On some qualitatively adequate discrete space–time models of epidemic propagation

Most of the models of epidemic propagations do not take into account the spatial distribution of the individuals. They give only the temporal change of the number of the infected, susceptible and recovered patients. In this paper we give some spatial discrete one-step iteration models for disease propagation and give conditions that guarantee some basic qualitative properties of the original process to the discrete models. Since the discrete models can be considered as the finite difference discretizations of continuous models of disease propagation given in the form of systems of partial differential equations, we can deduce conditions for the mesh size and the time step. Some of the results are demonstrated on numerical tests.