# Chronic Inflammation in the Epidermis: A Mathematical Model
## 2 Materials and Methods
### M1 Model  
In this section, we develop a mathematical model based on the schematic diagram in Figure 1. As indicated in Section 1, the key main players of the bacterial infection network in absence of competition with other bacteria are the following variables:\[B=\text{ density of harmful bacteria},\] \[P=\text{ concentration of protease},\] \[A_{I}=\text{ concentration of the intracellular transcription factor},\] \[A_{E}=\text{ concentration of extracellular cytokines},\]  
Figure 1A illustrates the dynamical regulation of immune activities in response to bacterial infection. Bacteria grow in the system with a carrying capacity and induce the secretion of proteases for enhanced bacterial invasion. These proteolytic activities are suppressed by protease inhibitors under normal conditions, but the perturbed balances between a protease and its inhibitors induce the activation of transcriptional factors, causing skin troubles, such as atopy. The upregulated transcriptional factors induce immune activities (extracellular cytokines), which in turn try to kill bacteria. In order to incorporate the biological interactions shown in Figure 1A into our model of bacteria-immune dynamics, we began by simplifying this network. Figure 1B shows a representation of Figure 1A. The kinetic interpretation of the arrows and hammerheads in the given network represents induction (arrow) and inhibition (hammerhead). We merged all complex networks between proteases and their inhibitors into one component (blue dotted box in Figure 1A), while we kept the components of bacteria, transcriptional factor and external allergic immune responses (cytokines) in one module (red dotted boxes in Figure 1A), respectively. The scheme includes bacterial growth, the secretion of proteases from bacterial invasion and stimulated transcriptional factors, the activation of the intracellular transcription factor from upregulated proteases and secreted cytokines, the activation of extracellular cytokines from the transcription factor, protein degradation of those key molecules and eradication of those bacteria by cytokines. Figure 1: A schematic of the M1 model. (**A**) A schematic of immune responses to bacterial infection; (**B**) the final network model that abstracts the key structure of the interaction network in (A). By merging a multi-species compartment (blue dashed box in (A) including proteases and their inhibitors) into a compartment (‘\(P\)’ in (B)), we get a simpler model in (B). Densities of bacteria, proteases, intracellular transcription factor and extracellular cytokines are represented by ‘\(B\)’, ‘\(P\)’, ‘\(A_{I}\)’ and ‘\(A_{E}\)’, respectively. Activation of transcription factors that are associated with immune responses, such as a member of the interferon regulatory factor (IRF) family, is often mediated by positive feedbacks among these transcription factors [27]. It is also known that activation of protease is mediated by molecular interactions among the members of the kallikrein family [28]. Production of the cytokines is also facilitated by a positive feedback, which is often referred to as a cytokine storm [29]. These activations with positive feedbacks can be modeled by Hill functions. In this work, we consider the following specific type of functional response known as the Hill function:  
\[\sigma_{\circ}(X):=\frac{m_{\circ}X^{n}}{a_{\circ}^{n}+(s_{\circ}X)^{n}}, \tag{1}\]  
where \(\circ\in\{\text{IP},\text{BP},\text{PI},\text{BI},\text{EI},\text{IE}\}\). Assume that the activation of protease is mediated by transcription factor \(A_{I}\) and bacteria \(B\). Based on these observations, we write the phenomenological equations for the rate change of those key players \((B,P,A_{I},A_{E})\) as follows:  
\[ \begin{split}\frac{dB}{dt}=&\lambda+r_{B}B\left(1- \frac{B}{K}\right)-\gamma A_{E}B,\\ \frac{dP}{dt}=&\sigma_{\text{IP}}(A_{I})+\sigma_{ BP}(B)-\delta_{P}P,\\ \frac{dA_{I}}{dt}=&\sigma_{PI}(P)+\sigma_{BI}(B)+ \sigma_{EI}(A_{E})-\delta_{I}A_{I},\\ \frac{dA_{E}}{dt}=&\sigma_{IE}(A_{I})-\delta_{E}A_ {E}.\end{split}