# 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
In Figure 8, we investigate the dynamics of the bi-stable competition system with the parameter set (Type II): \(r_{1}=1.5,\ r_{2}=1.0,\ K_{1}=2.0,\ K_{2}=1.5,\ \alpha_{12}=1.0,\ \ \alpha_{21}=1.0,\ \beta_{1}=1.0,\ \beta_{2}=0.1,\ \gamma=1.0\). Analysis in Section 3.2.2 indicates that \(E_{1}\) is stable if \(\delta>0.1\), whereas \(E_{2}\) is stable if \(\delta>0\). In comparison to the previous case in Figure 6, the system does not present the co-existence region (\(\mathbb{T}_{c}\)). Figure 8A shows the regions of harmful bacteria-persist (\(\mathbb{T}_{B}\)) and bacteria-free (\(\mathbb{T}_{F}\)) in the \(B_{1}-B_{2}\) phase-plane. While initial states (\(B_{1},B_{2}\)) in the region \(R_{B}\) (red) converge to the harmful bacteria-persist equilibrium (\(\mathbb{T}_{B}\), red circle), the initial states in the upper-left region (\(R_{F}\); blue) converge to the harmful bacteria-free equilibrium (\(\mathbb{T}_{F}\); red circle). For example, curves indicate the trajectories  
Figure 6: Co-existence and dynamics of harmful and good bacteria in response to various cytokine clearance levels (\(\delta\)) in the competition M2 model (3)–(5). **(A–C)** Trajectories (\(B_{1}(t),B_{2}(t)\)) of bacterial populations for various decay rates of cytokines (\(\delta=0.033\) **(A)**, 0.35 **(B)**, 0.7 **(C)**) with four initial conditions; **(D)** characterization of the dynamical system in the \(B_{1}\)-\(B_{2}\) plane. There exist three phenotypic regions: harmful bacteria-persist (\(\mathbb{T}_{B}\), lower pink box near \(x\)-axis), harmful bacteria-free (\(\mathbb{T}_{F}\), left blue box near \(y\)-axis) and co-existence (\(\mathbb{T}_{c}\), gray box in the center) regions. As \(\delta\) is increased (\(\delta=0.0033\to 0.35\to 0.7\)), the system undergoes the transition from \(\mathbb{T}_{B}\) to \(\mathbb{T}_{c}\) and then to \(\mathbb{T}_{F}\). **(E–G)** Trajectories of (\(B_{1}(t),A_{E}(t)\)) corresponding to (A–C), respectively. All other parameters are fixed as in Table 4 (Type I). \((B_{1}(t),B_{2}(t))\) for two very close initial conditions: \(B_{1}(0)=0.15,\ B_{2}(0)=0.713\) (blue curve) and \(B_{1}(0)=0.15,\ B_{2}(0)=0.71\) (red curve) near the boundary (green dotted curve) between \(R_{B}\) and \(R_{F}\). The initial condition of cytokines was set to be zero (\(A_{E}(0)=0\)). \(E_{1}=(0.9375,0),E_{2}=(0,0.3158)\). Figure 8B,C shows the trajectories in the \(B_{1}-A_{E}\) plane (Figure 8B) and time courses (Figure 8C) of bacterial populations \((B_{1},B_{2})\) and cytokine level \((A_{E})\) for two very close initial conditions in Figure 8A: \(B_{1}(0)=0.15,\ B_{2}(0)=0.713\) (dotted curves) and \(B_{1}(0)=0.15,\ B_{2}(0)=0.71\) (solid curves). Figure 8D shows the bi-stable nature of the dynamical system where one kind of the harmful or beneficial bacteria dies out and the other kind survives depending on the initial state \(B_{1}(0),\ B_{2}(0)\). Therefore, the initial status of exposure to both harmful and beneficial bacteria determines the dysbiosis or healthy tissue. In the next section, we investigate the effect of time delays in immune responses on the dynamics of the system. Figure 7: **(A)** Steady state solutions of bad bacteria (\(B_{1}^{*}\)) as a function of \(\delta\) corresponding to \(E_{1}\) (red), \(E_{2}\) (blue) and \(E_{+}\) (black) equilibria, respectively. Solid curve = stable, dotted curve = unstable. Green arrows indicate the points \(\delta=0.0033,0.35,0.7\) corresponding to Figure 6A–C, respectively. **(B)** Trajectories of \(B_{1}(t)\) in a \(B_{1}-\delta\) plane when \(\delta\) is a monotonic increasing function of time, satisfying \(\frac{d\delta}{dt}=0.00002\) with the initial condition \(B_{1}(0)=0.12,B_{2}(0)=0.01,A_{E}(0)=0,\delta(0)=0.01\) near the stable equilibrium point \(E_{1}=E_{1}(\delta=0.01)\). As \(\delta\) is increased, the system sequentially follows the stable branches in (A), leading to the transition from stable \(E_{1}\) branch (red solid curve in (A)) to stable \(E_{+}\) branch (black solid curve in (A)) and then to stable \(E_{2}\) branch (blue solid curve in (A)).