Algorithms for identifying the infection states of nodes in a network are crucial for understanding and containing infections. Often, however, only a relatively small set of nodes have a known infection state. Moreover, the length of time that each node has been infected is also unknown. This missing data -- infection state of most nodes and infection time of the unobserved infected nodes -- poses a challenge to the study of real-world cascades. In this work, we develop techniques to identify the latent infected nodes in the presence of missing infection time-and-state data. Based on the likely epidemic paths predicted by the simple susceptible-infected epidemic model, we propose a measure (Infection Betweenness) for uncovering these unknown infection states. Our experimental results using machine learning algorithms show that Infection Betweenness is the most effective feature for identifying latent infected nodes.