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<p>Fractional derivative modeling has become an important tool for studying and forecasting disease transmission dynamics. We propose a new mathematical model for Alzheimer’s disease, a condition in which dying and malfunctioning neurons impair memory. The model has a five-dimensional set of nonlinear fractional differential equations for microglia, amyloid-beta, tau protein, infected neurons, and functioning neurons. To further understand the dynamics of the proposed model, we demonstrated the solutions’ existence, uniqueness, positivity, and feasible domain. We used the next-generation technique to calculate the fundamental reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(({\mathscr {R}}_0)\)</EquationSource> </InlineEquation>, the threshold parameter of Alzheimer’s disease transmission. Two model equilibrium points have been found. The reproductive number parameters are subjected to sensitivity analysis in order to show how <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathscr {R}}_0\)</EquationSource> </InlineEquation> responds to parameter changes. The Ulam-Hyers-Rassias stability requirements have been confirmed. The suggested model is solved using the Newton polynomial interpolation method with the discretization of the Caputo fractional-order operator. Lastly, simulations are made to investigate the potential effects of factors that prevent the incidence of Alzheimer’s disease. The findings show how the proposed method may be able to provide deeper and possibly accurate predictions for the dynamics of Alzheimer’s disease, thus leading to more successful public health campaigns.</p>

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Utilizing fractional-order operator to Alzheimer’s disease dynamics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月7日

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  • Kottakkaran Sooppy Nisar,
  • Muhammad Farman

摘要

Fractional derivative modeling has become an important tool for studying and forecasting disease transmission dynamics. We propose a new mathematical model for Alzheimer’s disease, a condition in which dying and malfunctioning neurons impair memory. The model has a five-dimensional set of nonlinear fractional differential equations for microglia, amyloid-beta, tau protein, infected neurons, and functioning neurons. To further understand the dynamics of the proposed model, we demonstrated the solutions’ existence, uniqueness, positivity, and feasible domain. We used the next-generation technique to calculate the fundamental reproduction number \(({\mathscr {R}}_0)\) , the threshold parameter of Alzheimer’s disease transmission. Two model equilibrium points have been found. The reproductive number parameters are subjected to sensitivity analysis in order to show how \({\mathscr {R}}_0\) responds to parameter changes. The Ulam-Hyers-Rassias stability requirements have been confirmed. The suggested model is solved using the Newton polynomial interpolation method with the discretization of the Caputo fractional-order operator. Lastly, simulations are made to investigate the potential effects of factors that prevent the incidence of Alzheimer’s disease. The findings show how the proposed method may be able to provide deeper and possibly accurate predictions for the dynamics of Alzheimer’s disease, thus leading to more successful public health campaigns.