A model is formulated that describes the spatial propagation of a disease that can be transmitted between multiple species. The spatial component consists, for each species, of a certain number of patches that make up the vertices of a digraph, the arcs of which represent the movement of the various species between the patches. In each of the patches and for each species, a susceptible-exposed-infectious-recovered (SEIR) epidemic model describes the evolution of the disease status of individuals. Also in each patch, there is transmission of the disease from species to species. An analysis of the system is given, beginning with results on the mobility component. A formula is derived for the computation of the basic reproduction number R0 for sspecies and npatches, which then determines the global stability properties of the disease free equilibrium. Simulations for the spread of a disease in one species and two patches are presented.