# Global Dynamics of a Stochastic Viral Infection Model with Laternally Infected Cells
## 2 Model Formulation
Let \((\Omega,\widetilde{\mathfrak{F}},\{\widetilde{\mathfrak{F}}_{t}\}_{t\geq 0},P)\) be complete probability space with filtration \(\{\widetilde{\mathfrak{F}}_{t}\}_{t\geq 0}\) satisfies the usual conditions and see more details about [PERSON]'s formula (see [31; 32; 33]). **Definition 1**. : _Let \(\mathcal{U}\) denote the family of all continuous non-decreasing functions \(u_{1}:\mathbb{R}_{+}\to\mathbb{R}_{+}\) such that \(u_{1}(0)=0\) and \(u_{1}(r)>0\) if \(r>0\). For \(h>0\), let \(\mathbb{S}_{h}=\{y_{1}\in\mathbb{R}^{n}:|y_{1}|<h\}\). A continuous function \(V(y_{1},t)\) defined on \(\mathbb{S}_{h}\times[t_{0},\infty)\) is said be Lyapunov function if_  
1. \(V(0,t)=0\) _and for some_ \(u_{1}\in\mathcal{U}\)_,_ \(u_{1}(|y_{1}|)\leq V(y_{1},t)\)_, for every_ \((y_{1},t)\in\mathbb{S}_{h}\times[t_{0},\infty)\)_._
2. \(V(y_{1},t)\in C^{2,1}(\mathbb{S}_{h}\times[t_{0},\infty),\mathbb{R}_{+})\) _such that_ \(LV(y_{1},t)\leq 0\)_._  
Let \(y_{1}(t)\) be a regular time-homogeneous Markov process in \(\mathbb{R}^{n}\) defined by SDE  
\[dy_{1}(t)=f(y_{1}(t))dt+g(y_{1}(t))dW(t). \tag{3}\]  
The diffusion matrix of the process \(y_{1}(t)\) is described as \(\mathcal{A}(y_{1})=(b_{ij}(y_{1})),b_{ij}(y_{1})=g^{i}(y_{1})g^{j}(y_{1})\). **Lemma 1**. : _([34]) The Markov process \(y_{1}(t)\) has a unique ergodic stationary distribution \(\pi(\cdot)\) if there exists a bounded open domain \(\mathcal{D}\subset\mathbb{R}^{n}\) with regular boundary \(\Gamma\), having the following properties: (i). the diffusion matrix \(\mathcal{A}(y_{1})\) is strictly positive definite for all \(y_{1}\in\mathcal{D}\). (ii). there exists a non-negative \(C^{2}-\) function \(\mathcal{V}\) such that \(LV\) is negative for any \(\mathbb{R}^{n}\setminus\mathcal{D}\)._  
**Theorem 1**. : _Assume that \(g<\xi<h,\beta_{1}k<\mu+g,(\beta_{2}+1)<\frac{\mu}{k}\), the model (2) has a unique positive solution \((x(t),l(t),y(t),v(t),w(t))\) on \(t\geq 0\) with \((x(0),l(0),y(0),v(0),w(0))\in\mathbb{R}_{+}^{5}\), and the solution will remain in \(\mathbb{R}_{+}^{5}\) with probability \(1\)._  
Proof. : Basically, the coefficients of the system (2) satisfy the local Lipschitz condition, then (2) has a unique local solution \((x(t),l(t),y(t),v(t),w(t))\) on \([0,\tau_{\varepsilon}]\), where \(\tau_{\varepsilon}\) is an exposure time. Then, we prove that \(\tau_{\varepsilon}=+\infty\). Let us follow the similar proof of Theorem 3.1 in [32]. The major step is to describe a non-negative \(C^{2}\) function \(V:\mathbb{R}^{5}_{+}\to\mathbb{R}_{+}\) such that \(\lim_{h_{1}\to\infty,(x,l,y,v,w)\in\mathbb{R}^{5}_{+}}\cup_{h_{1}}\inf V(x,l,y,v,w )=+\infty,\quad\text{and}\quad Lv(x,l,y,v,w)\leq\mathcal{K}\), where \(D_{h_{1}}=(\frac{1}{h_{1}^{*}},h_{1})\times(\frac{1}{h_{1}},h_{1})\times(\frac{ 1}{h_{1}},h_{1})\times(\frac{1}{h_{1}},h_{1})\times(\frac{1}{h_{1}},h_{1})\), and \(\mathcal{K}\) is non-negative constant. Define a function \(V:\mathbb{R}^{5}_{+}\to\mathbb{R}_{+}\) as follows  
\[V(x,l,y,v,w)=(x-1-lnx)+(l-1-lnl)+(y-1-lny)+(v-1-lnv)+(w-1-lnv).\]  
The non-negativity of this function can be seen from \(\vartheta-1-ln\vartheta\geq 0\), \(\forall\vartheta>0\).