The Effective Reproduction Number in the Kermack-McKendrick model with age of infection and reinfection

Abstract: This study introduces a novel epidemiological model that expands upon the Kermack-McKendrick model by incorporating the age of infection and reinfection. By including infection age, we can classify participants, which enables a more targeted analysis within the modeling framework. The reinfection term addresses the real-world occurrences of secondary or recurrent viral infections. In the theoretical part, we apply the contraction mapping principle, the dominated convergence theorem, and the properties of Volterra integral equations to derive analytical expressions for the number of newly infected individuals denoted by $N(t)$. Then, we establish a Volterra integral equation for $N(t)$ and study its initial conditions for both a single cohort and multiple cohorts. From this equation, we derive a method for identifying the effective reproduction number, denoted as $\mathcal{R}(t)$. In the practical aspect, we present two distinct methods and separately apply them to analyze the daily new infection cases from the 2003 SARS outbreak in Singapore and the cumulative number of deaths from the COVID-19 epidemic in China. This work effectively bridges theoretical epidemiology and computational modeling, providing a robust framework for analyzing infection dynamics influenced by infection-age-structured transmission and reinfection mechanisms.