# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of Two Bacterial Strains Model (M2 Model)
#### 3.2.4 Effect of Time Delays in Immune Response on the Competition System  
In this section, we introduce time delays in the immune response for the reduced competition system (3)-(5). The governing equations for the simple model with time delays (\(\tau_{1},\tau_{2}\)) are given by:  
\[\frac{d}{dt}B_{1}(t) =r_{1}B_{1}(t)(1-\alpha_{11}B_{1}(t)-\alpha_{12}B_{2}(t))-\gamma A _{E}(t)B_{1}(t), \tag{41}\] \[\frac{d}{dt}B_{2}(t) =r_{2}B_{2}(t)(1-\alpha_{21}B_{1}(t)-\alpha_{22}B_{2}(t))-\gamma A _{E}(t)B_{2}(t),\] (42) \[\frac{d}{dt}A_{E}(t) =\beta_{1}B_{1}(t-\tau_{1})+\beta_{2}B_{2}(t-\tau_{2})-\delta A_{ E}(t), \tag{43}\]  
where \(\tau_{1},\tau_{2}\) are time delays in the immune response for harmful and beneficial bacteria attacks, respectively. Figure 8: Bi-stability dynamics of harmful and beneficial bacteria in response to the immune system in the two-species M2 model (3)–(5). **(A)** Dynamics 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 (\(\mathbb{T}_{B}\)), the initial states in the upper-left region (\(R_{F}\); blue) converge to the harmful bacteria-free equilibrium (\(\mathbb{T}_{F}\)). Blue and red curves indicate the trajectories (\(B_{1}(t),B_{2}(t)\)) 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 asterisk on the boundary (green dotted curve) between \(R_{B}\) and \(R_{F}\). \(A_{E}(0)=0\). * Filled red circle in (**A**,**B**) = stable S.S.: \(E_{1}=(0.9375,0),E_{2}=(0,0.3158)\). (**B**,**C) Trajectories (\(B_{1}(t),A_{E}(t)\)) and time courses (C) of bacterial populations (\(B_{1},B_{2}\)) and cytokine level (\(A_{E}\)) for two very close initial conditions in (A): \(B_{1}(0)=0.15\), \(B_{2}(0)=0.713\) (dotted curves) and \(B_{1}(0)=0.15\), \(B_{2}(0)=0.71\) (solid curves). (**D**) Characterization of the system: the dynamics adapts to the bi-stability system where the dynamical system chooses either harmful or good bacteria persisting tissue based on the initial exposure to those bacterial kinds. All other parameters are fixed as in Table 4 (Type II). In Figure 9, we investigate the effect of small time delays (\(\tau_{1}=\tau_{2}=1.5\)). In the absence of time delays (\(\tau_{1}=\tau_{2}=0\)), the population of bad bacteria (\(B_{1}\)) converges to zero (\(B_{1}^{s,2}=0\); blue solid line in Figure 9A), while the population of good bacteria (\(B_{2}\)) and the immune system (\(A_{E}\)) converge to positive equilibria \(B_{2}^{s,2}\) (blue solid line in Figure 9B) and \(A_{E}^{s,2}\) (blue solid line in Figure 9C), respectively. On the other hand, in the presence of time delays (\(\tau_{1}=\tau_{2}=1.5\)), the population of good bacteria (\(B_{2}\)) converges to zero (\(B_{2}^{s,1}=0\); red dashed line in Figure 9C), while the population of bad bacteria (\(B_{1}\)) and the immune system (\(A_{E}\)) persists with positive equilibria \(B_{1}^{s,1}\) (red dashed line in Figure 9B) and \(A_{E}^{s,1}\)(red dashed line in Figure 9C), respectively. Therefore, an introduction of weak time delays in the system induces a switch from the B2-dominant equilibrium (\(0,B_{2}^{s,2},A_{E}^{s,2}\)) to the B1-dominant equilibrium (\(B_{1}^{s,1},0,A_{E}^{s,1}\)). See Figure 9D,E. The initial condition was \(B_{1}(0)=0.15,\ B_{2}(0)=0.72,\ A_{E}(0)=0.0\). However, for larger time delays, the DDE system completely changes the dynamics. The system induces oscillations in both the population of bad bacteria (\(B_{1}\); Figure 10A) and the levels of immune cytokines (\(A_{E}\); Figure 10C) in the presence of larger time delays (\(\tau_{1}=\tau_{2}=3.5\)).