The effective reproduction number in the Kermack–McKendrick model with age of infection and reinfection

An age-of-infection structured epidemiological model with reinfection is proposed as an extension of the classical Kermack–McKendrick framework. The introduction of the infection-age variable allows individuals to be classified, leading to a more detailed investigation of the transmission process. The term reinfection addresses the real-world occurrence of secondary or recurrent viral infections. In our theory, we apply the contraction mapping principle, the dominated convergence theorem, and the properties of the Volterra integral equations to derive analytical expressions for the number of newly infected individuals denoted by [Formula: see text]. Then, we establish a Volterra integral equation for [Formula: see text] and study its initial conditions for both single and multiple cohorts. Based on this equation, we derive a method for identifying the effective reproduction number, denoted as [Formula: see text]. From a practical perspective, we present two distinct methods and separately apply them to analyze daily new infection cases from the 2003 Severe Acute Respiratory Syndrome (SARS) outbreak in Singapore and the cumulative number of deaths from the COVID-19 epidemic in China. This study effectively bridges theoretical epidemiology and computational modeling, providing a robust framework for analyzing infection dynamics influenced by infection-age-structured transmission and reinfection mechanisms.