# Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics
## 2 Materials and Models
### Associated Models  
In the following section, we use three different systems of equations in order to generate a better overall picture of how the mechanism of an infection works, being able to go from an overview into a more detailed version of the infection simulation. We start with a system containing three equations, where the only variables involved are the viral load, infected cells, and target cells. Then, for the second system, the cells are divided into dendritic activated and non-activated cells, healthy and infected hepatocytes, and cytotoxic T lymphocytes, providing a better understanding of the cells' interaction. The final system that is analyzed, the most complex one, takes into account the dynamics of the virus in the spleen, liver, and blood, as well as the infected and uninfected macrophages and dendritic cells.  
The first model of the HCV infection was proposed by [PERSON] et al. (2015), where the dynamics of the infection were analyzed through a system of three differential equations representing the target cells, infected cells, and viral load, as seen in Figure 5.  
\[\begin{split}\frac{dy_{1}}{dt}& =s-dy_{1}(t)-(1-\eta)\beta y_{3}(t)y_{1}(t)\\ \frac{dy_{2}}{dt}&=(1-\eta)\beta y_{3}(t)y_{1}(t)- \delta y_{2}(t)\\ \frac{dy_{3}}{dt}&=(1-\varepsilon)py_{2}(t)-cy_{3}( t)\end{split} \tag{6}\]  
This closed system takes into account the production rate of target cells (\(s\)) and their death rate (\(d\)). The cells become infected with the constant infection rate (\(\beta\)) and die with the constant rate (\(\delta\)). Hepatitis C virions are produced by infected cells at an average rate of p virions per cell per day and are cleared with clearance rate constant c. The possible effects of IFN in this model are to reduce either the production of virions from infected cells by a fraction (\(1-\varepsilon\)) or the de novo rate of infection by a fraction (\(1-\eta\)).  
Figure 5: The schematic interactions in the model (6).  
The classical HCV model is as follows [26]:  
\[\begin{split}\frac{dy_{1}}{dt}&{=}ry_{1}(t)\left(1{-} \frac{y_{1}(t){+}y_{2}(t)}{k}\right){-}\beta_{1}y_{1}(t)y_{2}(t)\\ \frac{dy_{2}}{dt}&{=}\beta_{1}y_{1}(t)y_{2}(t){-}d_{1 }y_{1}(t){-}\beta_{2}(t)y_{2}(t)y_{5}(t)\\ \frac{dy_{3}}{dt}&{=}\lambda{-}\delta_{1}y_{3}(t){-} \omega y_{3}(t)y_{2}(t),\\ \frac{dy_{4}}{dt}&{=}\omega y_{3}(t)y_{2}(t){-}\delta _{2}y_{4}(t)\\ \frac{dy_{5}}{dt}&{=}\
u y_{4}(t)y_{5}(t){-}\beta_{3 }y_{2}(t)y_{5}(t){-}\mu y_{5}(t).\end{split} \tag{7}\]  
The aim of this study focuses on replicating the initial response of the immune system with a particular emphasis on the macrophages and the plasmacytoid dendritic cells. This early response is triggered by the antiviral interferon \(\alpha\), whose quantities we want to monitor. The model we based our study on focuses on the compartmental model of the interferon response when it comes to contact with the hepatitis C virus (HCV), considering each of the variables as populations that interact with each other. The variables analyzed are the dynamics of the interferon, \(y_{1}=I(t)\), the dynamics of virus in the spleen \(y_{6}=VS(t)\), blood \(y_{7}=VB(t)\), and liver \(y_{8}=VL(t)\), and the infected/uninfected plasmacytoid dendritic cells and macrophages, as can be seen in Figure 6. This results in a non-linear dynamic system, whose evolution will be observed, while considering the delays with which the virus sets into the infected individual [27].  
\[\begin{split}\frac{dy_{1}}{dt}&{=}r_{1}{\cdot}y_{2} (t{-}\tau_{1}){+}r_{2}{\cdot}y_{3}(t{-}\tau_{2}){-}d_{2}y_{1}\\ \frac{dy_{2}}{dt}&{=}s_{1}{\