Optimal control of monomers and oligomers degradation in an Alzheimer’s disease model

The aggregation and accumulation of oligomers of misfolded A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-amyloids in the human brain is one of the possible causes for the onset of the Alzheimer's disease in the early stage. We introduce and study a new ODE model for the evolution of Alzheimer's disease based on the interaction between monomers, proto-oligomers, and oligomers of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} amyloid protein in a small portion of the human brain, based upon biochemical processes such as polymerization, depolymerization, fragmentation and concatenation. We further introduce the possibility of controlling the evolution of the system via a treatment that targets the monomers and/or the oligomers. We observe that a combined optimal treatment on both monomers and oligomers induces a substantial decrease of the oligomer concentration at the final stage. A single treatment on oligomers performs better than a single treatment on monomers. These results shed a light on the effectiveness of immunotherapy using anti-A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} antibodies, targeting monomers or oligomers. Several numerical simulations show how the oligomer concentration evolves without treatment, with single monomer/oligomer treatment, or with a combined treatment.