Computational Framework for Fractional Order Neurological Disorder Model under Interpreting Transmission Patterns

Kottakkaran Sooppy Nisar^1,*, Muhammad Farman^2,3,4, Ali Hasan³, Mohammed Altaf Ahmed⁵, Mohammad Tabish⁶
1 Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam Bin Abdulaziz University, Al Kharj, Saudi Arabia
2 Department of Mathematics, Mathematics Research Center, Near East University, Mersin 10, Turkey
3 Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
4 International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, Pakistan
5 Department of Computer Engineering, College of Computer Engineering & Sciences, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
6 Department of Pharmacology, College of Medicine, Shaqra University, Shaqra, Saudi Arabia
* Corresponding Author: Kottakkaran Sooppy Nisar. Email: email
(This article belongs to the Special Issue: Recent Developments on Computational Biology-II)

Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2026.080973

Received 20 February 2026; Accepted 21 April 2026; Published online 03 June 2026

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Abstract

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 ΠN, infection prevalence γ, and stimulation of microglial cells Θ. Numerical simulations support the long-run neuroinflammatory feedback mechanism, revealing that smaller values of fractional order η<1 reduce disease progression. This is based on the premise that increased memory (η 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.