Global stability analysis of a mathematical model from Alzheimer's disease

Abstract

This study focuses on a mathematical model of Alzheimer's disease involving $β$-amyloid, cellular prion protein and their complex. The global asymptotic stability of the model indicates that the complex continues to induce neuronal damage regardless of the initial states. To investigate the dynamics of this system, we have rigorously proved that when the formation rate of new plaques is zero, the system is unconditional globally asymptotically stable without any limitation proposed in previous work. Numerical simulations further validate the theoretical analysis, regardless of the random initial state, demonstrating that the system consistently converges to a unique positive equilibrium. From a therapeutic perspective, we propose targeted therapeutic strategies and verify their effectiveness through numerical simulations. These results provide a universal theoretical basis for understanding dynamic mechanisms of Alzheimer's disease and offer critical guidance for developing targeted therapeutics.