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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.
Mathematics Subject Classification: Primary: 35L60; Secondary: 92D30.
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