Global stability in a competitive infection-age structured model

We study a competitive infection-age structured SI model between two diseases. The well-posedness of the system is handled by using integrated semigroups theory, while the existence and the stability of disease-free or endemic equilibria are ensured, depending on the basic reproduction number $R_0^x$ and $R_0^y$ of each strain. We then exhibit Lyapunov functionals to analyse the global stability and we prove that the disease-free equilibrium is globally asymptotically stable whenever $\max\{R_0^x, R_0^y\}\leq 1$. With respect to explicit basin of attraction, the competitive exclusion principle occurs in the case where $R_0^x\neq R_0^y$ and $\max\{R_0^x,R_0^y\}>1$, meaning that the strain with the largest $R_0$ persists and eliminates the other strain. In the limit case $R_0^x=R^0_y>1$, an infinite number of endemic equilibria exists and constitute a globally attractive set.