Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach

Pages: 213 – 236



Paolo Freguglia (Department of Engineering, Information Sciences, and Mathematics, University of L’Aquila, Coppito AQ, Italy)

Andrea Tosin (Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Italy)


This paper deals with a Boltzmann-type kinetic model describing the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow, including the distribution of the driving risk along a road, are ultimately generated by one-to-one interactions among drivers, the model links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, it is finally possible to recover macroscopic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles.


fundamental and risk diagrams of traffic, safety and risk regimes, kinetic equations, stochastic microscopic interactions, Wasserstein spaces

2010 Mathematics Subject Classification

35Q20, 35Q70, 90B20

Published 10 January 2017