# Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics
## 3 Main Study and Results
### Analysis and Graph Interpretations
#### Hepatitis infection, model (8)  
In the first graph depicted below, Figure 10(a), an increase in the concentration of type I interferons is observed 18 h after infection with the virus in mice. In humans, the process may be prolonged due to the substantial increase in body weight and tissue surface area. This results in a higher time of lymph movement through the whole body, requiring a longer period of time to gather the virus and transport it to the lymph node where the lymphocytes eliminate it. This graph underlines the initiation of the immune response, particularly how long it takes for the interferons to be released and subsequently stimulate the B-cells and T-cells, as well as enhance the superior quality of macrophages and plasmacytoid dendritic cells.  
Figure 10: Evolution of functions \(y_{4}\), with different initial values of \(y_{2}\) in model (7).  
The following graph, Figure (b)b, illustrates the secretion of macrophages during the initial stage of infection, with a subsequent decrease in production due to their prompt immunological response. Macrophages play a crucial role in controlling the spread of the virus. They accomplish this by engulfing and breaking down both viruses and bacteria through a process called phagocytosis. Additionally, this type of cell emits anti-inflammatory cytokines later in the immune response to help reduce inflammation.  
A linear secretion pattern of plasmacytoid dendritic cells is illustrated in the following graph, Figure 12, with a significant increase in quantity during the initial 18 h. The quantity of infected pDCs stabilizes thereafter, which can be correlated to their primary function as'sensors' for viruses and bacteria. As the viral load decreases, the infected pDCs remain constant. It is anticipated that their secretion will then decrease as the infection is combated until complete elimination.  
Upon analyzing the graphs collectively, it can be concluded that infected macrophages and pDCs, 18 h after first exposure to the virus, release interferons that augment their ability to eradicate the virus. Additionally, these interferons activate crucial lymphocytes (B- and T-cells) that effectively eliminate the parasites. These graphs serve as mathematical evidence demonstrating that during the period of infection of pDCs and macrophages with the virus, they secrete the necessary interferons. Both cell types exhibit a linear increase in the first 18 h, while the concentration of interferons remains null. The active immune response, characterized by an increase in the IFN level, occurs only after this incubation and 'virus reconnaissance' period. Another key point is that plasmacytoid dendritic cells gain significance as the infection advances, because they persi  
Figure 11: Evolution of Interferon concentration and Infected macrophages over time, model (8).  
Figure 12: Infected plasmacytoid dendritic cells over time with time-varying conditions.  
T-cells in the lymph nodes. Meanwhile, macrophages play a crucial role in containing the virus and initiating the active immune response through processes such as phagocytosis and the release of other cytokines, particularly anti-inflammatory proteins that restrict the virus spread.