A Macroscopic Model for Bidirectional Pedestrian Flow

We present a macroscopic model for pedestrian dynamics in a corridor (or in any quasi one-dimensional system). The model is inspired from the Aw-Rascle model of car traffic but here, a two-directional flow is considered: in each point, two densities are defined, for left and right going pedestrians. The challenge is to bound the density even under congestion. This is enforced by a pressure term, modeling the interactions between pedestrians, that diverges when the density approaches the maximal density. The intensity of the divergence is controlled by a small parameter epsilon. In the limit where epsilon tends to zero, the system exhibits coexisting congested and uncongested phases separated by sharp interfaces. The lateral extension of the corridor can be taken into account through a multi-lane model, with appropriate lane changes. A characteristic of two-way models is that they can loose their hyperbolicity in some cases. Actually, this could be the counterpart of phenomena observed in real crowds, namely the instability of homogeneous flows towards lane-formation, or even crowd turbulence as observed at very high crowd densities.