When modeling the dynamics of infectious disease, the incorporation of contact network information allows for the capture of the non-randomness and heterogeneity of realistic contact patterns. Oftentimes, it is assumed that the underlying contact pattern is known with perfect certainty. However, in realistic settings, the observed data often serves as an imperfect proxy of the actual contact patterns in the population. Furthermore, the epidemic in the real world are often not fully observed; event times such as infection and recovery times may be missing. In order to conduct accurate inferences on parameters of contagion spread, it is crucial to incorporate these sources of uncertainty. In this paper, we propose the use of Mixture Density Network compressed ABC (MDN-ABC) to learn informative summary statistics for the available data. This method will allow for Bayesian inference on the epidemic parameters of a contagious process, while accounting for imperfect observations on the epidemic and the contact network. We will demonstrate the use of this method on simulated epidemics and networks, and extend this framework to analyze the spread of Tattoo Skin Disease (TSD) among bottlenose dolphins in Shark Bay, Australia.