# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of Two Bacterial Strains Model (M2 Model)
#### 3.2.3 Dynamics of the Competition Model M2 in Response to the Immune System  
We shall define three adaptive types of the competition system (harmful bacteria-persist (\(\mathbb{T}_{B}\)), harmful bacteria-free (\(\mathbb{T}_{F}\)) and co-existence (\(\mathbb{T}_{c}\))) corresponding to regions in the \(B_{1}\)-\(B_{2}\) plane, including equilibria points (\(E_{1},E_{2},E_{+}\)) discussed in the previous section:  
\[ \begin{split}\mathbb{T}_{B}&:=\{(B_{1},B_{2})\in \mathbb{R}^{2}:B_{1}>0,\ \ B_{2}=0\},(\text{corresponding to }E_{1})\\ \mathbb{T}_{F}&=\{(B_{1},B_{2})\in\mathbb{R}^{2}:B_{1 }=0,\ \ B_{2}>0\},(\text{corresponding to }E_{2})\\ \mathbb{T}_{c}&=\{(B_{1},B_{2})\in\mathbb{R}^{2}:B_{1 }>0,\ \ B_{2}>0\},(\text{corresponding to }E_{+})\end{split}  \tag{40}\]  
The basic parameter set for the M2 model is given in Table 4.  
In Figure 6, we investigate the dynamics of the two-species model (3)-(5) in the presence of the immune response for a base parameter set (Type I). \(r_{1}=1.5\), \(r_{2}=1.0\), \(K_{1}=0.5\), \(K_{2}=2.0\), \(\alpha_{12}=1.0\), \(\beta_{1}=0.1\), \(\beta_{2}=1.0\), \(\gamma=1.0\). Analysis in Section 3.2.2 indicates that: (i) \(E_{2}\) is stable if \(\delta>0.66666\), whereas \(E_{1}\) is stable if \(\delta<0.03333\); and (ii) \(E_{+}\) is expected to be stable if \(0.03333<\delta<\ 0.66666\). Figure 6A-C shows the trajectories \((B_{1}(t),B_{2}(t))\) of harmful and good bacteria populations for various decay rates of cytokines (\(\delta=0.033\) (Figure 6A), 0.35 (Figure 6B) and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Type I** & **Type II** \\ \hline \multicolumn{4}{c}{Inter- and intra-competition} \\ \hline \(a_{11}\) & Inter-specific competition coefficient & \(1/a_{11}=K_{1}=0.5\) & \(K_{1}=2.0\) \\ \(a_{22}\) & Inter-specific competition coefficient & \(1/a_{22}=K_{2}=2.0\) & \(K_{2}=1.5\) \\ \(a_{12}\) & Intra-specific competition coefficient & \(1.0\) & \(1.0\) \\ \(a_{21}\) & Intra-specific competition coefficient & \(1.0\) & \(1.0\) \\ \hline \multicolumn{4}{c}{Activation/production rates} \\ \hline \(r_{1}\) & Growth rate of harmful bacteria & \(1.5\) & \(1.5\) \\ \(r_{2}\) & Growth rate of good bacteria & \(1.0\) & \(1.0\) \\ \(\beta_{1}\) & activation of cytokines by harmful bacteria & \(0.1\) & \(1.0\) \\ \(\beta_{2}\) & activation of cytokines by good bacteria & \(1.0\) & \(0.1\) \\ \hline \multicolumn{4}{c}{Inhibition/decay Rates} \\ \hline \(\gamma\) & Per capita elimination rate of bacteria & \(1.0\) & \(1.0\) \\ \(\delta\) & Decay rate of cytokines & \(0.35\) & \(0.25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters used in the M2 model. 0.7 (Figure 6C)) with four initial conditions: \(B_{1}(0)=0.07\), \(B_{2}(0)=0.45\) (yellow curve); \(B_{1}(0)=0.38\), \(B_{2}(0)=0.58\) (green curve); \(B_{1}(0)=0.05\), \(B_{2}(0)=0.1\) (blue curve); \(B_{1}(0)=0.35\), \(B_{2}(0)=0.02\) (purple curve). The initial condition of the immune response was set to be zero (\(A_{E}(0)=0\)) for all cases. Figure 6E-F shows the trajectories of \((B_{1}(t),A_{E}(t))\) corresponding to Figure 6A-C, respectively. When the decay rate of cytokines is small (\(\delta=0.033\)), the system converges to \(E_{1}\) equilibrium where good bacteria are cleared out and harmful bacteria survive in a battle via the immune system (Figure 6A). Initial strong immune responses (black arrow in Figure 6E) due to the weak clearance rate (\(\delta\ll 1\)) significantly eliminate both harmful and good bacterial populations (black arrowhead in Figure 6A). While the good bacteria are totally eradicated by this strong immune response due to the relatively low growth rate, the harmful bacteria survive due to the higher growth rate and winning the competition battle with the good ones (blue arrow in Figure 6A). The system adapts a harmful bacteria-free equilibrium when \(\delta\) is large (\(\delta=0.7\) in Figure 6C). The strong clearance of immune activities in the system leads to relatively weak immune responses (black arrow in Figure 6G).