This paper develops and analyzes optimization models for rapid detection of viruses in large contact networks. In the model, a virus spreads in a stochastic manner over an undirected connected graph, under various assumptions on the spread dynamics. A decision maker must place a limited number of detectors on a subset of the nodes in the graph in order to rapidly detect infection of the nodes by the virus. The objective is to determine the placement of these detectors so as to either maximize the probability of detection within a given time period or minimize the expected time to detection. Previous work in this area assumed that the detectors are perfectly reliable. In this work, it is assumed that the detectors may produce false-negative results. In computational studies, the sample average approximation method is applied to solving the problem using a mixed-integer program and a greedy heuristic. The heuristic is shown to be highly efficient and to produce high-quality solutions. In addition, it is shown that the false-negative effect can sometimes be ignored, without significant loss of solution quality, in the original optimization formulation.