# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of Two Bacterial Strains Model (M2 Model)
#### 3.2.2 Stability of Equilibria  
Mathematical conditions for local stability of equilibria are derived based on the linearized equations around any of the equilibria \(E_{\circ}=(B_{1}^{\circ},B_{2}^{\circ},I^{\circ})\). The Jacobian matrix for \(E_{\circ}\) is given by:  
\[J(E_{\circ})=\left( \begin{array}{ccc}r_{1}-\frac{2r_{1}B_{1}^{\circ}}{K_{1} }-r_{1}\alpha_{12}B_{2}^{\circ}-\gamma I^{\circ}&-r_{1}\alpha_{12}B_{1}^{\circ }&-\gamma B_{1}^{\circ}\\ -r_{2}\alpha_{21}B_{2}^{\circ}&r_{2}-\frac{2B_{2}^{\circ}r_{2}}{K_{2}}-r_{2} \alpha_{21}B_{1}^{\circ}-\gamma I^{\circ}&-\gamma B_{2}^{\circ}\\ \beta_{1}&\beta_{2}&-\delta\end{array} \right). \tag{28}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & \(w_{1}<w_{2}\) & \(w_{2}<w_{1}\) \\ \hline \(D_{1}>0\) \& \(D_{2}>0\) & \(w_{2}<\delta\) & nonexistence \\ \(D_{1}<0\) \& \(D_{2}>0\) & \(w_{1}<\delta<w_{2}\) & \(w_{2}<\delta<w_{1}\) \\ \(D_{1}>0\) \& \(D_{2}<0\) & nonexistence & nonexistence \\ \(D_{1}<0\) \& \(D_{2}<0\) & nonexistence & \(w_{1}<\delta\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Existence condition of \(E_{+}\) when \(r_{1}>r_{2}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & \(w_{1}<w_{2}\) & \(w_{2}<w_{1}\) \\ \hline \(D_{1}>0\) \& \(D_{2}>0\) & nonexistence & \(w_{1}<\delta\) \\ \(D_{1}<0\) \& \(D_{2}>0\) & nonexistence & nonexistence \\ \(D_{1}>0\) \& \(D_{2}<0\) & \(w_{1}<\delta<w_{2}\) & \(w_{2}<\delta<w_{1}\) \\ \(D_{1}<0\) \& \(D_{2}<0\) & \(w_{2}<\delta\) & nonexistence \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Existence condition of \(E_{+}\) when \(r_{1}<r_{2}\). The Jacobi matrix for \(E_{0}\) is given by:  
\[J(E_{0})=\left( \begin{array}{ccc}r_{1}&0&0\\ 0&r_{2}&0\\ \beta_{1}&\beta_{2}&-\delta\end{array} \right). \tag{29}\]  
Since \(r_{1}>0\) and \(r_{2}>0\), \(E_{0}\) is always unstable. The Jacobi matrix for \(E_{1}\) is given by:  
\[J(E_{1})=\left( \begin{array}{ccc}-\frac{r_{1}r_{2}\delta}{r_{1}\delta+\gamma \beta_{1}K_{1}}&-\frac{r_{1}r_{2}^{2}\alpha_{12}K_{1}\delta}{r_{1}\delta+\gamma \beta_{1}K_{1}}&-\frac{\gamma K_{1}r_{1}\delta}{r_{1}\delta+\gamma\beta_{1}K_{ 1}}\\ 0&\frac{r_{1}r_{2}(1-\alpha_{21}K_{1})\delta+\gamma\beta_{1}K_{1}(r_{2}-r_{1}) }{r_{1}\delta+\gamma\beta_{1}K_{1}}&0\\ \beta_{1}&\beta_{2}&-\delta\end{array} \right). \tag{30}\]  
Characteristic equation \(P_{1}(\lambda)=0\) defined for \(J(E_{1})\) is given by:  
\[P_{1}(\lambda)=\left\{\lambda-\frac{r_{1}r_{2}(1-\alpha_{21}K_{1})\delta+ \gamma\beta_{1}K_{1}(r_{2}-r_{1})}{r_{1}\delta+\gamma\beta_{1}K_{1}}\right\} \left| \begin{array}{cc}\lambda+\frac{r_{1}r_{2}\delta}{r_{1}\delta+\gamma \beta_{1}K_{1}}&\frac{\gamma K_{1}r_{1}\delta}{r_{1}\delta+\gamma\beta_{1}K_{ 1}}\\ -\beta_{1}&\lambda+\delta\end{array} \right|. \tag{31}\]  
Let \(A_{1}(\lambda)\) be defined by:  
\[A_{1}=\left| \begin{array}{ccc}-\frac{r_{1}r_{2}\delta}{r_{1}\delta+\gamma \beta_{1}K_{1}}&-\frac{\gamma K_{1}r_{1}\delta}{r_{1}\delta+\gamma\beta_{1}K_{ 1}}\\ \beta_{1}&-\delta\end{array} \right|. \tag{32}\]  
Note that the trace and determinant of \(A_{1}\) satisfy \(\text{tr}(A_{1})<0\) and \(\text{det}(A_{1})>0\). Hence, \(E_{1}\) is locally asymptotically stable if:  
\[r_{1}r_{2}(1-\alpha_{21}K_{1})\delta+\gamma\beta_{1}K_{1}(r_{2}-r_{1})<0. \tag{33}\]  
Note that (33) is rewritten as:  
\[r_{1}r_{2}D_{1}\delta+\gamma\beta_{1}K_{1}(r_{1}-r_{2})>0. \tag{34}\]  
In other words, (34) is equivalent to one of the following two conditions:  
\[ \begin{array}{cc}(\text{E}_{1}\text{S}_{1})&r_{1}>r_{2}\text{ and }D_{1}>0,\\ (\text{E}_{1}\text{S}_{2})&r_{2}>r_{1}\text{, }D_{1}>0\text{ and }0<w_{1}<\delta. \end{array} \]  
Stability conditions of \(E_{2}\) are derived from the Jacobi matrix for \(E_{2}\):  
\[J(E_{2})=\left(