Stability and Bifurcation in a Delayed Malaria Model with Threshold Control

In this paper, we develop a delayed malaria model that integrates a discrete time delay and a non-smooth threshold-based control strategy. Using the time delay τ as a bifurcation parameter, we investigate the local stability of the endemic equilibrium through analysis of the characteristic equation. We establish sufficient conditions for the occurrence of Hopf bifurcation, demonstrating how stability switches emerge as τ varies. Furthermore, when the infected population exceeds a critical threshold Im, a sliding mode domain arises. We analyze the dynamics within this sliding region using the Utkin equivalent control method. Numerical simulations are provided to support the theoretical findings, illustrating the complex dynamical behaviors induced by both delay and threshold control.