# Global Dynamics of a Stochastic Viral Infection Model with Laternally Infected Cells
## 4 Extinction  
Now, we establish the conditions under which extinction of the disease.  
**Theorem 3**.: _Let \((x(t),l(t),y(t),v(t),w(t))\) be the solution of (2), \((x(0),l(0),y(0),v(0),w(0))\in\mathbb{R}_{+}^{5}\), if \(\tilde{R}_{0}^{s}:=\frac{3(\beta_{1}+\beta_{2})\lambda}{d_{1}((m+\frac{\sigma_ {2}^{2}}{2})\wedge((a-k)+\frac{\sigma_{3}^{2}}{2})\wedge((p+\frac{\sigma_{2}^{ 2}}{2}))}<1\), which implies that_  
\[\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}x(s)ds\leq\frac{\lambda}{d_{1}},\quad \lim_{t\to\infty}l(t)=0,\quad\lim_{t\to\infty}y(t)=0,\quad\lim_{t\to\infty}v(t )=0.\quad\text{a.s.}\]  
Proof.: By Theorem 1, the solution of (2) is positive,  
\[dx(t)\leq(\lambda-d_{1}x)dt+\sigma_{1}xdW_{1}(t).\]  
Consider the auxiliary stochastic equation of the above equation  
\[dx_{1}(t) =(\lambda-d_{1}x_{1})dt+\sigma_{1}x_{1}dW_{1}(t),\quad x_{1}(0)=x( 0)>0,\] \[\text{we get}\quad\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}x_{1}(s )ds =\frac{\lambda}{d_{1}}\quad\text{a.s.}\]  
Using the comparison theorem of stochastic equation [32], we get \(x(t)\leq x_{1}(t)a.s.\) Then  
\[\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}x(s)ds\leq\lim_{t\to\infty}\frac{1}{t} \int_{0}^{t}x_{1}(s)ds=\frac{\lambda}{d_{1}}.\quad\text{a.s.}\]  
Define \(ln(l(t)+y(t)+v(t))\) and apply the [PERSON]'s formula, assume \(k<a\), we get  
\[d(ln(l(t)+y(t)+v(t)))\] \[=\frac{1}{l(t)+y(t)+v(t)}\Big{\{}\frac{\beta_{1}xv}{1+v}+\frac{ \beta_{2}xy}{1+y}-ml-(a-k)y-\mu v-\xi vw\Big{\}}dt\] \[\quad-\frac{(\sigma_{2}^{2}l^{2}+\sigma_{3}^{2}y^{2}+\sigma_{4}^ {2}v^{2})}{2(l+y+v)^{2}}dt+\frac{\sigma_{2}l}{l+y+v}dW_{2}(t)+\frac{\sigma_{3} y}{l+y+v}dW_{3}(t)+\frac{\sigma_{4}v}{l+y+v}dW_{4}(t),\] \[\leq(\beta_{1}+\beta_{2})xdt-\frac{(m+\frac{\sigma_{2}^{2}}{2})l^ {2}+((a-k)+\frac{\sigma_{3}^{2}}{2})y^{2}+(\mu+\frac{\sigma_{2}^{2}}{2})v^{2} }{(l+y+v)^{2}}dt+\frac{\sigma_{2}l}{l+y+v}dW_{2}(t)+\] \[\quad+\frac{\sigma_{3}y}{l+y+v}dW_{3}(t)+\frac{\sigma_{4}v}{l+y+v }dW_{4}(t),\] \[\leq(\beta_{1}+\beta_{2})xdt-\frac{1}{3}((m+\frac{\sigma_{2}^{2}} {2})\wedge((a-k)+\frac{\sigma_{3}^{2}}{2})\wedge(\mu+\frac{\sigma_{4}^{2}}{2} ))dt+\frac{\sigma_{2}l}{l+y+v}dW_{2}(t)\] \[\quad+\frac{\sigma_{3}y}{l+y+v}dW_{3}(t)+\frac{\sigma_{4}v}{l+y+v }dW_{4}(t).\]Taking integration from \(0\) to \(t\) on both sides and divide by \(t\),  
\[\frac{ln(l(t)+y(t)+v(t))}{t}-\frac{ln(l(0)+y(0)+v(0))}{t}\] \[\leq\frac{\beta_{1}+\beta_{2}}{t}\int_{0}^{t}x(s)ds-\frac{1}{3}((m +\frac{\sigma_{2}^{2}}{2})\wedge((a-k)+\frac{\sigma_{3}^{2}}{2})\] \[\quad\wedge(\mu+\frac{\sigma_{4}^{2}}{2}))+\frac{\sigma_{2}}{t} \int_{0}^{t}\frac{l}{l+y+v}dW_{2}(s)+\frac{\sigma_{3}}{t}\] \[\quad\times\int_{0}^{t}\frac{y}{l+y+v}dW_{3}(s)+\frac{\sigma_{4}} {t}\int_{0}^{t}\frac{v}{l+y+v}dW_{4}(s),\] \[\lim_{t\to\infty}\sup\frac{ln(l(t)+y(t)+v(t))}{t}\] \[\leq(\beta_{1}+\beta_{2})\frac{\lambda}{d_{1}}-\frac{1}{3}((m+ \frac{\sigma_{2}^{2}}{2})\wedge((a-k)+\frac{\sigma_{3}^{2}}{2})\wedge(\mu+ \frac{\sigma_{4}^{2}}{2})),\] \[\leq\frac{1}{3}(R_{0}^{\varsigma}-1)((m+\frac{\sigma_{2}^{2}}{2}) \wedge((a-k)+\frac{\sigma_{3}^{2}}{2})\wedge(\mu+\frac{\sigma_{4}^{2}}{2}))<0 \quad\text{a.s.}\]  
It implies that \(\lim_{t\to\infty}l(t)=0\), \(\lim_{t\to\infty}y(t)=0\), \(\lim_{t\to\infty}v(t)=0\). \(a.s.\)