# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of the M1 Model
#### 3.1.1 Classification of Steady States  
There exist three types of equilibria for System (2). Let \(E_{B}:=(\bar{B},0,0,0)\) denote a steady state representing no protease and immune responses under bacterial persistence, where \(\bar{B}>K\) is a positive root of \(\bar{B}^{2}-K\bar{B}-K\lambda/r=0\). Let \(E^{*}:=(B^{*},P^{*},A_{I}^{*},A_{E}^{*})\) denote an inflammatory state that is maintained by additional external stimuli from bacteria. The components of \(E^{*}\) are determined by the solution of the following system of equations:  
\[ \begin{split}&\lambda+r_{B}B^{*}\left(1-\frac{B^{*}}{K}\right)- \gamma A_{E}^{*}B^{*}=0,\\ &\sigma_{IP}(A_{I}^{*})+\sigma_{BP}(B^{*})-\delta_{P}P^{*}=0,\\ &\sigma_{BI}(B^{*})+\sigma_{PI}(P^{*})+\sigma_{EI}(A_{E}^{*})- \delta_{I}A_{I}^{*}=0,\\ &\sigma_{IE}(A_{I}^{*})-\delta_{E}A_{E}^{*}=0.\end{split}  \tag{6}\]  
By substituting the first and third equations into the second equation of (6), we obtain the following equation with respect to \(A_{I}^{*}\):  
\[\delta_{I}A_{I}^{*}=\sigma_{PI}((\sigma_{IP}(A_{I}^{*})+\sigma_{BP}(B^{*}))/ \delta_{P})+\sigma_{EI}(\sigma_{IE}(A_{I}^{*})/\delta_{E})+\sigma_{BI}(B^{*}). \tag{7}\]  
It follows from the first and fourth equations of (6) that \(B^{*}\) is explicitly written as a positive root of the following quadratic equation with respect to \(B\):  
\[B^{2}-K\left\{1-\frac{\gamma}{r_{B}\delta_{E}}\sigma_{IE}(A_{I}^{*})\right\}B -\frac{K\lambda}{r_{B}}=0. \tag{8}\]  
Note that \(B^{*}>0\). \(A_{I}^{*}\) must satisfy:  
\[\sigma_{IE}(A_{I}^{*})<\frac{r_{B}\delta_{E}}{\gamma}. \tag{9}\]  
Then, there exists a unique positive root of (8), denoted by \(B=B^{*}>0\). Since \(\sigma_{IE}(A_{I})\) is continuous and monotonically increasing with respect to \(A_{I}\), (9) is rewritten as:  
\[A_{I}^{*}<\sigma_{IE}^{-1}\left(\frac{r_{B}\delta_{E}}{\gamma}\right). \tag{10}\]  
Then, (7) is rewritten as:  
\[ \begin{split}\delta_{I}A_{I}^{*}=&\sigma_{PI}(( \sigma_{IP}(A_{I}^{*})+\sigma_{BP}(K-\gamma K\sigma_{IE}(A_{I}^{*})/r_{B} \delta_{E}))/\delta_{P})\\ &+\sigma_{EI}(\sigma_{IE}(A_{I}^{*})/\delta_{E})+\sigma_{BI}(K- \gamma K\sigma_{IE}(A_{I}^{*})/r_{B}\delta_{E}).\end{split}  \tag{11}\]  
The existence of positive equilibrium \(E^{*}\) is determined by the root of (11) with Constraint (10). Let \(E_{L}^{*}\), \(E_{U}^{*}\) and \(E_{H}^{*}\) denote three equilibria of (2) ordered by the value of \(A_{I}\)s: \(A_{I,L}^{*}<A_{I,U}^{*}<~{}A_{I,H}^{*}\). From the biological point of view, \(A_{I,}^{*}\) represents the strength of inflammation triggered by bacterial antigenic stimuli. Figure 3 indicates that there are two possible cases: the existence of a unique equilibrium for weak activation of proteases (\(m_{IP}=8\); Figure 3A) or multiple equilibria for enhanced activation of proteolytic activation (\(m_{IP}=50\); Figure 3B). Here, \(m_{IP}\) is the activation rate of proteases from thetranscription factor in the cell. In the upper panels of Figure 3, the straight solid line (blue) and the dotted curve (green) show the left- and right-hand sides of (11) as a function of (\(A_{l}^{*}\)), respectively. The intersection of those two curves represents the equilibria (\(E_{L}^{*}\), \(E_{U}^{*}\) and \(E_{H}^{*}\)). Stability analysis indicates that: (i) when \(m_{IP}\) is small (\(m_{IP}=8.0\), the upper panel of Figure 3A), \(E_{L}^{*}\) (black filled circle) is stable; (ii) when \(m_{IP}\) is relatively large (\(m_{IP}=50.0\), the upper panel of Figure 3B), two steady states (\(E_{L}^{*}\) and \(E_{U}^{*}\); empty circles) are unstable, but one steady state (\(E_{H}^{*}\); black filled circle) is stable. Figure 3A shows the emergence of the bacterial persistence phenotype in response to the weak activation of proteases (\(m_{IP}=8.0\)). The unique positive stable equilibrium \(E_{L}^{*}\) resides in the region (pink box) where the transcription factor activities are suppressed and bacterial growth is active.