Monitoring and investigation to control the brain network disease under immunotherapy by using fractional operator.

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.