In this work, we study the SIR-Network model proposed by Stolerman, Coombs, and Boatto in 2015, which describes the dynamics of an infectious disease in a city where each neighborhood is a node of a network, and the edges of a network represent the daily flux of people between the nodes. In their original study, Stolerman and colleagues established a flux-driven epidemic control by analyzing epidemic thresholds for fully-connected networks where a single node has a different infection rate. Inspired by this result, we analyze a larger class of networks establishing new epidemic thresholds using the basic reproduction number R_0 obtained from the classic next-generation matrix. We find a family of networks (star-class) with the same kind of epidemic control inspired by the star-shaped networks, providing analytical estimates in both general (any network size) and particular (fixed network size) cases. In addition, cycle-shaped networks exhibit a flux-driven epidemic control but with different epidemic thresholds compared to the star-class networks. Finally, we numerically integrate our system to gain an intuition of where the theoretical estimates are challenging and explore the temporal dynamical behavior for the different classes of networks.