# Global Dynamics of a Stochastic Viral Infection Model with Laternally Infected Cells
## 2 Model Formulation  
In the process of viral infection, the immune system plays a critical role. Viral dynamics can be modeled properly to provide insights into understanding the disease and the clinical treatments used to treat it. In adaptive immune responses, lymphocytes are responsible for specificity and memory. The two main types of lymphocytes are B cells and T cells. The function of T cells is to recognize and kill infected cells, while the function of B cells is to produce antibodies to neutralize the viruses. Researchers have studied the effects of immune responses such as CTL responses and antibody responses [23; 24; 25; 26; 27]. Some other researchers have also taken into account the effect of CTL responses and intracellular delays [28; 29; 5]. The mathematical model that describes the effect of humoral immune response on virus dynamics is presented in [27]  
\[ \begin{split}\dot{x}(t)&=\lambda-d_{1}x(t)-\beta_{1 1}x(t)v(t)-\beta_{22}x(t)y(t),\\ \dot{y}(t)&=\beta_{1}x(t)v(t)+\beta_{2}x(t)y(t)-ay( t),\\ \dot{v}(t)&=ky(t)-\mu v(t)-\dot{\xi}v(t)w(t),\\ \dot{w}(t)&=gv(t)w(t)-hw(t).\end{split}  \tag{1}\]  
This model consists of uninfected target cells \(x(t)\), actively infected cells \(y(t)\), free viruses \(v(t)\), and antibodies/B cells \(w(t)\). The uninfected cells \(x(t)\) are produced at a rate constant \(\lambda\) and death rate \(d_{1}\). \(\beta_{11}\) is the infection rate by free virus and uninfected cells being converted to productively infected ones at a rate \(\beta_{22}\) per both cells. \(a,\mu,h\) are the death rate of infected cells, free virus particles, and antibodies/B cells, respectively. Free virus particle are produced from productively infected cells at the rate \(k\) and \(\xi\) is the rate of neutralization by antibodies. \(g\) is the rate of antibodies activated against the virus. Herein, we upgrade model (1) to include the latent infection component. We assume that the uninfected cell \(x(t)\) gets infected by a free virus \(v(t)\) or by direct contact with an infected cell \(y(t)\) at the rate \(\dfrac{\beta_{1}x(t)v(t)}{1+v(t)}+\dfrac{\beta_{2}x(t)y(t)}{1+y(t)}\) with a Holling type II functional response. \(\beta_{1}>0\) and \(\beta_{2}>0\) represent the virus to cell infection rate and cell to cell transmission rate, respectively [30]. We also assume that \((1-\varphi)\) and \(\varphi\in(0,1)\) are the proportions of infection that lead to latency and productivity, respectively. Additionally, in order to reflect a more realistic situation of disease development, we incorporated the effect of randomization within the host by introducing nonlinear perturbations on the natural death rate with white noise into each equation. Therefore, the extended model takes the following form:  
\[ \begin{split} dx(t)&=(\lambda-d_{1}x(t)-\dfrac{ \beta_{1}x(t)v(t)}{1+v(t)}-\dfrac{\beta_{2}x(t)y(t)}{1+y(t)})dt+\sigma_{1}x(t) dW_{1}(t),\\ dl(t)&=((1-\varphi)\Big{(}\dfrac{\beta_{1}x(t)v(t) }{1+v(t)}+\dfrac{\beta_{2}x(t)y(t)}{1+y(t)}\Big{)}-(m+\gamma)l(t))dt+\sigma_{2 }l(t)dW_{2}(t),\\ dy(t)&=(\varphi\Big{(}\dfrac{\beta_{1}x(t)v(t)}{1+v (t)}+\dfrac{\beta_{2}x(t)y(t)}{1+y(t)}\Big{)}+\gamma l(t)-ay(t))dt+\sigma_{3}y (t)dW_{3}(t),\\ dv(t)&=(ky(t)-\mu v(t)-\xi v(t)w(t))dt+\sigma_{4}v(t )dW_{4}(t),\\ dw(t)&=(gv(t)w(t)-hv(t))dt+\sigma_{5}w(t)dW_{5}(t), \end{split}  \tag{2}\]  
with initial values \(x(0)>0,l(0)>0,y(0)>0,v(0)>0,w(0)>0\). \(l(t)\) denotes the concentrations of infected cells in latent stage at \(t\). \(m\) be the death rate of \(l(t)\) and latent infection become productively infected cells at the rate \(\gamma\). \(\sigma_{2}^{2},\sigma_{2}^{2},\sigma_{3}^{2},\sigma_{4}^{2},\sigma_{5}^{2}\) are intensities of the environmental white noises. \(W_{1}(t),W_{2}(t),W_{3}(t),W_{4}(t),W_{5}(t)\) are mutually independent standard Brownian motions with \(W_{i}(0)=0\) (\(i=1,2,3,4,5\)). We define the basic concepts of probability theory and SDEs.