Global stability of ${\mathbb{S}\mathbb{E}\mathbb{I}}_A{\mathbb{I}}_S\mathbb{V}$ epidemic models with two discrete time delays and control action

This article investigates two mathematical models describing the transmission dynamics of infectious diseases, with a particular focus on the impact of vaccination. Both models partition the total population into five compartments: susceptible individuals $\mathbb{S}(t)$, exposed individuals $\mathbb{E}(t)$, asymptomatic infected individuals ${\mathbb{I}}_A(t)$, symptomatic infected individuals ${\mathbb{I}}_S(t)$, and vaccinated individuals $\mathbb{V}(t)$. The first model incorporates discrete time delays to explore their effect on the stability of disease-free and endemic equilibrium. The second model introduces a control intervention to assess its influence on disease mitigation. For both frameworks, we establish the non-negativity and boundedness of solutions, ensuring biological feasibility. We then derive the basic reproduction number ${\mathfrak{R}}_0$ to characterize the local and global stability of the equilibrium. Global asymptotic stability is proven by constructing appropriate Lyapunov functions. Finally, numerical simulations are presented to illustrate and support the theoretical results, emphasizing the influence of time delays and vaccination strategies on the long-term behavior of the models.