Statistical fluctuations in pedestrian evacuation times and the effect of social contagion

Mathematical models of pedestrian evacuation and the associated simulation software have become essential tools for the assessment of the safety of public facilities and buildings. While a variety of models is now available, their calibration and test against empirical data are generally restricted to global averaged quantities; the statistics compiled from the time series of individual escapes (“microscopic” statistics) measured in recent experiments are thus overlooked. In the same spirit, much research has primarily focused on the average global evacuation time, whereas the whole distribution of evacuation times over some set of realizations should matter. In the present paper we propose and discuss the validity of a simple relation between this distribution and the microscopic statistics, which is theoretically valid in the absence of correlations. To this purpose, we develop a minimal cellular automaton, with features that afford a semiquantitative reproduction of the experimental microscopic statistics. We then introduce a process of social contagion of impatient behavior in the model and show that the simple relation under test may dramatically fail at high contagion strengths, the latter being responsible for the emergence of strong correlations in the system. We conclude with comments on the potential practical relevance for safety science of calculations based on microscopic statistics.

Figure 1

Survival functions $P(\tau >\mathrm{\Delta }t)$ for $L_d=1$ and different cooperativeness levels: strongly competitive (red), moderately competitive (green), and cooperative (blue), from top to bottom. The curves have been shifted vertically to improve the visibility. The dashed black lines are power-law fits, with the exponents indicated in Table 1. The dotted lines are the survival functions obtained by replacing the distribution of propensities with Dirac functions peaked at their mean values.

Figure 2

Survival functions $P(\tau >\mathrm{\Delta }t)$ for different door widths $L_d=1$, 2, 3, 4, from top to bottom, with a strongly competitive crowd.

Figure 3

Distribution of burst sizes for a door width $L_d=2$ and a strongly competitive crowd.

Figure 4

Distribution of global evacuation times in the absence of contagion, for a strongly competitive crowd, $L_d=1$, and $L=25$. The solid red curve is the Gaussian distribution predicted on the basis of Eq. (3).

Figure 5

Potential $V$ for ${\mathrm{\Pi }}^{\left(\mathrm{intr}\right)}=0.93$, $z=4$, and different coupling parameters $J$, as indicated in the legend.

Figure 6

Color maps of the propensities ${\mathrm{\Pi }}_j\left(t\right)$ in the crowd at time $t=4000$ for ${\mathrm{\Pi }}^{\left(\mathrm{intr}\right)}=0.94\pm {10}^{-7}$ and $J=2.96$ (left), $J=2.9704$ (middle), and $J=2.98$ (right). Vacant sites appear in black.

Figure 7

Evacuations in the presence of behavioral contagion for $J=3.35$ and ${\mathrm{\Pi }}^{\left(\mathrm{intr}\right)}=0.94$ in a system of size $L=25$. Shown on the left is the survival function and on the right the normalized global distribution of evacuation times per pedestrian $\frac{T_{\mathrm{esc}}}{N}$, with the simulations in light blue, the expectations from the micro-macro relation in hatched pink (see the text), and the Gaussian distribution of Eq. (3) as a solid red curve.

Figure 8

Evacuations in the presence of behavioral contagion for $J=453$ and ${\mathrm{\Pi }}^{\left(\mathrm{intr}\right)}=0.99905$ in a system of size $L=30$. The rest is the same as in Fig. 7.

Figure 9

Graphical representations of the distribution function obtained with the contagion-free model for a strongly competitive crowd, where the heterogeneous distribution of ${\mathrm{\Pi }}^{\mathrm{intr}}$ is replaced by a Dirac peak. Shown on the left is the probability distribution function in a logarithmic plot and on the right the survival function in a semilogarithmic plot.