Unveiling the spread of epidemics involving partial immunity and reinfection: insights from a time-delayed mathematical model

In recent times, several infectious diseases that breakthrough contagiousness in vaccinated people and reinfections in primarily infected counterparts have become more common. In most cases, reinfection occurs as a result of viral mutations. The mutated pathogens may possess alternate modes of transmission, and even the exposure period may vary in some cases. Hence, to gain a deeper understanding of disease transmission dynamics, it becomes eminent to incorporate a dedicated compartment for reinfection within a mathematical model. To account for these vital factors, this research presents an epidemic model that includes a specific compartment for individuals experiencing reinfection. The model also incorporates two separate time delay parameters that account for the incubation periods for the initial infection and the subsequent reinfection. The study reveals that the system exhibits forward bifurcation with respect to the parameter β _1 in the absence of time delays. However, in the presence of time delay, Hopf bifurcation is identified when the delays exceed corresponding cut-off values. The research further highlights that an extension in the exposure period leads to a rise in the reinfected population, thereby contributing to the persistence of the disease over an extended timeframe. To demonstrate the practical applicability of the proposed model, we have fitted our model with the data of real-time vaccination cases for COVID-19 in India. Notably, the constructed model can be readily adapted for other infectious diseases by selecting appropriate parameter values based on pertinent real-time data. The theoretical results are validated via numerical simulation.