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We consider the spread of an infectious disease in a heterogeneous environment modeled as a network of patches. We focus on the invasibility of the disease, as quantified 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 we illustrate our theoretical results by simulations with the exact R0. [ABSTRACT FROM AUTHOR]

In this paper, the necessary and sufficient conditions for the existence of traveling wave solutions are derived for a class of diffusive disease-transmission models with network structures. The existence of traveling semifronts is obtained by Schauder's fixed-point theorem, and these traveling semifronts are shown to be bounded by transforming the boundedness problem into the classification problem of nonnegative solutions to a linear elliptic system on R. To overcome the reducibility problem arising in the proofs, Harnack's inequality for positive supersolutions on R is proved. [ABSTRACT FROM AUTHOR]