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For many infectious diseases, including malaria and COVID-19, the host may experience more than one episode of infection, where reinfection occurs due to waning immunity. In this paper, we propose a new age-structured epidemic model to investigate the dynamics of such diseases with multiple infections. The model is based on a system of partial differential equations that describes the interplay between completely susceptible individuals, temporarily immune individuals, and infected individuals at different stages. The model incorporates both time and age-dependent variables and parameters. We derive the basic reproduction number and conduct rigorous analyses on the equilibrium solutions and their stability properties. Specifically, we study the global asymptotic stability of the disease-free equilibrium and obtain the explicit conditions for the occurrence of a backward bifurcation. Our findings could provide useful insights into the effects of disease prevention and intervention strategies such as vaccination campaigns.