Young Researchers Workshop: Kinetic models in biology and social sciences

A pedestrian flow model with stochastic velocities: microscopic and macroscopic approaches

Stephan Knapp

University of Mannheim


A well known phenomenon in crowds is the sudden presence of non-moving persons which look at their cell phones, reorientate themselves or watch at a point of interest. This leads to changes in the velocities of the surrounding neighbors, evasive maneuver and hence to bottlenecks depending on the crowd density. To include this behavior into a mathematical model, we consider the well-known social force model by D. Helbing and P. Molnár. This, in the literature well studied microscopic model describes the movement of each pedestrian according to his/her destination as well as interaction and obstacle forces. We embed the mentioned stochastic behavior into the microscopic pedestrian model by redefining a time-discrete stochastic process which simultaneously implies a microscopic simulation algorithm. Due to the high computational costs of approximating the crowd density for a large number of people, we derive a scalar-type model which approximates the evolution of the crowd density. Finally, the microscopic and the scalar-type model are compared by numerical results in several examples.