Time delayed fractional diabetes mellitus model and consistent numerical algorithm

Area of Science:

• Mathematical Biology
• Epidemiology
• Fractional Calculus

Background:

• Diabetes Mellitus (DMM) is a significant health concern with complex dynamics.
• Traditional integer-order models may not fully capture the intricate behaviors of DMM.
• Incorporating fractional calculus and time delays offers a more nuanced approach to modeling disease progression.

Purpose of the Study:

• To develop and analyze a time-delayed fractional-order diabetes mellitus model (DMM).
• To investigate the existence, uniqueness, positivity, and boundedness of solutions for the generalized DMM.
• To analyze the stability of equilibrium states and the role of the basic reproduction number (R0).

Main Methods:

• Fractional order Caputo differential operator and delay factor integration.
• Mathematical analysis to prove unique, positive, and bounded solutions.
• Calculation of the basic reproduction number (R0) and stability analysis at equilibrium points.
• Formulation of a hybridized finite difference numerical method.

Main Results:

• The generalized fractional DMM possesses unique, positive, and bounded solutions.
• Identified disease-free and endemic equilibrium states.
• Determined the basic reproduction number (R0) and its impact on disease dynamics and stability.
• Validated the numerical method's physical characteristics and biological behavior through simulations.

Conclusions:

• The time-delayed fractional-order DMM provides a robust framework for understanding diabetes dynamics.
• The basic reproduction number (R0) is a critical indicator for disease control and system stability.
• The developed numerical method accurately simulates the biological behavior of the fractional DMM.