One of the most critical issues in epidemiology revolves around the bridging of the diverse space and time scales stretching from the microscopic scale, where detailed knowledge on the immune mechanisms, host-microbe and host-host interactions is often available, to the macroscopic population-scale where the epidemic emerges, the questions arise and the answers are required. In this paper we show how the so called Equation-Free approach, a novel computational framework for multi-scale analysis, can be exploited to efficiently analyze the macroscopic emergent behavior of complex epidemic models on certain type of networks by acting directly on the multi-scale simulation. The methodology can be used to bypass the need of derivation of closures for the emergent population-level equations providing a systematic computational strict approach for macroscopic-level analysis. We illustrate the methodology through a stochastic individual-based model with agents acting on two different networks: a random regular and an Erdős-Rényi network. We construct the macroscopic bifurcation diagrams and locate the critical points that mark the onset of emergent hysteresis behavior which are associated with disease outbreaks. Finally, we perform a rare-events analysis that may in principle be used to estimate the mean time of possible outbreaks of phenomenologically latent infectious diseases.