Analyzing the Effects of Lifestyle Modifications on Type 2 Diabetes in India through Atangana-Baleanu Fractional Differential Equations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-205
Published: Jul 6, 2026
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Masira Khan, Amjad Shaikh, Sachin Gunjal, Kottakkaran Sooppy Nisar, Analyzing the Effects of Lifestyle Modifications on Type 2 Diabetes in India through Atangana-Baleanu Fractional Differential Equations, Int. J. Anal. Appl., 24 (2026), 205.

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Masira Khan, Amjad Shaikh, Sachin Gunjal, Kottakkaran Sooppy Nisar

Abstract

This study introduces an innovative fractional-order mathematical model aimed at examining the dynamics of type 2 diabetes mellitus through the application of the Atangana–Baleanu–Caputo (ABC) fractional derivative. The proposed model enhances the traditional integer-order diabetes framework by integrating memory and genetic factors, providing a more accurate depiction of disease progression and the impact of lifestyle interventions. The existence and uniqueness of the fractional system are confirmed via the fixed point theorem, thereby affirming its mathematical validity. Approximate analytical solutions are derived through the iterative Laplace transform method (ILTM), while numerical simulations are performed utilizing Mathematica software. Alongside numerical simulations, the analytical properties of the model are confirmed by deriving the basic reproduction number, establishing disease-free equilibrium, and ensuring the positivity of solutions, which guarantees both mathematical consistency and epidemiological feasibility. Simulation results indicate that the fractional order parameter \(\mu\) significantly affects the temporal dynamics of all compartments—susceptible, afflicted, treated, lifestyle-modified, and prevented. Reduced \(\mu\) values (enhanced memory effects) postpone illness progression and lifestyle changes, whereas elevated \(\mu\) values (diminished memory effects) hasten transitions but result in less enduring consequences. The model offers quantitative information regarding the efficacy of lifestyle interventions, allowing policymakers to formulate data-driven plans for diabetes prevention and management. The fractional ABC diabetes model provides a comprehensive and adaptable framework for elucidating the influence of memory on chronic disease dynamics and informing evidence-based public health strategies.

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