THE SPREAD OF PRION DISEASES IN THE BRAIN — MODELS OF REACTION AND TRANSPORT ON NETWORKS

Abstract

In this paper we will present a modeling approach to describe the progression and the spread of prion diseases in the brain. Although there exist a number of mathematical models for the interaction of prions with their native counterpart, prion transport and spread is usually neglected. The concentration dynamics of prions, and thus the dynamics of the disease progression, however, are influenced by prion transport, especially in a medium as complex as the brain. Therefore, we focus here on the interaction between prion concentration dynamics and prion transport. The model is constructed by combining a model of prion-prion interaction with transport on networks. The approach leads to a system of reaction-diffusion equations, whereby the diffusion term is discrete. The equations are solved numerically on domains given as large networks. We show that the prion concentration grows faster on networks characterized by a higher degree heterogeneity. Furthermore, we introduce cell death as a consequence of increasing prion concentration, leading to network decomposition. We show that infectious diseases destroy networks similarly to targeted attacks, namely by affecting the nodes with the highest degree first. Relating the incubation period and disease progression to the process of network decomposition, we find that, interestingly, a long incubation time followed by sudden onset and fast progression of the disease does not need to be reflected in the overall concentration dynamics of the infective agent.