The demand for certain types of health care services depends on decisions of both the individual and the health care provider. This paper studies the conditions under which it is possible to separately identify the parameters driving the two decision processes using only count data on the total demand. It is found that the frequently used hurdle models may not be adequate to describe this type of demand, especially when the assumption of a single illness spell per observation period is violated. A test for the single spell hypothesis is developed and alternative modelling strategies are suggested, including one that allows for correlated unobserved heterogeneity. The results of the paper are illustrated with an application.