Analysis of a stochastic Alzheimer’s disease model with β-amyloid oligomer effect: Stationary distribution and extinction

Abstract

$\beta$-amyloid (A$\beta$) oligomers have been increasingly shown to produce the crucial cytotoxicity during the progression of Alzheimer’s disease (AD). In this paper, we develop a stochastic AD model with A$\beta$ oligomer effect, where Black–Karasinski process is introduced to describe the random fluctuations in neurobiological environment. First, the well-posedness and Markov–Feller property of the solution of the model are proved. By Lyapunov functional approach and stochastic stability theory, we establish sufficient conditions for the existence of a stationary distribution. Moreover, the exponential extinction of A$\beta$ oligomers is provided.