Mathematical analysis for stochastic model of Alzheimer's disease

Alzheimer's disease is a worldwide disease of dementia and is characterized by beta-amyloid plaques. Increasing evidences show that there is a positive feedback loop between the level of beta-amyloid and the level of calcium. In this paper, stochastic noises are incorporated into a minimal model of Alzheimer's disease which focuses upon the evolution of beta-amyloid and calcium. Mathematical analysis indicates that solutions of the model without stochastic noises converge either to a unique equilibrium or to bistable equilibria. Analytical conditions for the stochastic P-bifurcation are derived by means of technique of slow-fast dynamical systems. A formula is presented to approximate the mean switching time from a normal state to a pathological state. A disease index is also proposed to predict the risk to transit from a normal state to a disease state. Further numerical simulations reveal how the parameters influence the evolutionary outcomes of beta-amyloid and calcium. These results give new insights on the strategies to slow the development of Alzheimer's disease. (C) 2020 Elsevier B.V. All rights reserved.