This paper makes three contributions. First, it uses copula functions to obtain a flexible bivariate parametric model for nonnegative integer-valued data (counts). Second, it recovers the distribution of the difference in the two counts from a specified bivariate count distribution. Third the methods are applied to counts that are measured with error. Specifically we model the determinants of the difference between the self-reported number of doctor visits (measured with error) and true number of doctor visits (also available in the data set used).