Farman Muhammad, Talib Ammara, Jamil Khadija, Nisar Kottakkaran Sooppy, Sambas Aceng, Bayram Mustafa, Hafez Mohamed
Faculty of Arts and Sciences, Department of Mathematics, Near East University, Northern Cyprus, Turkey.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, 22200, Campus Besut, Terengganu, Malaysia.
Sci Rep. 2025 Aug 18;15(1):30161. doi: 10.1038/s41598-025-15307-y.
Parkinson's disease (PD) is one of the well-known neurodegenerative diseases. The main reason is the death of dopaminergic neurons that release dopamine in the brain region known as the Substantia Nigra pars Compacta (SNc). In this study, we developed a mathematical model of Parkinson's disease incorporating a fractal-fractional operator with the Mittag-Leffler kernel to capture the complex, memory-dependent dynamics of the disease. We conduct a qualitative analysis to explore the existence and uniqueness of solutions and examine both disease-free and endemic equilibrium states. Stability conditions are explored using fixed-point theory and Lyapunov functions, while the dynamics are further analyzed through sensitivity analysis to identify the parameters most influential to the basic reproduction number. Additionally, chaos control is investigated using PID feedback strategies, and a Newton polynomial-based numerical method is implemented to simulate the system's behavior. This approach enhances our understanding of Parkinson's disease progression and offers a foundation for developing personalized therapeutic strategies.
帕金森病（PD）是一种广为人知的神经退行性疾病。主要原因是在称为黑质致密部（SNc）的脑区中释放多巴胺的多巴胺能神经元死亡。在本研究中，我们开发了一种帕金森病数学模型，该模型结合了具有米塔格 - 莱夫勒核的分形 - 分数算子，以捕捉该疾病复杂的、依赖记忆的动态变化。我们进行了定性分析，以探索解的存在性和唯一性，并研究无病和地方病平衡状态。使用不动点理论和李雅普诺夫函数探索稳定性条件，同时通过敏感性分析进一步分析动态变化，以确定对基本再生数最有影响的参数。此外，使用PID反馈策略研究混沌控制，并实施基于牛顿多项式的数值方法来模拟系统行为。这种方法增强了我们对帕金森病进展的理解，并为制定个性化治疗策略提供了基础。
