In this talk, an SIR model with spatial dependence is discussed and results regarding its stability and numerical approximation are presented. SIR models have been used to describe epidemic propagation phenomena, and one of the first models is derived by Kermack and McKendrick in 1927. In such models, the population is spit into three classes: $S$ is the group of species susceptible to infection, $I$ is the compartment of the ill species, and $R$ the class in of recovered species. 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 qualitatively properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are deployed, 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.