MODELING AND STABILITY ANALYSIS OF NEUROLOGICAL DISORDER UNDER ALZHEIMER'S DISEASE WITH NOVEL FRACTIONAL TECHNIQUE

In this study, we developed a novel fractional-order mathematical model to understand the dynamics of Alzheimer's disease progression based on different factors by using the Caputo fractional derivative. The framework is applied to simulate key pathological markers, such as amyloid-beta accumulation, tau protein hyperphosphorylation, and neuronal degeneration. The uniqueness and exitance of the solution in this study are mathematically supported by the Banach contraction principle. Stability and controllability analyses are performed, including the implementation of a fractional-order PID controller to manage chaos and ensure the system's equilibrium. Numerical simulations across fractional orders reveal that lower fractional values significantly delay the buildup of Aβ, dampen neurodegeneration, and slow cognitive decline. The simulations reveal the model's capability to capture the complex dynamics of Alzheimer's disease progression, offering insights into its nonlinear characteristics and potential therapeutic interventions. The findings support personalized treatment planning through data-driven computational approaches, particularly in light of newly approved disease-modifying therapies.