Mathematical model of Alzheimer’s disease with prion proteins interactions and treatment

Abstract

The accumulation of β-amyloid (Aβ) is one of the most important pathogenic factors in the occurrence of Alzheimer’s disease (AD). Studies have shown that oligomers are more toxic in the process of Aβ aggregation because oligomers can interact with receptors such as prion proteins (Prpc) and the interaction causes Prpc to be misfolded into pathogenic oligomeric prion proteins (Prpol). In this paper, we propose an AD model including two types of Aβ, the interaction of oligomers with Prpc and anti-Aβ drugs treatment. The existence, uniqueness, and non-negativity of the solutions are analyzed. Furthermore, we prove that the model admits a unique globally asymptotically stable equilibrium, which means the drug cannot cure AD completely. Finally, we present some numerical simulations and investigate the relative contributions to AD with two types of Aβ, Aβ40 and Aβ42. In addition, elastic analysis description Aβ42 paranuclei can be used as a therapeutic target.