Summary This paper makes three contributions. Firstly, it uses copula functions to obtain a flexible bivariate parametric model for non-negative integer-valued data (counts). Secondly, it recovers the distribution of the difference in the two counts from a specified bivariate count distribution. Thirdly, 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 used).