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## Section: New Results

### Alzheimer’s disease and prion: An in vitro mathematical model

Alzheimer's disease ($AD$) is a fatal incurable disease leading to progressive neuron destruction. $AD$ is caused in part by the accumulation in the brain of $A\beta$ monomers aggregating into oligomers and fibrils. Oligomers are amongst the most toxic structures as they can interact with neurons via membrane receptors, including $Pr{P}^{c}$ proteins. This interaction leads to the misconformation of $Pr{P}^{c}$ into pathogenic oligomeric prions, $Pr{P}^{ol}$. In [14], we develop a model describing in vitro $A\beta$ polymerization process. We include interactions between oligomers and $Pr{P}^{c}$, causing the misconformation of $Pr{P}^{c}$ into $Pr{P}^{ol}$. The model consists of nine equations, including size structured transport equations, ordinary differential equations and delayed differential equations. We analyse the well-posedness of the model and prove the existence and uniqueness of solutions of our model using Schauder fixed point theorem and Cauchy-Lipschitz theorem. Numerical simulations are also provided to give an illustration of the profiles that can be obtained with this model.