An epidemiological model with waning immunity and reinfection

Abstract

Waning immunity and reinfection are critical features of many infectious diseases, but epidemiological models often fail to capture the intricate interaction between an individual's history of immunity and their current infection status; when they do, the approach is usually overly simplistic. We develop a novel dual-age structured model that simultaneously tracks immunity age (time since the last recovery from infection) and infection age (time since infection) to analyze epidemic dynamics under conditions of waning immunity and reinfection. The model is formulated as a system of age-structured partial differential equations that describe susceptible and infected populations stratified by both immunity and infection ages. We derive basic reproduction numbers associated with the model and numerically solve the system using a second-order Runge-Kutta scheme along the characteristic lines. We further extend the model to explore vaccination interventions, specifically targeting individuals according to their immunity age. Numerical results reveal that higher contact rates produce larger amplitude oscillations with longer interepidemic periods. The relationship between initial infection levels and long-term epidemic behavior is nonmonotonic. Vaccination efficiency depends critically on the viral load profile across immunity and infection age, with more pronounced viral load distributions requiring higher vaccination rates for disease elimination. Most efficient vaccination strategies begin with intermediate immunity ages rather than targeting only fully susceptible individuals. The structured dual-age framework provides a flexible approach to analyzing the dynamics of reinfection and evaluating targeted vaccination strategies based on the history of immunity.