Fractional-order dynamical modelling of neurophysiological reaction-diffusion model of adiposity-induced cognitive...

Obesity, hyperglycemia, and aging are major risk factors for dementia-related disorders, particularly Alzheimer's disease. We develop a time-fractional reaction-diffusion mathematical framework integrating nine interacting biological compartments governed by Michaelis-Menten kinetics and Hill functions to characterize the nonlinear interplay between metabolic dysfunction and neurodegeneration. Rigorous analysis establishes boundedness, stability, and the existence of a global attractor describing long-term system behavior. To enhance biological realism and predictive capability, we implement fractional physics-informed neural networks that embed fractional partial differential equations within deep learning architectures. This approach captures memory-dependent and spatiotemporal diffusion dynamics characteristic of neurological disorders while enabling estimation of time-varying parameters, fractional orders, and latent states. Global sensitivity analysis using partial-rank correlation coefficients identifies key drivers of tau pathology, insulin dysregulation, and dementia progression. Simulations reveal heterogeneous diffusion patterns linking obesity to accelerated neurodegenerative processes. An optimal control framework incorporating behavioral interventions and personalized anti-amyloid therapies is proposed to delay obesity-induced cognitive decline. The model highlights the critical roles of neuroinflammation and blood-brain barrier permeability in disease acceleration. The proposed FPI-NN framework achieves high predictive accuracy, with $R^2$ values of 0.95, 0.93, 0.92, and 0.92 for ${\text{Y}}_1$ , ${\text{W}}_{\beta }$ , ${\text{W}}_{\tau }$ , and ${\text{V}}_{\text{d}}$ , respectively. The corresponding RMSE values are 0.078, 0.094, 0.098, and 0.099, while the MAE values are 0.065, 0.079, 0.082, and 0.084, respectively. The MAPE values are $6.50\%$ , $7.90\%$ , $8.20\%$ , and $8.40\%$ , respectively. Cross-validation further confirms the stability and robustness of the proposed framework. This work provides a reproducible and interpretable computational platform for modeling fractional-order neurodegenerative dynamics and optimizing therapeutic strategies in obesity-associated dementia.