Operations researchers worldwide rely extensively on quantitative simulations to model alternative aspects of the COVID-19 pandemic. Proper uncertainty quantification and sensitivity analysis are fundamental to enrich the modeling process and communicate correctly informed insights to decision-makers. We develop a methodology to obtain insights on key uncertainty drivers, trend analysis and interaction quantification through an innovative combination of probabilistic sensitivity techniques and machine learning tools. We illustrate the approach by applying it to a representative of the family of susceptible-infectious-recovered (SIR) models recently used in the context of the COVID-19 pandemic. We focus on data of the early pandemic progression in Italy and the United States (the U.S.). We perform the analysis for both cases of correlated and uncorrelated inputs. Results show that quarantine rate and intervention time are the key uncertainty drivers, have opposite effects on the number of total infected individuals and are involved in the most relevant interactions.