Dynamics study of an SEIS epidemic model incorporating nonlinear innate immunity and media awareness with optimal control

In this study, we present and study an Susceptible-Exposed-Infectious-Susceptible (SEIS) epidemic model that considers nonlinear innate immunity and integrates media awareness. We investigated equilibrium points and stability criteria of the model. The basic reproduction number, [Formula: see text], is determined using the next-generation matrix method, indicating that the disease-free equilibrium (DFE) is locally stable for [Formula: see text] and unstable for [Formula: see text]. The global stability of the DFE is established under specific conditions using the LaSalle–Lyapunov theory, whereas the global stability of the endemic equilibrium for [Formula: see text] is determined through a geometric approach. The possible existence of two endemic equilibria when [Formula: see text] leads to a scenario of backward bifurcation at the threshold of [Formula: see text], as confirmed by the center manifold theory, which is notably influenced by media-related parameters. To further address the spread of the disease, we formulated an optimal control problem incorporating a parameter of media intervention [Formula: see text], as a control variable. This problem was solved analytically using Pontryagin’s maximum principle. The influence of media awareness on infectious disease dynamics was illustrated using various numerical simulations to showcase the analytical findings of this study. Finally, the paper concludes with a detailed discussion.