Optimal Solution of a Fractional HIV/AIDS Epidemic Mathematical Model.

Area of Science:

• Mathematical Biology
• Epidemiology
• Fractional Calculus

Background:

• The spread of HIV/AIDS remains a significant global health challenge.
• Mathematical models are crucial for understanding and predicting epidemic dynamics.
• Fractional calculus offers a more nuanced approach to modeling complex biological systems.

Purpose of the Study:

• To develop and analyze a novel fractional mathematical model for HIV/AIDS transmission.
• To introduce an efficient optimization technique for solving the proposed fractional model.
• To investigate the existence, uniqueness, and convergence of the numerical method.

Main Methods:

• A five-compartment fractional mathematical model using the Caputo derivative was formulated.
• An optimization technique based on generalized shifted Jacobi polynomials (GSJPs) was constructed.
• Coefficients and parameters were determined using matrix operations and Lagrange multipliers.

Main Results:

• The existence, uniqueness, and convergence of the proposed numerical method were theoretically established.
• The generalized shifted Jacobi polynomials provided accurate approximations for the fractional HIV/AIDS model solutions.
• Illustrative examples demonstrated the effectiveness and performance of the developed method.

Conclusions:

• The fractional mathematical model with the Caputo derivative provides a robust framework for studying HIV/AIDS.
• The GSJPs-based optimization technique is efficient and accurate for solving such fractional models.
• This approach can aid in understanding disease dynamics and informing public health strategies.