Theory and simulations of delayed stochastic and deterministic models of prion diseases.

Neurodegenerative diseases (NDs), such as Alzheimer's, Parkinson's, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein ( ${\text{PrP}}^{\text{C}}$ ), produced by neurons, misfolds into a toxic form, known as scrapie prion protein ( ${\text{PrP}}^{\text{Sc}}$ ). ${\text{PrP}}^{\text{Sc}}$ induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial spread of toxic proteins over the brain connectome. Our first objective is to establish the existence and uniqueness of a global positive solution to the prion disease models. Afterwards, we analyze the asymptotic behavior of the models by identifying regimes of persistence and extinction of toxic proteins. For the deterministic delayed systems, we perform a stability analysis for the persistence and demonstrate that the system undergoes a Hopf bifurcation. We also study the intensity of fluctuations of the equilibrium state of the stochastic model. Additionally, we present numerical simulations to illustrate the model dynamics using biologically relevant parameters.