  In this paper, an epidemic model with spatial dependence is studied and
results regarding its stability and numerical approximation are presented. We
consider a generalization of the original Kermack and McKendrick model in which
the size of the populations differs in space. The use of local spatial
dependence yields a system of partial-differential equations with integral
terms. The uniqueness and qualitative properties of the continuous model are
analyzed. Furthermore, different spatial and temporal discretizations are
employed, and step-size restrictions for the discrete model's positivity,
monotonicity preservation, and population conservation are investigated. We
provide sufficient conditions under which high-order numerical schemes preserve
the stability of the computational process and provide sufficiently accurate
numerical approximations. Computational experiments verify the convergence and
accuracy of the numerical methods.
