Since the birth of human beings, the spreading of epidemics such as COVID-19 affects our lives heavily and the related studies have become hot topics. All the countries are trying to develop effective prevention and control measures. As a discipline that can simulate the transmission process, complex networks have been applied to epidemic suppression, in which the common approaches are designed to remove the important edges and nodes for controlling the spread of infection. However, the naive removal of nodes and edges in the complex network of the epidemic would be practically infeasible or incur huge costs. With the focus on the effect of epidemic suppression, the existing methods ignore the network connectivity, leading to two serious problems. On the one hand, when we remove nodes, the edges connected to the nodes are also removed, which makes the node is isolated and the connectivity is quickly reduced. On the other hand, although removing edges is less detrimental to network connectivity than removing nodes, existing methods still cause great damage to the network performance in reality. Here, we propose a method to measure edge importance that can protect network connectivity while suppressing epidemic. In the real-world, our method can not only lower the government’s spending on epidemic suppression but also persist the economic growth and protect the livelihood of the people to some extent. The proposed method promises to be an effective tool to maintain the functionality of networks while controlling the spread of diseases, for example, diseases spread through contact networks.