We present in this paper a model of vehicular traffic flow for a multimodal transportation road network. We introduce the notion of class of vehicles to refer to vehicles of different transport modes. Our model describes the traffic on highways (which may contain several lanes) and network transit for pubic transportation. The model is drafted with Eulerian and Lagrangian coordinates and uses a Logit model to describe the traffic assignment of our multiclass vehicular flow description on shared roads. The paper also discusses traffic streams on dedicated lanes for specific class of vehicles with event-based traffic laws. An Euler-Lagrangian-remap scheme is introduced to numerically approximate the model’s flow equations.