In this paper, an SIR 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 integro-differential equations. The uniqueness and qualitative
properties of the continuous model are analyzed. Furthermore, different choices
of spatial and temporal discretizations are employed, and step-size
restrictions for population conservation, positivity, and monotonicity
preservation of the discrete model are investigated. We provide sufficient
conditions under which high order numerical schemes preserve the discrete
properties of the model. Computational experiments verify the convergence and
accuracy of the numerical methods.Comment: 33 pages, 5 figures, 3 table