Epidemiological model of neurodegenerative disease using Homotopy perturbation method and Tamimi-Ansari method

Abstract

In this study, a mathematical model describing the progression of neurodegenerative diseases is analyzed to understand the complex interactions among functional neurons, infected neurons, extracellular $\alpha$-synuclein, microglia, and T-cells. The proposed Neurodegenerative Disease Model is solved using two powerful semi-analytical approaches-the Homotopy Perturbation Method (HPM) and the Tamimi-Ansari Method (TAM). A comparative error analysis with the numerical \texttt{ODE45} solver demonstrates that both HPM and TAM provide accurate and reliable approximations for the system’s dynamics. Parametric investigations are performed to explore how variations in key biological parameters influence neuronal infection, immune activation, and protein aggregation processes. The results reveal that neuronal degeneration is highly sensitive to the infection and clearance dynamics of $\alpha$-synuclein and immune cell interactions. These findings highlight potential biological mechanisms that can serve as therapeutic targets for mitigating neurodegenerative diseases such as Parkinson’s and Alzheimer’s.