This study develops a fractional-order model of Alzheimer’s disease using a \((q,\tau )\) -generalized Atangana-Baleanu-Caputo (ABC) operator to capture the spatiotemporal dynamics of amyloid-beta and tau protein spread, coupled with a neuron regeneration mechanism. The fractional parameters \( \alpha \) , \( q \) , and \( \tau \) control memory depth, deformation of the kernel, and temporal scaling, respectively. Numerical simulations demonstrate that: (i) intermediate fractional orders \( 0.6 \le \alpha \le 0.9 \) produce biologically realistic propagation delays, (ii) lower \( q \) values enhance nonlocal interactions and accelerate tau diffusion across the connectome, and (iii) increasing the scaling parameter \( \tau \) slows accumulation, mimicking effective clearance or treatment response. Incorporating a treatment term with drug diffusion and decay reveals that sustained low decay rates ( \( \mu < 0.3 \) ) markedly reduce tau concentrations and protect neuron populations. These findings show that the \((q,\tau )\) -ABC framework not only captures the hereditary and memory effects of Alzheimer’s progression but also provides a flexible platform for simulating therapeutic interventions and predicting disease trajectories using real brain connectome data.
