Pandemic and epidemic diseases such as CoVID-19, SARS-CoV2, and Ebola have spread to multiple countries and infected thousands of people. Such diseases spread mainly through person-to-person contacts. Health care authorities recommend contact tracing procedures to prevent the spread to a vast population. Although several mobile applications have been developed to trace contacts, they typically require collection of privacy-intrusive information such as GPS locations, and the logging of privacy-sensitive data on a third party server, or require additional infrastructure such as WiFi APs with known locations. In this paper, we introduce CONTAIN, a privacy-oriented mobile contact tracing application that does not rely on GPS or any other form of infrastructure-based location sensing, nor the continuous logging of any other personally identifiable information on a server. The goal of CONTAIN is to allow users to determine with complete privacy if they have been within a short distance, specifically, Bluetooth wireless range, of someone that is infected, and potentially also when. We identify and prove the privacy guarantees provided by our approach. Our simulation study utilizing an empirical trace dataset (Asturies) involving 100 mobile devices and around 60000 records shows that users can maximize their possibility of identifying if they were near an infected user by turning on the app during active times.