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

Area of Science:

• Mathematical epidemiology
• Fractional calculus
• Disease modeling

Background:

• Rubella outbreaks present significant global health, social, and economic challenges.
• Effective disease control and understanding are vital for eradication efforts.
• Classical epidemiological models may not fully capture complex disease dynamics, including memory effects.

Purpose of the Study:

• To develop and analyze a nonlinear Rubella model using the Atangana-Baleanu derivative in Caputo framework (ABC) to incorporate memory and hereditary effects.
• To investigate transmission modes, risk factors, and long-term effects of Rubella using a fractional-order approach.
• To formulate and solve a fractional optimal control problem for Rubella management.

Main Methods:

• Development of a fractional-order SEITR model with ABC derivative.
• Analysis of model properties: existence, uniqueness, positivity, boundedness.
• Derivation of the basic reproduction number and stability analysis of equilibria using Lyapunov theory.
• Bifurcation analysis and sensitivity analysis to identify critical thresholds and influential parameters.
• Formulation and solution of a fractional optimal control problem using Pontryagin's Maximum Principle.
• Numerical simulations using the Toufik-Atangana method.

Main Results:

• The fractional model successfully captures memory and hereditary effects in Rubella transmission.
• Analytical and numerical methods confirmed the stability of Rubella-free and endemic states.
• Sensitivity analysis identified key parameters influencing disease spread.
• Time-dependent optimal control strategies (vaccination and treatment) proved more effective and cost-efficient than constant controls.
• Numerical simulations demonstrated significant reductions in infection rates and costs with increased intervention coverage.

Conclusions:

• Fractional-order models, particularly with the ABC derivative, offer a more realistic approach to understanding and managing Rubella.
• Optimal control strategies, especially time-dependent ones, are crucial for cost-effective epidemic control.
• The study highlights the importance of integrating advanced mathematical tools for enhanced public health interventions against Rubella.