We consider the spread of an infectious disease in a heterogeneous environment, modelled as a network of patches. We focus on the invasibility of the disease, as quantifi ed by the corresponding value of an approximation to the network basic reproduction number, R0, and study how changes in the network structure affect the value of R0. We provide a detailed analysis for two model networks, a star and a path, and discuss the changes to the corresponding network structure that yield the largest decrease in R0. We develop both combinatorial and matrix analytic techniques, and illustrate our theoretical results by simulations with the exact R0.