The spread of infectious disease is an inherently stochastic process. As such, real-time control and prediction methods present a significant challenge. For diseases that spread through direct human interaction, the contagion process can be modeled on a social contact network where individuals are represented as nodes, and contact between individuals is represented as links. The objective of the model described in this paper is to infer the most likely path of infection through a contact network for an ongoing outbreak. The problem is formulated as a linear integer program. Specific properties of the problem are exploited to develop a much more efficient solution method than solving the linear program directly. The model output can provide insight into future epidemic outbreak patterns and aid in the development of intervention strategies. The model is evaluated for a combination of network structures and sizes, as well as various disease properties and potential human error in assessing these properties. The model performance varies based on these parameters, but it is shown to perform best for heterogeneous networks, which are consistent with many real world systems.