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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 wheneverdoi:10.1051/mmnp/2020007 fatcat:ls4r6776zncf5cc3n33l3a43eu