Optimal Control of an Impulsive VS-EIAR Epidemic Model with Application to COVID-19

Abstract: In this work, we investigate a VS-EIAR epidemiological model that incorporates vaccinated individuals ${V_i : i = 1, \ldots, n}$, where $n \in \mathbb{N}^{*}$. The dynamics of the VS-EIAR model are governed by a system of ordinary differential equations describing the evolution of vaccinated, susceptible, exposed, infected, asymptomatic, and deceased population groups. Our primary objective is to minimize the number of susceptible, exposed, infected, and asymptomatic individuals by administering vaccination doses to susceptible individuals and providing treatment to the infected population. To achieve this, we employ optimal control theory to regulate the epidemic dynamics within an optimal terminal time $\tau^{*}$. Using Pontryagin's Maximum Principle (PMP), we establish the existence of an optimal control pair $(v^{*}(t), u^{*}(t))$. Additionally, we extend the model to an impulsive VS-EIAR framework, with particular emphasis on the impact of immigration and population movement. Finally, we present numerical simulations to validate the theoretical results and demonstrate their practical applicability.