Evaluate prediction results (intervals)

We are doing a research. We have a dataset with 5 fact. For each fact we predicted an interval. Given the theoretical intervals, I want to evaluate the goodness we are fitting the theoretical results with our predictions..

Intervals represent the initial and final day of month for each fact.

We have tried transforming the intervals into numeric (average of days) and euclidean distance between predicted and theoretical value. But we are not getting good results.
Interval has (min, max). What if test the agreements of min and max between predicted and theoretical separately? But, overlap could be better...