Comfort of pedestrians from a mathematical viewpoint: Kernel estimate approach

This article discusses techniques for converting key concepts from the field of crowd dynamics into a mathematical format. The terms converted are pedestrian density, pedestrian flow, instantaneous speed and pedestrian comfort. The text is based on the concept of so-called density kernels, which are an ideal choice for the above-mentioned purposes because they are mathematically well quantifiable and at the same time they keep the intuitive essence of all the above-mentioned quantities. In addition, the mathematical reformulation of these terms allows for the description of basic laws of pedestrian flow using elegant mathematical formulas. Moreover, we present a relatively large collection of specific density kernels and investigate qualitatively their topological properties (symmetry, eccentricity, support symmetry). Furthermore, we analytically derive the equation of continuity for pedestrian flow and discuss the limits of its validity. We also define quantitatively the intuitively-used concept of pedestrian comfort and validate the entire concept on the original experimental data. Analysis of the leave-the-room experiment shows that high values of the comfort coefficient (detected immediately after the pedestrians enter the room) are sharply reduced in the middle part of the room where walkers accumulate. In the vicinity of the exit – where intensity of pedestrians is increasing – the comfort coefficient grows up. Furthermore, the detected comfort is only weakly dependent on the specific pedestrian whose comfort is being analyzed. The minimum level of comfort coefficient enumerated for the entire room reveals areas where comfort is strongly dampened (the area located symmetrically along an axis perpendicular to the exit), as well as areas where moving pedestrians feel significantly more comfortable. In addition, the average value of the comfort coefficient measured near the exit-door depends significantly on the angle at which pedestrians enter the door. The achieved outputs confirm that the mathematically established concept reliably replicates the intuitive idea of limiting the comfort of pedestrians by the movement and location of other pedestrians.