Analyzing optimal control techniques in a nonlinear fractional Rubella model with the Atangana-Baleanu derivative.

Rubella outbreaks have posed serious health, social, and economic challenges worldwide, straining public health systems and economies. Effective understanding and control of the disease remain crucial to prevent its spread, reduce its impact, and support global eradication efforts. This study presents a nonlinear Rubella model using the Atangana-Baleanu derivative in Caputo framework (ABC) to account for memory and hereditary effects in disease dynamics. We explore various transmission modes, identify key risk factors, and examine the long-term effects associated with the disease through this fractional-order approach. The model extends a classical SEITR framework and introduces a fractional approach to analyze key mathematical properties including existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number is derived, and both Rubella-free and endemic equilibria were determined and their local and global stability was established using Lyapunov theory. The model underwent a bifurcation analysis to understand critical thresholds for disease persistence. Sensitivity analysis identified key parameters that significantly influence disease transmission, guiding effective intervention strategies. Adjusting time-invariant treatment and vaccination efforts is shown to accelerate epidemic control. Further, a fractional optimal control problem is formulated and solved using Pontryagin's Maximum Principle and the ABC framework. Results showed that time-dependent vaccination and treatment significantly reduce infections and associated costs more effectively than constant controls. Numerical simulations are performed using the Toufik-Atangana method, showing that increased treatment and vaccination coverage significantly reduce infection rates and overall cost. The employed numerical method supports our analytical results, preserving positivity, boundedness and stability of obtained solutions, which emphasize the importance of the ABC derivative. The model outcomes provide valuable insights into how fractional-order models and optimal control strategies can enhance epidemic management, especially in designing cost-effective interventions for Rubella. To the best of our knowledge, this is the first study to apply the ABC fractional derivative to the considered Rubella model with optimal control. The novelty lies in integrating ABC fractional calculus with both constant and time-dependent optimal controls, supported by modern analytical and numerical techniques for a more realistic and cost-effective approach to disease management.