This paper studies the problem of detecting asymptomatic cases in a temporal contact network in which multiple outbreaks have occurred. We show that the key to detecting asymptomatic cases well is taking into account both individual risk and the likelihood of disease-flow along edges. We consider both aspects by formulating the asymptomatic case detection problem as a directed prize-collecting Steiner tree (Directed PCST) problem. We present an approximation-preserving reduction from this problem to the directed Steiner tree problem and obtain scalable algorithms for the Directed PCST problem on instances with more than 1.5M edges obtained from both synthetic and fine-grained hospital data. On synthetic data, we demonstrate that our detection methods significantly outperform various baselines (with a gain of 3.6 × ). We apply our method to the infectious disease prediction task by using an additional feature set that captures exposure to detected asymptomatic cases and show that our method outperforms all baselines. We further use our method to detect infection sources ("patient zero") of outbreaks that outperform baselines. We also demonstrate that the solutions returned by our approach are clinically meaningful by presenting case studies.