Finite-time stability and optimal control of a stochastic reaction-diffusion model for Alzheimer's disease with impulse and time-varying delay

Abstract

With increased longevity in a nowadays society, Alzheimer's disease (AD), an incurable neurodegenerative disease, becomes a serious threat to human health. To address the challenges to understanding and predicting AD development and treatment, we focus on the development of a stochastic reaction-diffusion model with time-varying delay and impulsive perturbations, which is driven by Le´vy jump process to model the in vivo progression of AD. Moreover, based on an bounded impulsive interval method, certain sufficient conditions of finite-time stability are provided. Further, from a cost-benefit perspective, an optimal control problem of AD is formulated to minimize the pathogenic proteins and control cost. Several illustrative examples are presented to demonstrate and verify theoretical results of this study.