The epidemiology of person-to-person communicable disease in large,homogeneous populations is modeled by three ordinary differential equations.These represent the susceptible,infectious and recovered individuals.For smaller populations an agent-based model can capture the inhomogeneous nature of human interactions by modeling each individual as a node in a network,or vertex in a graph.In this model we look at a set of large homogeneous populations,each modeled as a classic SIR epidemic (the patches),but which have immigration between them (the graph).It is a mathematically interesting and epidemiologically useful question to ask which aspects of disease transmission are controlled by the local properties of each SIR model,and which are controlled by the global connectivity of the graph.In this talk we will answer a piece of this question for a particular patch SIR model on any graph where each vertex has k neighbors.