Transitions between disease-free and endemic states in a controlled epidemic system with non-monotonic incidence framework

Epidemic outbreaks continue to pose serious global health challenges, demanding, in particular, efficient mathematical tools for understanding and controlling disease transmission. This study employs a non-monotonic incidence rate combined with optimal control theory to capture realistic behavioral responses during epidemics. The work integrates parameter estimation, sensitivity analysis, and bifurcation analysis to identify the key factors affecting disease spread and stability transitions. The basic reproduction number is derived, and its sensitivity to key parameters is quantified. Using Pontryagin’s Maximum Principle, optimal strategies are developed to minimize infections and control costs. The model is calibrated with real-world data from selected countries, confirming its relevance to actual epidemics. The analysis reveals that a forward bifurcation governs the transition between disease-free and endemic equilibria, providing crucial insight into stability changes as parameters vary. Overall, the results indicate how optimized control strategies can reduce infection burdens, offering valuable guidance for public health policy and long-term epidemic management.