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Time delayed fractional diabetes mellitus model and consistent numerical algorithm.

Mudassar Rafique1, Muhammad Aziz Ur Rehamn1, Muhammad Rafiq2,3

  • 1Department of Mathematics, University of Management and Technology, Lahore, Pakistan.

Scientific Reports
|October 12, 2024
PubMed
Summary
This summary is machine-generated.

This study transforms the integer-order diabetes mellitus model (DMM) into a time-delayed fractional-order model. The research analyzes the model

Keywords:
Fractional time delayed differential equationGL non-standard finite difference schemesGlobal stability.LaSalle principalVolterra type Lyapunov function

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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.