Herd immunity is a critical concept in epidemiology, describing a threshold at which a sufficient proportion of a population is immune, either through infection or vaccination, thereby preventing sustained transmission of a pathogen. In the classic susceptible-infectious-recovered (SIR) model, which has been widely used to study infectious disease dynamics, the achievement of herd immunity depends on key parameters, including the transmission rate ( \(\beta \) ) and the recovery rate ( \(\gamma \) ), where \(\gamma \) represents the inverse of the mean infectious period. While the transmission rate has received substantial attention, recent studies have underscored the significant role of \(\gamma \) in determining the timing and sustainability of herd immunity. Additionally, it is becoming increasingly evident that assuming \(\gamma \) as a constant parameter might oversimplify the dynamics, as variations in recovery times can reflect diverse biological, social, and healthcare-related factors. In this paper, we investigate how heterogeneity in the recovery rate affects herd immunity. We show empirically that the mean of the recovery rate is not a reliable metric for determining the achievement of herd immunity. Furthermore, we provide a theoretical result demonstrating that it is instead the mean recovery time, which is the mean of the inverse \(1/\gamma \) of the recovery rate that is critical in deciding whether herd immunity is achievable within the SIR framework. A similar result is proved for the SEIR model. These insights have significant implications for public health interventions and theoretical modeling of epidemic dynamics.
