We consider derivative-free optimization, and in particular black box optimization, where the functions to minimize and the functions representing the constraints are given by black boxes without derivatives. Two fundamental families of methods are available: Model-based methods and directional direct search algorithms. This work exploits the flexibility of the second type of method in order to integrate to a limited extent the models used in the first family. Intensive numerical tests on two sets of forty-eight and one hundred and four test problems illustrate the efficiency of this hybridization and show that the use of models improves significantly the mesh adaptive direct search algorithm.