# Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics
## 3 Main Study and Results
### Variation of infection rates for the model (8)  
For the subsequent numerical simulations, the focus was on analyzing how the number of monocytes changes in response to variations in parameters that could feasibly be modified in real-world scenarios. The infection rate, which represents the efficiency with which a virus infects target cells, is influenced by a combination of biological, environmental, and therapeutic factors [28].  
To investigate this, the parameters \(s_{1}\) and \(s_{2}\) were varied, representing the infection rates of macrophages and plasmacytoid dendritic cells (pDCs), respectively. Variances of \(10^{-5}\) and \(10^{-7}\) were selected, as these values are in proximity to the typical range of \(10^{-6}\) under normal conditions.  
In the first graph (Figure 11a), the relationship between interferon concentration and monocyte infection rate is depicted. When monocytes such as macrophages and pDCs become infected, they release interferons as part of the immune response to combat viruses or bacteria. Consequently, the graph representing higher infection rates (represented by the red line) illustrates that interferon concentration exceeds 2500 pg/mL after 30 h. In contrast, the graph for lower infection rates (represented by the blue line) shows a concentration below this threshold at the same time point. These results suggest a direct proportionality between infection rates and interferon concentrations, as a higher number of infected monocytes release greater amounts of interferons.  
Figure 12 focuses on the changes in pDC concentrations under higher infection rates. Small differences are detected when having different infection rates. The orange graph, which is the one with a higher plasmacytoid dendritic cells infection rate has a much more abrupt surge in infected cell concentration compared to the blue graph with the lower infection rate. Both graphs seem to indicate a stabilization around the same value after slightly more than 30 h. However, no noticeable differences are observed in these graphs, which is likely due to the macrophages initiating the immune response as the first line of defense upon encountering the virus [29]. Macrophages are widely distributed across various tissues, including the lungs [30], whereas pDCs are predominantly located in lymphoid organs and lymph nodes [31].  
This distinction is further supported by Figure 11, which displays the concentration of infected macrophages over time. The graphs reveal that with higher infection rates, represented by the orange graph, a slight increase in the number of infected macrophages is evident after 25 h. Additionally, the macrophages start at a value greater than zero because of the population of infected macrophages already in the system, since they are the first immune cells to react to any already existing pathogen. For the first 18 h of the infection, both graphs show a linear rise for both infection rates (the orange and blue lines), implying that macrophage growth follows the same pattern at this early stage regardless of what the infection rate is. This highlights the role of macrophages in stabilizing the virus and initiating the immune attack [32], which is subsequently carried out by other immune cells, including pDCs.