A mathematical model for predicting and controlling COVID-19 transmission with impulsive vaccination

This study examines an epidemiological model known as the susceptible-exposed-infected-hospitalized-recovered (SEIHR) model, with and without impulsive vaccination strategies. First, the model was analyzed without impulsive vaccination in the presence of a reinfection effect. Subsequently, it was studied as part of a periodic impulsive vaccination strategy targeting the susceptible population. These vaccination impulses were administered in very brief intervals at specific time instants, with a fixed time gap between each impulse. The two approaches can be modified to respond to different amounts of susceptibility, with control efforts intensifying as susceptibility levels rise. The model's analysis includes crucial aspects such as the non-negativity of solutions, the existence of steady states, and the stability corresponding to the basic reproduction number. We demonstrate that when vaccination measures are taken into account, the basic reproduction number remains as less than one. Therefore, the disease-free equilibrium in the case of vaccination could still be asymptotically stable at the higher disease transmission rate, as compared to the case of no vaccination in which the disease-free equilibrium may no longer be asymptotically stable. Furthermore, we show that when the disease-free equilibrium is stable, the endemic equilibrium cannot be attained, and that when the reproduction number rises above unity, the disease-free equilibrium becomes unstable while the endemic equilibrium becomes stable. We have also derived conditions for the global stability of both equilibriums. To support our theoretical results, we have constructed a time series of numerical simulations and compared them with real-world data from the ongoing SARS-CoV-2 (COVID-19) pandemic.