A discrete mathematical model for the dynamics of a crowd of gazing pedestrians with and without an evolving environmental awareness

Abstract

In this article, we present a microscopic-discrete mathematical model describing crowd dynamics in no panic conditions. More specifically, pedestrians are set to move in order to reach a target destination and their movement is influenced by both behavioral strategies and physical forces. Behavioral strategies include individual desire to remain sufficiently far from structural elements (walls and obstacles) and from other walkers, while physical forces account for interpersonal collisions. The resulting pedestrian behavior emerges therefore from non-local, anisotropic and short/long-range interactions. Relevant improvements of our mathematical model with respect to similar microscopic-discrete approaches present in the literature are: (i) each pedestrian has his/her own dynamic gazing direction, which is regarded to as an independent degree of freedom and (ii) each walker is allowed to take dynamic strategic decisions according to his/her environmental awareness, which increases due to new information acquired on the surrounding space through their visual region. The resulting mathematical modeling environment is then applied to specific scenarios that, although simplified, resemble real-word situations. In particular, we focus on pedestrian flow in two-dimensional buildings with several structural elements (i.e., corridors, divisors and columns, and exit doors). The noticeable heterogeneity of possible applications demonstrates the potential of our mathematical model in addressing different engineering problems, allowing for optimization issues as well.

Appendix: parameter estimate and sensitivity analysis

As already explained, the model parameters can be distinguished in physical quantities and merely technical coefficients. When possible, we opt to use a parameter estimate inferred from the existing literature. In the other cases, our choice is done after preliminary simulations, that help us to select the more realistic parameter setting. In this respect, this Appendix gives details both on how we establish the set of parameter values and on how our estimates affect the simulation outcomes.

Fig. 11

Effect of the extension of the visual region on pedestrian dynamics. The initial distribution of the group of individuals is the same for all realizations. All walkers initially know only the position of door 2, but they have an evolving environmental awareness (i.e., they are allowed to opt for the nearest known door)

Variation of the visual region extension Although the physical significance of both $R^{\mathrm{vis}}_i$ and $\theta _i$ is sufficient to define their values, it is worth noting that the extension of the visual region can definitely change pedestrian behavior. In order to analyze this aspect, we refer to the simulation setting of Fig. 8 in Sect. 5, i.e., a group of 10 individuals distributed in a square room with three exits that want to reach the nearest door. In particular, all pedestrians are set to initially know only the position of exit 2 but have a variable environmental awareness (i.e., each of them is allowed to opt for an alternative door if he/she actually sees it). Keeping fixed the other model parameters, we vary either the visual depth $R^{\mathrm{vis}}_i=R^{\mathrm{vis}}$ and the visual angle $\theta _i=\theta$ (we recall that both are in common for all individuals). As captured by the trajectories reported in Fig. 11, very low values of $R^{\mathrm{vis}}$ and $\theta$ result in a significantly limited visual region for the pedestrians, that therefore typically maintain door 2 as target destination (they are in fact not able to see any other exits). The model outcomes in such a range of parameters could be also obtained by employing a discrete approach that does not include a variable environmental awareness. The capability of pedestrians to individuate (and therefore to choose) alternative targets then increases with the overall extension of their visual region, i.e., it is enhanced by increments in the values of $R^{\mathrm{vis}}$ and $\theta$. In particular, for any $R^{\mathrm{vis}}$ higher then the size of the domain, the walker behavior is entirely determined by the extension of his/her visual angle $\theta$, see the right panels in Fig. 11. Given these considerations, we opt to estimate $R^{\mathrm{vis}}= 50$ m, which is in the range of the characteristic dimensions of the domains employed in this work, and $\theta = 85^{\circ }$, a value consistent with the literature, see Bruno et al. (2011).

For the sensitivity analysis of the remaining model parameters, we hereafter employ the “two-adjacent-rooms-without-column” domain presented in Sect. 3.3 [cf. Fig. 7 (top-right panel)]. In particular, a group of 200 individuals is initially located in the left room and wants to reach the door placed in the right room. Starting from the very same initial distribution of pedestrians, we then singularly vary selected model parameters and analyze the corresponding simulation outcomes both by monitoring the evacuation time of the crowd and by observing the emergence of characteristic collective dynamics.

Variation of the wall repulsion parameters The wall repulsion radius $R^{\mathrm{wall}}$ is estimated on the basis of empirical considerations. We first assume that a person within a building takes into account the presence of a wall only when he/she is approaching it. In this respect, given $R^{\mathrm{body}} = 0.3$ m, we set $R^{\mathrm{wall}}=0.4$  m, i.e., an individual tries to maintain a distance of about $R^{\mathrm{wall}}-R^{\mathrm{body}}= 0.1$ m from the nearest wall. Lower values of $R^{\mathrm{wall}}$ would result in fact in decrements of the evacuation time (see the corresponding panel in Fig. 12) but also in unphysical dynamics: the pedestrians would be in fact dramatically pressed along the domain boundary or along the internal walls. On the opposite, for $R^{\mathrm{wall}}>0.4$ m, no variations in the evacuation time occur. This is a further justification for the chosen parameter value. The coefficients A and B determine instead the exact form of the exponential function that defines the wall repulsion velocity. Keeping all the other parameters fixed, A does not have a significant impact on the overall dynamics, see Fig. 12: indeed, we opt for an intermediate value $A=1$ m/s. Variations of B play instead a critical role in the model outcomes. For $B< 10^{-2}$ m, the evacuation time is almost constant: however, it decreases when $B \in (10^{-2}$ m, 1 m), until reaching a threshold for $B> 1$ m. However, high enough values of B (i.e., $>1$ m) correspond to the unrealistic situation of pedestrians that constantly move along the boundary of the domain (not shown). We indeed opt to set $B=10^{-2}$ m.

Variation of the interpersonal interaction parameters By setting the pedestrian body radius $R^{\mathrm{body}}$ equal to 0.3 m, the contact radius $R^{\mathrm{cont}}$ is necessarily equal to $2\,R^{\mathrm{body}}=0.6$ m for any pair of interacting pedestrians. The evacuation time of the crowd decreases for $C\in (10^{-1}$ s$^{-1}$, 30 s$^{-1}$), see Fig. 12. However, we observe that, for too low values (i.e., $<$5 s$^{-1}$), the walkers collide too frequently. On the opposite, high enough values of C (i.e., $>$30 s$^{-1}$) result in unrealistically dynamics, as the pedestrians “rebound” one on each other (due to the too high contribution of the contact component of the velocity) thereby barely reaching the target door. Given such considerations, we estimate C equal to 25 s$^{-1}$. The evacuation time has instead a biphasic behavior with respect to variations of D, as it is constant for $D<10$ s$^{-1}$, increases for $D\in (10$ s$^{-1}, 10^{3}$ s$^{-1}$), until reaching a threshold for $D>10^{3}$ s$^{-1}$. In this respect, we opt for the intermediate estimate $D=50$ s$^{-1}$.

The interpersonal repulsion radius is not determined by pedestrian physical characteristics (unless the reasonable conditions $R^{\mathrm{cont}} < R^{\mathrm{rep}} < R^{\mathrm{vis}}$). It in fact defines the minimal distance below which a walker starts to deviate his/her motion to avoid possible collisions with another individual, i.e., it has a sort of psychological nature. In particular, the top-central panel in Fig. 12 shows that variations of $R^{\mathrm{rep}}$ within the above-cited range of values do not significantly affect the simulation outcomes (in terms of evacuation time of the crowd). However, from a phenomenological point of view, it is clear that the comfort interpersonal distance, evaluated by $R^{\mathrm{rep}}$ depends on the specific situation. In this respect, it is consistent to assume $R^{\mathrm{rep}}=1$ m in the case of evacuation scenarios (as panicking individuals tend to remain close one to another) and a higher $R^{\mathrm{rep}}=3$ m in the other cases. The coefficient E does not affect the dynamics of the crowd within a given range of values (i.e., $E < 1$ m/s, see again Fig. 12). However, outside such a set of values, i.e., for E high enough, we observe a blow up in the evacuation time: this is due to the fact that the repulsive component overcomes the target velocity $\mathbf v ^{\mathrm{targ}}$ and therefore the individuals try to keep the desired interpersonal distance rather than to reach the exit door. Given these observations, we opt for $E=1$ m/s. On the opposite, variations of F do not play a critical role in walker behavior (refer to Fig. 12 (bottom-central graph)): indeed, an intermediate $F=0.5$ m is chosen.

Fig. 12

Effect of variations of selected model parameters. The graphs are obtained by monitoring the evacuation time in the case of the “two-adjacent-rooms-without-column” simulation setting. In all cases the initial pedestrian distribution is the same. Within each panel, we identify the parameter estimate used in the previous sections

Variation of the gazing evolution parameter The estimate of coefficient G is inferred by qualitative and physical observations: too low values of G (i.e., $<$ 1 (rad s)/m) generate unacceptable extremely rapid and uncontrollable rotations of pedestrian’s head (not shown). On the opposite, too high values (i.e., $>$3 (rad s)/m) obviously force each walker to perfectly align his/her gaze to his/her direction of motion, which is unrealistic. We indeed set an intermediate $G = 2$ (rad s)/m.

Variation of the number of pedestrians The number of pedestrian dramatically affects the dynamics of the system. As reproduced in Fig. 12 (bottom-right panel), aside slight perturbations due to numerical effects, the evacuation time is in fact directly proportional to the number of individuals N. In this respect, $N=200$ corresponds to a value that allows to have a reasonable pedestrian density, i.e., of 1 individual/m$^2$, in the case of the domain setting used for this parameter sensitivity. The same considerations hold for the number of walker used in the simulations presented in the previous sections.

Summing up the sensitivity analysis proposed in this section, it is possible to conclude that pedestrian behavior strongly depends both on the extension and on the dynamics of the visual region (which are defined by the visual depth/angle and by the coefficient G, respectively). However, empirical considerations allow us to infer a consistent estimate of the values of such parameters. On the opposite, the wall repulsion radius, as well as the interpersonal repulsion distance, does not affect the model outcomes. The rationale is that the corresponding velocity contributions are negative exponential functions: indeed, rather than the point at which a pedestrian starts to experience interpersonal interactions, it is fundamental the exact form of the exponential laws, given by the relative coefficients (A, B, C, D, E, F). However, physical observations help us to provide a realistic set also of these parameters. Finally, the overall simulation results are obviously affected by the total number of individuals taken into account, that obviously determines the evacuation time of the overall crowd. In this respect, it is necessary to account a realistic density of pedestrians for unit of area (i.e., $\approx$1 m$^{-2}$).