Multivariate count data are defined as the number of items of different categories issued from sampling within a population, which individuals are grouped into categories. The analysis of multivariate count data is a recurrent and crucial issue in numerous modelling problems, particularly in the fields of biology and ecology (where the data can represent, for example, children counts associated with multitype branching processes), sociology and econometrics. We focus on I) Identifying categories that appear simultaneously, or on the contrary that are mutually exclusive. This is achieved by identifying conditional independence relationships between the variables; II)Building parsimonious parametric models consistent with these relationships; III) Characterising and testing the effects of covariates on the joint distribution of the counts. To achieve these goals, we propose an approach based on graphical probabilistic models, and more specifically partially directed acyclic graphs.