# Chronic Inflammation in the Epidermis: A Mathematical Model
## 3 Results
### Dynamics of Two Bacterial Strains Model (M2 Model)
#### 3.2.6. Therapeutic Approaches  
Results in the previous section indicate that harmful bacteria may not be completely removed by typical immune responses due to recurrence in the presence of time delays, and one has to introduce a drug that enhances the immune system for the eradication of harmful bacteria. We introduce a drug (\(D\)) under the following assumptions: (i) drugs enhance the immune activities of extracellular cytokines by inhibiting signaling networks involving the intracellular transcription factors and proteaseactivation; (ii) the drug is administrated with either a constant rate or periodic injection of drug; (iii) drug compounds have a natural decay. The governing equations then are given by:  
\[\frac{d}{dt}B_{1}(t) =r_{1}B_{1}(t)(1-\alpha_{11}B_{1}(t)-\alpha_{12}B_{2}(t))-\gamma A_ {E}(t)B_{1}(t), \tag{52}\] \[\frac{d}{dt}B_{2}(t) =r_{2}B_{2}(t)(1-\alpha_{21}B_{1}(t)-\alpha_{22}B_{2}(t))-\gamma A _{E}(t)B_{2}(t),\] (53) \[\frac{d}{dt}A_{E}(t) =D(t)+\beta_{1}B_{1}(t-\tau_{1})+\beta_{2}B_{2}(t-\tau_{2})-\delta A _{E}(t),\] (54) \[\frac{d}{dt}D(t) =\lambda_{D}-\delta_{D}D(t). \tag{55}\]  
where \(\tau_{1}\), \(\tau_{2}\) are time delays in the immune response for harmful and beneficial bacteria, respectively, as in the previous section, \(\lambda_{D}\) is the injection rate of drugs and \(\delta_{D}\) is the decay rate of the drug. In Figure 12, we investigate the effect of constant drug injection on the dynamics of the DDE system (52)-(55) in the presence of large time delays (\(\tau_{1}=\tau_{2}=3.5\)). Figure 12A-C shows populations of harmful (\(B_{1}\), red curve) and beneficial (\(B_{2}\), blue curve) bacteria in the upper panels and the levels of cytokines (\(A_{E}\), solid black curve) and drugs (\(D\), dotted green curve) in the lower panels for low (\(\lambda_{D}=0.01\)), intermediate (\(\lambda_{D}=0.03\)) and large (\(\lambda_{D}=0.1\)) injection rates, respectively. A small injection (\(\lambda_{D}=0.01\)) of drugs is not so effective to remove the harmful bacteria and still maintain the oscillatory patterns of both harmful bacteria (\(B_{1}\)) and immune cytokines (\(A_{E}\)). This still leads to the extinction of beneficial bacteria (\(B_{2}(t)\to 0\) as \(t\rightarrow\infty\)). See Figure 12A. For an intermediate level of injection (\(\lambda_{D}=0.03\); Figure 12B), drugs annihilate the oscillations of the harmful bacteria population and immune responses, leading to the infection-persist state (\(B_{2}^{s}=0,\ B_{1}^{s}>0\)). On the other hand, a large amount of drug injection (\(\lambda_{D}=0.1\); Figure 12C) significantly enhances the immune  
Figure 11: Characterization of the simpler delay differential equation (DDE) immune system (45). (**A**–**D**) Time courses of harmful bacteria population (\(B_{1}\)) in the absence (\(\tau=0.0\) in (A)) and presence of different time delays (\(\tau=1.0\) in (B); \(\tau=1.417\) in (C), and \(\tau=1.4181\) in (D)). The stable equilibrium becomes unstable as \(\tau\) passes through the Hopf bifurcation point \(\tau^{*}\simeq 1.418\). activity and eliminates both harmful and beneficial pathogens (\(B_{1}^{s}=0,\ B_{2}^{s}=0\)). Figure 12D-E shows the trajectories in the \(B_{1}\)-\(B_{2}\) (Figure 12D) and \(B_{1}\)-\(A_{E}\) (Figures 12E) planes, respectively, for the corresponding \(\lambda_{D}\)'s (\(\lambda_{D}\) = 0.01 (red), 0.03 (blue) and 0.1 (black)). In real intravenous injection, the drug is administrated in a periodic infusion; we investigate the effect of drugs in a more realistic situation in the clinical setting. For this, we replace Equation (55) with the following:  
\[\frac{d}{dt}D(t)=\sum_{i=1}^{N_{D}}\lambda_{D}I_{[t_{i},t_{i}+t_{d}]}(t)-\delta _{D}D(t). \tag{56}\]  
where \(\lambda_{D}\) is the injection rate, \(\delta_{D}\) is the decay rate of drugs, \(N_{D}\) is the number of drug injections and \(I_{[t_{i},t_{i}+t_{d}]}(\cdot)\) is the characteristic equations that give one over the time interval \([t_{j},t_{i}+t_{d}]\) with duration \(t_{d}\) and zero otherwise.