  We introduce and study a new model for the progression of Alzheimer's disease
incorporating the interactions of A$\beta$-monomers, oligomers, microglial
cells and interleukins with neurons through different mechanisms such as
protein polymerization, inflammation processes and neural stress reactions. In
order to understand the complete interactions between these elements, we study
a spatially-homogeneous simplified model that allows to determine the effect of
key parameters such as degradation rates in the asymptotic behavior of the
system and the stability of equilibriums. We observe that inflammation appears
to be a crucial factor in the initiation and progression of Alzheimer's disease
through a phenomenon of hysteresis, which means that there exists a critical
threshold of initial concentration of interleukins that determines if the
disease persists or not in the long term. These results give perspectives on
possible anti-inflammatory treatments that could be applied to mitigate the
progression of Alzheimer's disease. We also present numerical simulations that
allow to observe the effect of initial inflammation and concentration of
monomers in our model.
