Neurodegenerative diseases, such as Alzheimer’s and Parkinson’s, pose a significant global health burden, progressively impairing cognitive and motor functions. The complex interplay between neuronal health, immune response, and pathological protein accumulation necessitates advanced mathematical modeling for better understanding and intervention strategies. In order to analyze brain diseases, we have developed a five-compartment nonlinear mathematical model to observe the dynamics and focuses on extracellular \(\alpha\) -synuclein function, functioning neurons, infected neurons, activated microglia density, and \(T\) -cells, which may contribute to neuroinflammation in neurodegenerative contexts. Since our model hasn’t been put forth in the literature before, it is novel. In addition, we modified the recently created model by adding fractional order derivatives to better comprehend the relationship between immune response dynamics and neuronal health in brain disorders. To better comprehend these intricate processes and advance medical therapies, our study combines novel mathematical methods with computer simulations. Stability analysis confirms the existence of a feasible disease-free equilibrium. In contrast, sensitivity analysis highlights the critical influence of parameters such as neuron production \((\Pi_N)\) , infection rate \((\gamma)\) , and microglial activation \((\Theta)\) . Numerical simulations reveal that lower fractional orders \((\nu < 1)\) slow disease progression, indicating the long-term impact of neuroinflammatory feedback mechanisms. To assess the computational efficiency and accuracy of the proposed fractional-order model, we compare numerical solutions obtained using the Lagrange interpolation method and the ODE45 solver. The results demonstrate that the Lagrange interpolation method exhibits superior accuracy and stability in capturing the long-term behavior of neurodegenerative progression, whereas ODE45, a classical numerical approach, struggles with fractional dynamics due to its dependence on integer-order derivatives. The findings of this study provide valuable insights into the progression of neurodegenerative diseases and offer a framework for exploring targeted therapeutic strategies. By refining fractional-order parameters and integrating real-world clinical data, future research can enhance the predictive power of these models, aiding in early diagnosis and optimized treatment strategies for neurodegenerative disorders.
