Deterministic and stochastic dynamic analysis of a Parkinson’s disease model

Abstract

Parkinson’s disease (PD) is closely related to high level of reactive oxygen species (ROS) and misfolded $\alpha$-synaptic protein (${\alpha \text{SYN}}^{\ast }$). A deterministic model of ROS and ${\alpha \text{SYN}}^{\ast }$ was proposed by Cloutier et al, who analyzed the effect of different stress signals on a switch from low level to high one for ROS. In this paper, we further investigate the existence and stability of a positive equilibrium of the deterministic model and derive the conditions on which the model experiences saddle–node bifurcation inducing a bistability with low and high levels. Then, a stochastic model of ROS and ${\alpha \text{SYN}}^{\ast }$ is formulated through considering Gaussian white noise into the deterministic one. The existence of global unique positive solution is analyzed and sufficient conditions for the existence of stationary distribution are provided for the stochastic model. Furthermore, noise-induced transition between the bistability is explored through confidence ellipse for the same noise intensity and the average number of alternations between the bistability and the average dominance duration that the model spends on a stable steady state for different noise intensity. Our results reveal that ROS displays bistability with low and high levels under moderate stress. In the presence of noise, the decreasing of stress and the increasing of noise intensity easily induce the transition from high stable steady state to low one to relieve the disease. In addition, smaller stress is an important factor in suppressing the transition from low stable steady state to high one, which also can be prevented by decreasing noise intensity for larger stress. Therefore, disease state can switch to healthy state through regulating noise intensity. Our results may provide a new idea to control noise to alleviate PD through physical therapy.