Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment

Infectious diseases continue to be a significant threat human health and civilization, finding effective methods combat them is crucial. In this paper, we investigate the impact of awareness campaigns optimal control techniques on infectious without proper vaccines. Specifically, develop an SIRS-type mathematical model that incorporates through media treatment for disease transmission dynamics control. The displays two equilibria, disease-free equilibrium endemic equilibrium, exhibits Hopf bifurcation when parameter exceeds its critical value, causing switch in stability system. We also propose problem minimizes cost measures while achieving desired level By applying minimum principle problem, obtain analytical numerical results show how infection rate affects system can maintain This study highlights importance controlling demonstrates effectiveness theory with minimal cost.