Epidemics theory has been used in different contexts in order to describe the propagation of diseases, human interactions or natural phenomena. In computer science, virus spreading has been also characterized using epidemic models. Although in the past the use of epidemic models in telecommunication networks has not been extensively considered, nowadays, with the increasing computation capacity and complexity of operating systems of modern network devices (routers, switches, etc.), the study of possible epidemic-like failure scenarios must be taken into account. When epidemics occur, such as in other multiple failure scenarios, identifying the level of vulnerability offered by a network is one of the main challenges. In this paper, we present epidemic survivability, a new network measure that describes the vulnerability of each node of a network under a specific epidemic intensity. Moreover, this metric is able to identify the set of nodes which are more vulnerable under an epidemic attack. In addition, two applications of epidemic survivability are provided. First, we introduce epidemic criticality, a novel robustness metric for epidemic failure scenarios. A case study shows the utility of this new metric comparing several network topologies and epidemic intensities. Then, two immunization strategies are proposed: high epidemic survivability (HES) and high epidemic survivability adaptive (HESA). The presented results show that network vulnerability can be significantly reduced by using our proposals, compared to other well-known existing methods.