Figure 3.

The diagrams display the role of ${\mathcal{R}}_0^{com}$ in appearance/non-appearance of the trivial solution and disease-free equilibrium. (a) By fixing the parameter values Table 3 in model (14), the commensal bacteria in the mucosa appear when ${\mathcal{R}}_0^{com}>1$. Therefore, as long as ${\mathcal{R}}_0^{com}<1$, only the trivial solution for the model exists. Since the trivial solution is always unstable, the extinction of all populations is never achieved. (b) By fixing the parameter values Table 3 in model (14), the commensal bacteria in the lumen appear when ${\mathcal{R}}_0^{com}>1$. Therefore, as long as ${\mathcal{R}}_0^{com}<1$, only the trivial solution for the model exists. Since the trivial solution is always unstable, the extinction of all populations is never achieved.