A Mathematical Model for a Disease Outbreak Considering Waning-Immunity Class with Nonlinear Incidence and Recovery Rates

In this paper, we constructed and analyzed a modified SIR-type model for the epidemic problem which considering waning-immunity class in the population, with the nonlinear incidence and recovery rates. Depending on the basic reproduction ratio, we investigated conditions for both non-endemic and co-existing cases. The RouthHurwitz criteria is used to verify the local stability of equilibria, whereas for the global stability, the suitable Lyapunov function is selected to analyze the behavioral stability for each equilibria. Moreover, we studied the optimal control problem for this case. Numerically, we give some simulations to support our analitycal findings.