# Chronic Inflammation in the Epidermis: A Mathematical Model
## 2 Materials and Methods
### M2 Model  
In this section, we consider two different bacterial strains. Figure 2A illustrates the dynamical regulation of bacterial infection and immune responses. There exists competition between harmful and good bacteria for bacterial growth. Bacterial infection induces upregulation of the transcriptional factor within the cell for immune activity and the secretion of proteases for enhanced bacterial invasion. Induced extracellular cytokines from the transcription factor then suppress both of those bacteria. In order to incorporate the interaction network shown in Figure 2B into our model of bacterial competition and inflammation, we began by simplifying this network. As indicated in Section 1, five main players of the bacterial infection network are harmful bacteria, good bacteria, protease, intracellular transcription factors and extracellular cytokines. Let the variables \(B_{1}\), \(B_{2}\), \(P\), \(A_{1}\) and \(A_{E}\) be the densities or concentrations of harmful bacteria, good bacteria, protease, transcription factor and cytokines, respectively. Figure 2B shows a representation of Figure 2A. The kinetic interpretation of the arrows and hammerheads in the given network represents induction (arrow) and inhibition (hammerhead). The scheme includes bacterial growth of both harmful and good bacteria, mutual inhibition between harmful bacteria and good bacteria, secretion of proteases from those bacterial invasions, activation of the intracellular transcription factor from these bacterial infections, activation of extracellular cytokines via the transcription factor, protein degradation of those key molecules and eradication of those bacteria by cytokines. It has been known that: (i) the half-life of proteases (\(P\)) is short, which indicates the large decay rate of \(P\) and that protease reactions occur quickly; and (ii) typical chemical reactions among proteins and genes at a fast time scale lead to the fast internal dynamics. This allows us to use quasi-steady state approximation (QSSA) to simplify the complex models (or interaction networks in Figure 2B). Based on the topological structure and uni-directional activation flows in the immune reaction module (transcription factor activities (\(A_{1}\)), proteolytic activities (\(P\)), cytotoxic levels (\(A_{E}\)); gray box in Figure 2B) and the corresponding QSSA, we merged all complex networks between transcriptional factors (\(A_{1}\)), proteases (\(P\)) and extracellular cytokines (\(A_{E}\)) into one component (gray box in Figure 2B,C), while we kept the harmful (\(B_{1}\)) and bad (\(B_{2}\)) bacterial components in one module (yellow dotted boxes in Figure 2B), respectively. Based on these observations, we write the phenomenological equations for the rate change of those key players (\(B_{1},B_{2}\), \(A_{E}\)) as follows:  
\[\frac{dB_{1}}{dt} =r_{1}B_{1}\left(1-\alpha_{11}B_{1}-\alpha_{12}B_{2}\right)- \gamma A_{E}B_{1}, \tag{3}\] \[\frac{dB_{2}}{dt} =r_{2}B_{2}\left(1-\alpha_{21}B_{1}-\alpha_{22}B_{2}\right)- \gamma A_{E}B_{2},\] (4) \[\frac{dA_{E}}{dt} =\beta_{1}B_{1}+\beta_{2}B_{2}-\delta A_{E}. \tag{5}\]  
where \(r_{1}\) and \(r_{2}\) are the growth rate of harmful and good bacteria, respectively, \(\alpha_{ii}\) (\(i=1,2\)) and \(\alpha_{ij}\) (\(i\
eq j,i,j=1,2\)) represent intra-specific and inter-specific competition coefficients between harmful and good bacteria, respectively, \(\gamma\) is the killing rate of those bacteria by the extracellular cytokines, \(\beta_{1}\) and \(\beta_{2}\) are the activation of immune responses (level of cytokines) from harmful and good bacteria,