Abstract
A prion differential equation model motivated by Parkinson's disease (PD) is studied. A fractional-order form of this model is proposed. After that, we discretized fractional-order Parkinson's disease model. A sufficient condition for the existence and the uniqueness of a solution to the system is obtained. The stability of the fixed points of the system is achieved by using the Jury test. The impacts of varying the parameters of the system are examined. Under certain conditions, the system undergoes some kinds of bifurcations. We observe that the model loses its stability through double-period bifurcation to chaotic behavior as the growth rate increases. Also, the system stabilizes by increasing the memory parameter, and the contact rate between the two types of prions increases. The system shows rich dynamical behavior for a wide range of the values of the parameters.