Stability and bifurcation analysis of Alzheimer's disease model with diffusion and three delays

A reaction-diffusion Alzheimer's disease model with three delays, which describes the interaction of beta-amyloid deposition, pathologic tau, and neurodegeneration biomarkers, is investigated. The existence of delays promotes the model to display rich dynamics. Specifically, the conditions for stability of equilibrium and periodic oscillation behaviors generated by Hopf bifurcations can be deduced when delay sigma (sigma = sigma(1)+ sigma(2)) or sigma(3) is selected as a bifurcation parameter. In addition, when delay sigma and sigma(3) are selected as bifurcation parameters, the stability switching curves and the stable region are obtained by using an algebraic method, and the conditions for the existence of Hopf bifurcations can also be derived. The effects of time delays, diffusion, and treatment on biomarkers are discussed via numerical simulations. Furthermore, sensitivity analysis at multiple time points is drawn, indicating that different targeted therapies should be taken at different stages of development, which has certain guiding significance for the treatment of Alzheimer's disease.

Automated summary

This study investigates a reaction-diffusion Alzheimer's disease model with three delays, which captures the interaction among beta-amyloid deposition, pathologic tau, and neurodegeneration biomarkers. The presence of delays leads to rich dynamics in the model. The stability of equilibrium and periodic oscillation behaviors can be determined by selecting certain delays as bifurcation parameters. Numerical simulations are conducted to explore the effects of time delays, diffusion, and treatment on biomarkers. Sensitivity analysis at multiple time points suggests the need for different targeted therapies at different stages, providing valuable guidance for Alzheimer's disease treatment.