Given a noisy or sampled snapshot of an infection in a large graph, can we automatically and reliably recover the truly infected yet somehow missed nodes? And, what about the seeds, the nodes from which the infection started to spread? These are important questions in diverse contexts, ranging from epidemiology to social media.In this paper, we address the problem of simultaneously recovering the missing infections and the source nodes of the epidemic given noisy data. We formulate the problem by the Minimum Description Length principle, and propose NetFill, an efficient algorithm that automatically and highly accurately identifies the number and identities of both missing nodes and the infection seed nodes.Experimental evaluation on synthetic and real datasets, including using data from information cascades over 96 million blog posts and news articles, shows that our method outperforms other baselines, scales near-linearly, and is highly effective in recovering missing nodes and sources.