Epidemic containment is a major concern when confronting large-scale infections in complex networks. Many works have been devoted to analytically understand how to restructure the network to minimize the impact of major outbreaks of infections at large scale. In many cases, the strategies consist in the isolation of certain nodes, while less attention has been paid to the intervention on links. In epidemic spreading, links inform about the probability of carrying the contagion of the disease from infected to susceptible individuals. Note that these states depend on the full structure of the network, and its determination is not straightforward from the knowledge of nodes' states. Here, we confront this challenge and propose a set of discrete-time governing equations \rev{which} can be closed and analyzed, assessing the contribution of links to spreading processes in complex networks. Our approach allows a scheme for the \rev{containment} of epidemics, based on deactivating the most important links in transmitting the disease. The model is validated in synthetic and real networks, obtaining an accurate determination of the epidemic incidence and the critical thresholds. Epidemic containment based on links' deactivation promises to be an effective tool to maintain functionality on networks while controlling the spread of diseases, as for example in air transportation networks.