Metrics such as the outbreak size in an epidemic process on a network are fundamental quantities used in public health analyses.The datasets used in such models used in practice, e.g., the contact network and disease states, are sensitive in many settings.We study the complexity of computing epidemic outbreak size within a given time horizon, under edge differential privacy.These quantities have high sensitivity, and we show that giving algorithms with good utility guarantees is impossible for general graphs. To address these hardness results, we consider a smaller class of graphs with similar properties as social networks (called expander graphs) and give a polynomial-time algorithm with strong utility guarantees. Our results are the first to give any non-trivial guarantees for differentially private infection size estimation.