Stability and Optimal Control Analysis of an SEIQR Epidemic Model with Saturated Incidence Rate

Abstract

In this article, we proposed a new mathematical model to investigate the dynamics of the infectious disease, control, and general disease transmission. The model exhibits two distinct non-trivial equilibrium states. As a fundamental prerequisite for stability analysis, we first derive the epidemiological threshold parameter R0 through next-generation matrix methodology.
According to our investigation, R0 plays an essential role in describing the model’s dynamics. We demonstrate that in the case when R0 takes values less or greater than unity, the endemic (disease-free) condition is asymptotically stable both locally and globally. To try to stop the general disease from spreading throughout a community, we add control parameters, create a control model, and suggest control techniques. The maximum principle of Pontryagin is used to derive the optimality system. Finally, the numerical simulations are performed using the fourth-order Runge-Kutta technique to validate and confirm our analytical conclusions. Phase portrait analysis further illustrates the convergence of system trajectories toward disease-free or endemic equilibria under different control scenarios, reinforcing the stability criteria derived for R0.