Mathematical analysis of a PDE epidemiological model applied to scrapie transmission

Abstract

The aim of this paper is to analyse a dynamic model which describes the spread of scrapie in a sheep flock. Scrapie is a transmissible spongiform encephalopathy, endemic in a few European regions and subject to strict control measures. The model takes into account various factors and processes, including seasonal breeding, horizontal and vertical transmission, genetic susceptibility of sheep to the disease, and a long and variable incubation period. Therefore the model, derived from a classical SI (susceptible-infected) model, also incorporates a discrete genetic structure for the flock, as well as a continuous infection load structure which represents the disease incubation. The resulting model consists of a set of partial differential equations which describe the evolution of the flock with respect to time and infection load. To analyse this model, we use the semigroup and evolution family theory, which provides a flexible mathematical framework to determine the existence and uniqueness of a solution to the problem. We show that the corresponding linear model has a unique classical solution and that the complete nonlinear model has a global solution.