In the realm of urban dynamics, mathematical modelling plays a major role in the description of pedestrian behaviour in crowds. These pedestrian dynamics models serve as essential tools for comprehending and improving urban flow, safety, and efficiency. They are relevant in the design of virtually every complex architecture in which many people want to move from one point to another, and building and evacuation plans rely on their accuracy.

In order to find a reliable mathematical model for pedestrian dynamics, approaches have been made from all kinds of different directions. Some models are based on the premise that interactions between people can be modelled similarly to repelling magnets, some model pedestrians as particles obeying the laws of molecular dynamics, and others translate rational agents’ decisions into laws of dynamics.

A new article in the European Journal of Applied Mathematics, “Gradient-based parameter calibration of an anisotropic interaction model for pedestrian dynamics” by Zhomart Turarov and Claudia Totzeck addresses and extends one model that is based on social forces between pedestrians.

The novelty of this article’s model lies in the consideration of agents’ body sizes. The model that the authors build upon describes the movement of all pedestrians by a set of ordinary differential equations (ODE). Every pedestrian has a desired velocity, i.e., direction and speed, at which they want to move through the crowd.

When two pedestrians hit, of course, they have to sidestep. This is introduced by rotating the velocity vector of two pedestrians once they are too close to each other. The resulting system of ODEs can be solved by standard methods to generate trajectories of the pedestrians in the model.

The authors show numerical results of their proposed model in several situations of pedestrian crowds in exemplary hallway or crossing situations. Interestingly, the agents soon start to show collective behaviour and movement patterns repeatedly form, depending on several model parameters that steer components like the attraction and repulsion forces and the introduced body sizes.

Different values for these parameters lead to different characteristics in the solutions. For example, the authors observed that while pedestrians tend to form ‘traffic lanes’ in hallways (lines in which pedestrians with similar desired velocities organize), the distribution of these lanes depends heavily on the body sizes and the assumed attraction and repulsion forces. The dynamics’ fundamental diagrams agree with previous results on similar models and experimental data.

Apart from extending the model, the main contribution of the article is the analysis and solution of the parameter calibration problem for given pedestrians’ trajectory data. Instead of solving the model ODE system for a given set of parameters, in this task, one tries to find in turn the set of parameters for which the model describes some given measured trajectory data best.

Computationally, this task is more challenging since every evaluation of the cost function requires the solution of the ODE system for the input parameters. The cost function needs to be optimized over the whole set of possible parameters.

The authors prove the existence of a solution to this calibration problem and derive its first-order optimality condition. By some computations, they are eventually able to arrive at a stochastic gradient descent scheme, which allows them to compute the optimally calibrated parameters in a numerical setting. For open-sourced experimental data of real pedestrians participating in a test in a corridor and at a crossing, the model is fitted to the measured trajectories.

Gradient-based parameter calibration of an anisotropic interaction model for pedestrian dynamics. Zhomart Turarov, and Claudia Totzeck (2023). 

This open access paper appears in European Journal of Applied Mathematics.

Since 2008 EJAM surveys have been expanded to cover Applied and Industrial Mathematics. Coverage of the journal has been strengthened in probabilistic applications, while still focusing on those areas of applied mathematics inspired by real-world applications, and at the same time fostering the development of theoretical methods with a broad range of applicability. Survey papers contain reviews of emerging areas of mathematics, either in core areas or with relevance to users in industry and other disciplines. Research papers may be in any area of applied mathematics, with special emphasis on new mathematical ideas, relevant to modelling and analysis in modern science and technology, and the development of interesting mathematical methods of wide applicability.