A global health concern, neurodegenerative disorders like Parkinson's and Alzheimer's impact both mental and physical functioning. The complex interplay among immunological response, protein accumulation, and brain health necessitates sophisticated mathematical modeling. This study introduces a fractional-order mathematical model using the Mittag-Leffler derivative to describe the dynamics of neurodegeneration, incorporating key biological factors such as functioning and infected neurons, extracellular alpha-synuclein, microglia, and T-cells. A fundamental assumption of the model is that neuronal deterioration is influenced by memory effects, where past states impact current disease progression, making fractional-order calculus more suitable than traditional integer-order models. The model accounts for the secretion and clearance of alpha-synuclein, the activation of immune responses, and the role of microglia in mitigating or exacerbating neuronal damage. Sensitivity analysis emphasizes the crucial role of factors like neuronal cells production Pi(N), infection prevalence gamma, and stimulation of microglial cells Theta. Numerical simulations support the long-run neuroinflammatory feedback mechanism, revealing that smaller values of fractional order eta < 1 reduce disease progression. This is based on the premise that increased memory (eta values less than one) leads to slower transmission of pathological protein aggregation. The study demonstrates that building a surrogate machine learning model of the NARX-BRBNN type, calibrated using numerical solver output, not only decreases computing complexity but also accurately replicates the dynamics of the fractional equation. This comparison underscores the necessity of employing fractional-order numerical schemes for accurately modeling complex neurobiological systems. The study proposes focused treatment approaches and provides insightful information on the course of neurodegenerative diseases.
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