Stability and chaos analysis of neurological disorder of complex network with fractional order comparative study

Alzheimer's disease, a prevalent neurological condition causing dementia, is characterized by hyperphosphorylated tau protein intracellular neurofibrillary tangles and amyloid beta-peptide extracellular plaques. Currently, medication cannot cure, stop, or slow its progression. This paper presents a novel epidemiological model for Alzheimer's disease, considering both integer and fractional order operators to characterize the temporal dynamics of key cell populations. The analysis reveals a single solution to a proposed system with a positively invariant region. The Banach fixed-point theorem and Krasnoselskii type are used to investigate the uniqueness of the model. Local and global stability characteristics are analyzed, and the disease impact on compartments and community-wide transmission rates is evaluated. The study conducts a sensitivity analysis of parameters using chaos analysis. A linear feedback control strategy is employed to manage chaotic situations by directing system dynamics towards equilibrium points. A numerical scheme utilizing the Newton polynomial interpolation method is presented in the study, along with its solutions and MATLAB simulations that demonstrate the behavior of the model. With different results for varying values, the fractal fractional order model performs better in medical interventions than the conventional integer order model, pointing to new possibilities for advancement. The study improves our understanding of the progression and recurrence of neurological diseases, which will benefit in disease spread investigation and the development of control techniques.