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Alzheimer's disease (AD) is a common neurodegenerative disorder nowadays.Amyloid-beta (A$\beta$) and tau proteins are among the main contributors to thedevelopment or propagation of AD. In AD, A$\beta$ proteins clump together toform plaques and disrupt cell functions. On the other hand, the abnormalchemical change in the brain helps to build sticky tau tangles that block theneuron's transport system. Astrocytes generally maintain a healthy balance inthe brain by clearing the A$\beta$ plaques (toxic A$\beta$). However,over-activated astrocytes release chemokines and cytokines in the presence ofA$\beta$ and react to pro-inflammatory cytokines, further increasing theproduction of A$\beta$. In this paper, we construct a mathematical model thatcan capture astrocytes' dual behaviour. Furthermore, we reveal that the diseasepropagation depends on the current time instance and the disease's earlierstatus, called the ``memory effect''. We consider a fractional order networkmathematical model to capture the influence of such memory effect on ADpropagation. We have integrated brain connectome data into the model andstudied the memory effect, the dual role of astrocytes, and the brain'sneuronal damage. Based on the pathology, primary, secondary, and mixedtauopathies parameters are considered in the model. Due to the mixed tauopathy,different brain nodes or regions in the brain connectome accumulate differenttoxic concentrations of A$\beta$ and tau proteins. Finally, we explain how thememory effect can slow down the propagation of such toxic proteins in thebrain, decreasing the rate of neuronal damage.

Complex non-Markovian dynamics and the dual role of astrocytes in Alzheimer's disease development and propagation