# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of Two Bacterial Strains Model (M2 Model)
#### 3.2.1 Existence of Equilibria  
By replacing the variable \(A_{E}\) with \(I\) for notational purposes, the M2 model system is given by:  
\[\begin{split}\frac{dB_{1}}{dt}&=r_{1}B_{1}\left(1- \frac{B_{1}}{K_{1}}-\alpha_{12}B_{2}\right)-\gamma B_{1}I,\\ \frac{dB_{2}}{dt}&=r_{2}B_{2}\left(1-\frac{B_{2}}{K _{2}}-\alpha_{21}B_{1}\right)-\gamma B_{2}I,\\ \frac{d}{dt}I(t)&=\beta_{1}B_{1}+\beta_{2}B_{2}- \delta I.\end{split} \tag{13}\]  
Figure 5: Dynamics of the M1 model (2). (**A–C**) Dynamics of the M1 system in the \(B\)-\(A_{E}\) phase plane to a low (\(m_{IP}=30\); (**A**)), intermediate (\(m_{IP}=50\); (**B**)) and high (\(m_{IP}=80\); (**C**)) activation rate (\(m_{IP}\)) of protease by the transcription factor. * Filled red circles in (**A–C**) = stable steady state (S.S.), empty black circle in (**B**) = unstable S.S. (**D**) A schematic of two adaptive types of bacterial infection systems (bacteria persist (\(\mathbb{T}_{B}\)) and immune boosting (\(\mathbb{T}_{I}\)) systems): \(\mathbb{T}_{B}=\{(B,A_{E})\in\mathbb{R}^{2}:B>th_{B},\ \ 0\leq A_{E}<th_{AE}\}\), \(\mathbb{T}_{I}=\{(B,A_{E})\in\mathbb{R}^{2}:0\leq B<th_{B},\ \ A_{E}>th_{AE}\}\). All other parameters are fixed as in Table 1. (**E–G**) Time courses of the main variables (\(B,P,A_{I},A_{E}\)) for various activation rates (\(m_{IP}=30,50,80\)) with the initial condition: \(B(0)=1.7,P(0)=0.1,A_{I}(0)=0.1,A_{E}(0)=0.1\). All other parameters are fixed as in Table 1.  
and:  
\[r_{1}r_{2}(K_{2}\alpha_{12}-1)\delta-\gamma\beta_{2}K_{2}(r_{1}-r_{2})>0\text{ and} \tag{23}\]  
\[r_{1}r_{2}(K_{1}\alpha_{21}-1)\delta-\gamma\beta_{1}K_{1}(r_{2}-r_{1})>0,\]  
respectively. Define \(D_{1}\), \(D_{2}\), \(w_{1}\) and \(w_{2}\) by:  
\[D_{1}:=K_{1}\alpha_{21}-1,\ \ D_{2}:=K_{2}\alpha_{12}-1. \tag{24}\]  
\[w_{1}=\frac{\gamma\beta_{1}K_{1}(r_{2}-r_{1})}{r_{1}r_{2}D_{1}},\ \ w_{2}:=\frac{\gamma\beta_{2}K_{2}(r_{1}-r_{2})}{r_{1}r_{2}D_{2}}. \tag{25}\]  
Finally, we consider the conditions for the existence of \(E_{+}\). It follows from (22) and (23) that \(E_{+}\in\mathbb{R}_{+}^{3}\) if and only if:  
\[r_{1}r_{2}D_{1}\delta+\gamma\beta_{1}K_{1}(r_{1}-r_{2})<0\text{ and }r_{1}r_{2}D_{2} \delta+\gamma\beta_{2}K_{2}(r_{2}-r_{1})<0 \tag{26}\]  
or:  
\[r_{1}r_{2}D_{1}\delta+\gamma\beta_{1}K_{1}(r_{1}-r_{2})>0\text{ and }r_{1}r_{2}D_{2} \delta+\gamma\beta_{2}K_{2}(r_{2}-r_{1})>0. \tag{27}\]  
Note that the stability conditions of \(E_{1}\) and \(E_{2}\) are given by (34) and (37), respectively (see the next subsection for details). Conditions (34) and (37) are mutually exclusive with (26), but are identical to (27). In other words, coexistent equilibrium \(E_{+}\) exists only if both \(E_{1}\) and \(E_{2}\) are unstable or locally stable. In the later case, the system would be expected to exhibit bistability between \(E_{1}\) and \(E_{2}\).  
In summary, the existence conditions of internal equilibrium \(E_{+}\) are classified in Tables 2 and 3 according to the sign of \(r_{1}-r_{2}\) and \(w_{1}-w_{2}\).  
In the next section, we check the stability of the equilibria of the M2 model (13).