Analysis of a Novel Time‐Continuous Dimer Growth Model Describing One Possible Cause of Alzheimer's Disease and a Time‐Discretization for Its Numerical Simulation

ABSTRACT Because oligomers of the amyloid‐ () protein can possibly be regarded as one main cause for progressive development of Alzheimer's disease, different mathematical models for its emergence have been proposed by different scientific groups. In this work, we suggest a combined dimer growth model of oligomers of the ‐protein based on two recent scientific articles. For that purpose, we combine one model using a constant monomer production rate while the other one assumes a logistic monomer production rate. In contrast to both articles which mainly discuss biological implications of their non‐linear systems of ordinary differential equations, we present, as one of our main contributions, thorough derivations of certain mathematical properties such as non‐negativity, boundedness, existence globally in time, uniqueness globally in time and discuss local asymptotic stability of the unique equilibrium state of monomer (), dimer (), oligomer () and fibril () amounts of substances. Additionally, we demonstrate that this unique equilibrium state can also be calculated by a classical Newton‐type approach and we develop a new non‐standard finite‐difference‐method, founded on non‐local approximations of the right‐hand‐side vectorial field, to preserve as many desirable properties of the time‐continuous case as possible. Finally, we support our theoretical findings with numerical simulations.