{0}+\frac{\stackrel{⌣}{S}}{\stackrel{^}{\gamma }}\right){\int }_{0}^{T}\text{ }{\text{e}}^{-\stackrel{^}{c}\left(t-s\right)}\text{d}s\\ \le \epsilon ,\text{ }\omega \in {\Omega }_{0}.\end{array}$ (2.13)

$\underset{t\to \infty }{lim}\left(\frac{1}{t}{\int }_{0}^{t}\text{ }{I}_{s}\text{d}s\right)=0,\text{ }\text{a}\text{.s}\text{.}\text{\hspace{0.17em}}\text{ }\underset{t\to \infty }{lim}\left(\frac{1}{t}{\int }_{0}^{t}\text{ }{V}_{s}\text{d}s\right)=0,\text{ }\text{a}\text{.s}.$ (2.14)

$\begin{array}{l}\frac{1}{t}\left\{{\int }_{0}^{t}\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\wedge 2\right)\end{array}$ 之间，通过BDG不等式和(4.1)式可知，存在常数

${\stackrel{˜}{M}}_{p},{\stackrel{^}{M}}_{p}>0$，使得

$\begin{array}{c}E\left(\underset{t\in \left[k,k+1\right]}{\mathrm{sup}}{|{\int }_{k}^{t}\frac{{\nu }_{{\alpha }_{s}}^{e}{W}_{s}}{{W}_{\mathrm{max}}}\text{d}{B}_{s}^{\left(2\right)}|}^{p}\right)\le {\stackrel{˜}{M}}_{p}\frac{{\stackrel{⌣}{\nu }}^{e}}{{W}_{\mathrm{max}}}E{\left({\int }_{k}^{k+1}\text{ }\text{ }{W}_{s}^{2}\text{d}s\right)}^{\frac{p}{2}}\\ \le {\stackrel{˜}{M}}_{p}\frac{{\stackrel{⌣}{\nu }}^{e}}{{W}_{\mathrm{max}}}E\left(\underset{k\le s\le k+1}{\mathrm{sup}}{W}_{s}^{p}\right)\\ \le {\stackrel{^}{M}}_{p}.\end{array}$ (4.11)

${B}_{k,C}:=\left\{\frac{1}{k}\underset{t\in \left[k,k+1\right]}{\mathrm{sup}}|{\int }_{k}^{t}\frac{{\nu }_{{\alpha }_{s}}^{e}{W}_{s}}{{W}_{\mathrm{max}}}\text{d}{B}_{s}^{\left(2\right)}|\ge C\right\}$

$ℙ\left({B}_{k,C}\right)\le \frac{1}{{k}^{p}{C}^{p}}E\left(\underset{t\in \left[k,k+1\right]}{sup}{|{\int }_{k}^{t}\frac{{\nu }_{{\alpha }_{s}}^{e}{W}_{s}}{{W}_{\mathrm{max}}}\text{d}{B}_{s}^{\left(2\right)}|}^{p}\right)\le \frac{{\stackrel{^}{M}}_{p}}{{k}^{p}{C}^{p}}$

$ℙ\left(\underset{k\to \infty }{\mathrm{lim}}\mathrm{sup}{B}_{k,C}\right)=0.$

$\underset{k=1}{\overset{\infty }{\sum }}\text{ }E\left({|{u}_{k}|}^{p}|{\mathcal{F}}_{k-1}\right){k}^{-p}\le {\stackrel{^}{M}}_{p}\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }{k}^{-p}<\infty$ (4.12)

$\underset{t\to \infty }{\mathrm{lim}}\frac{1}{t}\left({\int }_{0}^{t}\left({b}_{{\alpha }_{t}}-{a}_{{\alpha }_{t}}\right){W}_{s}\text{d}{B}_{s}^{\left(1\right)}\right)=0;\underset{t\to \infty }{\mathrm{lim}}\frac{1}{t}\left({\int }_{0}^{t}\frac{{a}_{{\alpha }_{t}}{I}_{s}}{{W}_{\mathrm{max}}}\text{d}{B}_{s}^{\left(2\right)}\right)=0;$ (4.13)

$\underset{t\to \infty }{\mathrm{lim}}\frac{1}{t}\left({\int }_{0}^{t}\left(1-{\eta }_{{\alpha }_{s}}\right){\beta }_{{\alpha }_{s}}{V}_{s}{W}_{s}\text{d}{B}_{s}^{\left(3\right)}\right)=0;\underset{t\to \infty }{\mathrm{lim}}\frac{1}{t}\left({\int }_{0}^{t}\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right){I}_{s}\text{d}{B}_{s}^{\left(1\right)}\right)=0;$

$\underset{t\to \infty }{\mathrm{lim}}\frac{1}{t}\left({\int }_{0}^{t}\text{ }{c}_{{\alpha }_{s}}{V}_{s}\text{d}{B}_{s}^{\left(1\right)}\right)=0;\underset{t\to \infty }{\mathrm{lim}}\frac{{N}_{t}}{t}=0;$

(a) 若 $i↦{\Phi }_{i}^{*}:={\Phi }_{i}-\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)$ 为非减的，且有

${\Delta }_{9}:=\frac{{\sum }_{i\in \mathcal{M}}{\stackrel{˜}{\pi }}_{i}{\Phi }_{i}}{{\sum }_{i\in \mathcal{M}}{\stackrel{˜}{\pi }}_{i}\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)}<1.$ (4.14)

$\underset{t\to \infty }{\mathrm{lim}}{I}_{t}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{t\to \infty }{\mathrm{lim}}{V}_{t}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}\text{.}$ (4.15)

$\begin{array}{c}\underset{i\in \mathcal{M}}{\sum }\text{ }{\pi }_{i}^{*}{S}_{i}\le \underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left(\frac{1}{t}{\int }_{0}^{t}\left\{\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\right\}\text{d}s\right)\\ \le \underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\left(\frac{1}{t}{\int }_{0}^{t}\left\{\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\right\}\text{d}s\right)\le \underset{i\in \mathcal{M}}{\sum }\text{ }{\stackrel{˜}{\pi }}_{i}{S}_{i}.\end{array}$ (4.16)

(b) 若 $i↦{\Phi }_{i}^{*}:={\Phi }_{i}-\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)$ 为非增的，且有

${\Delta }_{10}:=\frac{{\sum }_{i\in \mathcal{M}}{\pi }_{i}^{*}{\Phi }_{i}}{{\sum }_{i\in \mathcal{M}}{\pi }_{i}^{*}\left({\omega }_{i}+{q}_{i}-{\nu }_{i}\right)}<1.$ (4.17)

$\underset{t\to \infty }{lim}{I}_{t}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}d\underset{t\to \infty }{lim}{V}_{t}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}\text{.}$

$\begin{array}{c}\underset{i\in \mathcal{M}}{\sum }\text{ }{\stackrel{˜}{\pi }}_{i}{S}_{i}\le \underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left(\frac{1}{t}{\int }_{0}^{t}\left\{\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\right\}\text{d}s\right)\\ \le \underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\left(\frac{1}{t}{\int }_{0}^{t}\left\{\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\right\}\text{d}s\right)\le \underset{i\in \mathcal{M}}{\sum }\text{ }{\pi }_{i}^{*}{S}_{i}.\end{array}$ (4.18)

${I}_{1}\left(t\right)=-{\int }_{0}^{t}{\mu }_{{\alpha }_{s}}^{e}\text{d}{B}_{s}^{\left(1\right)};{I}_{2}\left(t\right)=-\frac{1}{t}{\int }_{0}^{t}\frac{{\nu }_{{\alpha }_{s}}{W}_{s}}{{W}_{max}}\text{d}{B}_{s}^{\left(2\right)}.$

$\begin{array}{l}\frac{1}{t}\mathrm{ln}\left({I}_{t}/{I}_{0}\right)\\ =\frac{1}{t}{\int }_{0}^{t}\left\{-{\nu }_{{\alpha }_{s}}\frac{{W}_{s}+{I}_{s}}{{W}_{\mathrm{max}}}+\frac{\left(1-{\eta }_{{\alpha }_{s}}\right){\beta }_{{\alpha }_{s}}{V}_{s}{W}_{s}}{{I}_{s}}-\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right)-\frac{1}{2}{\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{2}{\left(\frac{{\nu }_{{\alpha }_{s}}{W}_{s}}{{W}_{\mathrm{max}}}\right)}^{2}\right\}\text{d}s-\frac{1}{t}{\int }_{0}^{t}\left({\omega }_{{\alpha }_{s}}^{e}+{q}_{{\alpha }_{s}}^{e}-{\nu }_{{\alpha }_{s}}^{e}\right)\text{d}{B}_{s}^{\left(1\right)}-\frac{1}{t}{\int }_{0}^{t}\frac{{\nu }_{{\alpha }_{s}}{W}_{s}}{{W}_{\mathrm{max}}}\text{d}{B}_{s}^{\left(2\right)}\\ \le \frac{1}{t}{\int }_{0}^{t}\left\{\frac{\left(1-{\eta }_{{\alpha }_{s}}\right){\beta }_{{\alpha }_{s}}{V}_{s}{W}_{s}}{{I}_{s}}-\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right)\right\}\text{d}s+\frac{{I}_{1}\left(t\right)}{t}+{I}_{2}\left(t\right)\\ \le \frac{1}{t}{\int }_{0}^{t}\left\{{\Psi }_{s}+{\Phi }_{{\alpha }_{s}}-\left({\omega }_{{\alpha }_{s}}+{q}_{{\alpha }_{s}}-{\nu }_{{\alpha }_{s}}\right)\right\}\text{d}s+\frac{{I}_{1}\left(t\right)}{t}+{I}_{2}\left(t\right).\end{array}$ (4.19)

$\underset{t\to \infty }{lim}{I}_{2}\left(t\right)=0\text{ }\text{a}\text{.s}\text{.}$ (4.20)

${〈{I}_{1}〉}_{t}$${I}_{1}\left(t\right)$ 的二次变差，则有

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}{〈{I}_{1}〉}_{t}\le {\left({\stackrel{⌣}{\mu }}^{e}\right)}^{2}.$

$\underset{t\to \infty }{lim}\frac{1}{t}{I}_{1}\left(t\right)=0\text{ }\text{a}\text{.s}\text{.}$ (4.21)

$\underset{t\to \infty }{lim}{I}_{t}=0\text{ }\text{a}\text{.s}\text{.}$

${I}_{t}\left(\omega \right)\le \epsilon ,\text{ }t\ge {T}_{1},\text{ }\omega \in {\Omega }_{1}$ (4.22)

${\xi }_{i}:={\omega }_{i}+{\iota }_{i}+{\left({\mu }_{i}^{e}\right)}^{2}/2,i\in \mathcal{M}$，且对任意 $0\le s\le t$

${S}_{s,t}:={\int }_{s}^{t}{\mu }_{{\alpha }_{u}}^{e}\text{d}{B}_{u}^{\left(2\right)};\text{ }{\Phi }_{s,t}:=\mathrm{exp}\left(-{\int }_{s}^{t}{\xi }_{{\alpha }_{u}}\text{d}u-{S}_{s,t}\right).$

$\underset{t\to \infty }{lim}\frac{1}{t}ln{\Phi }_{0,t}<0,\text{ }\text{a}\text{.s}\text{.}$

${\Phi }_{0,t}\left(\omega \right)\le \epsilon ,\text{ }t\ge {T}_{2},\text{ }\omega \in {\Omega }_{2}.$ (4.23)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{{S}_{s,t}}{\sqrt{2{〈S〉}_{s,t}\mathrm{log}\mathrm{log}{〈S〉}_{s,t}}}=-1\text{ }\text{a}\text{.s}.$

$-\left(1+\epsilon \right)\sqrt{2{〈S〉}_{s,t}\mathrm{log}\mathrm{log}{〈S〉}_{s,t}}\le {S}_{s,t}\le \left(-1+\epsilon \right)\sqrt{2{〈S〉}_{s,t}\mathrm{log}\mathrm{log}{〈S〉}_{s,t}},t,s\ge {T}_{2},\epsilon \in {\Omega }_{3}.$ (4.24)

$\begin{array}{c}{V}_{t}\left(\omega \right)={\Phi }_{0,t}\left(\omega \right)\left\{{V}_{0}+{\int }_{0}^{t}\text{ }\text{ }{\Phi }^{-1}\left(s\right)\left(1-{ϵ}_{{\alpha }_{s}}\right){p}_{{\alpha }_{s}}{I}_{s}\left(\omega \right)\text{d}s\right\}\\ ={\Phi }_{0,t}\left(\omega \right){V}_{0}+{\Phi }_{s,t}\left(\omega \right){\int }_{0}^{t}\left(1-{ϵ}_{{\alpha }_{s}}\right){p}_{{\alpha }_{s}}{I}_{s}\left(\omega \right)\text{d}s\\ \le {V}_{0}\epsilon +{\Phi }_{T,t}\left(\omega \right)\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{0}^{T}{I}_{s}\left(\omega \right){\Phi }_{s,T}\left(\omega \right)\text{d}s+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{T}^{t}{I}_{s}\left(\omega \right){\Phi }_{s,t}\left(\omega \right)\text{d}s\\ \le {V}_{0}\epsilon +\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}\epsilon {\int }_{T}^{t}\mathrm{exp}\left(-{\int }_{s}^{t}\text{ }{\xi }_{{\alpha }_{u}}\text{d}u+\left(1+\epsilon \right)\sqrt{2{〈S〉}_{s,t}\mathrm{log}\mathrm{log}{〈S〉}_{s,t}}\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(1-\stackrel{^}{ϵ}\right)\stackrel{⌣}{p}{\int }_{0}^{T}{I}_{s}\left(\omega \right){\Phi }_{s,T}\left(\omega \right)\text{d}s\mathrm{exp}\left(-{\int }_{T}^{t}\text{ }\text{ }{\xi }_{{\alpha }_{u}}\text{d}u+\left(1+\epsilon \right)\sqrt{2{〈S〉}_{T,t}\mathrm{log}\mathrm{log}{〈S〉}_{T,t}}\right).\end{array}$ (4.25)

$\left(1+\epsilon \right)\sqrt{2{〈S〉}_{s,t}\mathrm{log}\mathrm{log}{〈S〉}_{s,t}}\le c+q{\int }_{s}^{t}\text{ }{\xi }_{{\alpha }_{u}}\text{d}u.$ (4.26)

$\underset{t\to \infty }{lim}\left(\frac{1}{t}{\int }_{0}^{t}\left\{{S}_{{\alpha }_{s}}-\left({b}_{{\alpha }_{s}}-{a}_{{\alpha }_{s}}\right){W}_{s}+\frac{{b}_{{\alpha }_{s}}{W}_{s}^{2}}{{W}_{\mathrm{max}}}\text{d}s\right\}\right)=0$ (4.27)

[1] Neumann, A., Lam, N., Dahari, H., et al. (1998) Hepatitis C Viral Dynamics in Vivo and the Antiviral Efficacy of In-terferon-Alpha Therapy. Science, 282, 103-107.
https://doi.org/10.1126/science.282.5386.103

[2] Bostan, N. and Mahmood, T. (2010) An Overview about Hepatitis C: A Devastating Virus. Critical Reviews in Microbiology, 6, 91-133.
https://doi.org/10.3109/10408410903357455

[3] Akhtar, S. and Carpenter, T. (2013) Stochastic Modelling of Intra-Household Transmission of Hepatitis C Virus: Evidence for Substantial Non-Sexual Infection. Journal of Infection, 2, 179-183.
https://doi.org/10.1016/j.jinf.2012.10.020

[4] Qesmi, R., Wu, J., Wu, J. and Heffernan, J. (2010) Influence of Backward Bifurcation in a Model of Hepatitis B and C Viruses. Mathematical Biosciences, 224, 118-125.
https://doi.org/10.1016/j.mbs.2010.01.002

[5] Jopling, C., Yi, M., Lancaster, A., Lemon, S. and Sarnow, P. (2005) Modulation of Hepatitis C Virus RNA Abundance by a Liver-Specific Micro-RNA. Science, 309, 1577-1581.
https://doi.org/10.1126/science.1113329

[6] Shi, R. and Cui, Y. (2016) Global Analysis of a Mathematical Model for Hepatitis C Virus Transmissions. Virus Research, 217, 8-17.
https://doi.org/10.1016/j.virusres.2016.02.006

[7] Bao, J. and Shao, J. (2018) Asymptotic Behavior of SIRS Models in State-Dependent Random Environments.

[8] Blé, G., Esteva, L. and Peregrino, A. (2018) Global Analysis of a Mathematical Model for Hepatitis C Considering the Host Immune System. Journal of Mathematical Analysis and Applications, 461, 1378-1390.
https://doi.org/10.1016/j.jmaa.2018.01.050

[9] Anderson, W. (1991) Continuous-Time Markov Chains. Springer, Berlin.
https://doi.org/10.1007/978-1-4612-3038-0

[10] Bellet, L. (2006) Ergodic Properties of Markov Process. In: Open Quantum Systems II, Springer, Berlin, 1-39.
https://doi.org/10.1007/3-540-33966-3_1

[11] Khasminskii, R., Zhu, C. and Yin, G. (2007) Stability of Re-gime-Switching Diffusions. Stochastic Processes and Their Applications, 117, 1037-1051.
https://doi.org/10.1016/j.spa.2006.12.001

[12] Mao, X. and Yuan, C. (2006) Stochastic Differential Equations with Markovian Switching. Imperial College Press, London.
https://doi.org/10.1142/p473

[13] Chen, X., Chen, Z., Tran, K. and Yin, G. (2019) Recurrence and Ergodicity for a Class of Regime-Switching Jump Diffusions. Applied Mathematics & Optimization, 80, 415-444.
https://doi.org/10.1007/s00245-017-9470-9

[14] Yin, G. and Zhu, C. (2010) Hybrid Switching Diffusions Properties and Applications. Springer, Berlin.
https://doi.org/10.1007/978-1-4419-1105-6

[15] Chow, Y. (1965) Local Convergence of Martingales and the Law of Large Numbers. The Annals of Mathematical Statistics, 36, 552-558.
https://doi.org/10.1214/aoms/1177700166

[16] Da Prato, G. and Zabczyk, J. (1996) Ergodicity for Infinite Di-mensional Systems. Cambridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511662829

[17] Ikeda, N. and Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam.

Top