# Microscopic dynamics at the Running of the Bulls (San Fermín Festival) in
the context of the Social Force Model

F.E. Cornes  G.A. Frank  C.O. Dorso  Departamento de Física, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad
Universitaria, 1428 Buenos Aires, Argentina.  Instituto de Física de Buenos
Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.  Unidad
de Investigación y Desarrollo de las Ingenierías, Universidad Tecnológica
Nacional, Facultad Regional Buenos Aires, Av. Medrano 951, 1179 Buenos Aires,
Argentina.

###### Abstract

This research explores the dynamics of the emergency evacuation during the
“Running of the Bulls” festival (Spain, 2013). As people run to escape from
danger, many pedestrians stumble and fall down, while others will try to pass
over them. We carefully examined three specific recordings of the running,
that show this kind of behavior. We developed a microscopic model mimicking
the stumbling mechanism in the context of the Social Force Model (SFM). In our
model, “moving” individuals can suddenly switch to a “fallen” state when they
are in the vicinity of a fallen individual. We arrived to the conclusion that
the presence of a fallen pedestrian increases dramatically the falling
probability of the pedestrians nearby. Also, the product between the local
density gradient and the velocity of each pedestrian appears as a relevant
indicator for an imminent fall. We call this the pedestrian “falling
susceptibility ($f_{s}$)”.

###### keywords:

Pedestrian dynamics , Social Force Model , Stumbling mechanism

###### PACS:

45.70.Vn , 89.65.Lm

##  1 Introduction

The matter of emergency evacuation is of obvious importance in common life. In
the last decades, a growing interest appeared in the research of the
pedestrian dynamics during an emergency evacuation. The proper understanding
of the evacuation dynamics will allow safer facilities when designing common
spaces.

The evacuation through narrow pathways or doorways appears as one of the most
studied scenarios in the literature. Many experiments and numerical
simulations have been carried out for a better understanding of the crowd
behavior in a variety of situations. These investigations focus on the door
width as the main cause of the overcrowding during the evacuation process [1,
2, 3, 4, 5, 6, 7, 8, 9]. Therefore, it is pointed as the key magnitude
affecting the evacuation performance.

The dynamic of emergency evacuations through narrow doors is dominated by
“blockings”. Research has shown that a small group of pedestrians close to the
door can be responsible for blocking the way to the rest of the crowd by
forming an arch-like metastable structure, or blocking cluster, yielding long
lasting delays [10, 11, 12, 13]. Also, the pressure inside the bulk is
considered to be of decisive importance in the breaking process of the
blocking clusters [14, 10, 12, 13, 15, 16, 17, 18, 19, 20].

The pedestrian’s anxiety [13], the presence of obstacles [21] and the
possibility of fallings [22] are all relevant issues that affect the formation
of blocking clusters in the context of narrow doors (bottlenecks). However,
blocking clusters are expected to become irrelevant out of this context [10],
and therefore, the connection between the aforementioned issues and the
evacuation performance appears somehow obscure.

The emergency evacuations in the absence of blocking clusters are, from our
point of view, an unattended matter throughout the literature. Thus, we
decided to place the attention on those situations where the blocking clusters
are quite improbable (i.e. wide openings), but fallings are still a relevant
issue. Specifically, our investigation focuses on three real-life situations
(empirical approach) of the “Running of the Bulls” event (2013). The videos
capture the instances where participants stumble and fall during the rush in
different wide door scenarios [23, 24, 25].

Our first objective is to acquire reliable data from real stumbling events,
and use this data to build a formalism to recognize situations in which
tripping becomes highly probable within the context of the SFM. We assume that
pedestrians behave as either “moving” or “fallen” individuals. The transition
probabilities from the “moving” behavior to the “fallen” one corresponds to
those from the video recordings. No transition from “fallen” to “moving” is
permitted under this approach.

We emphasize the critical attitude that we sustained along the investigation.
Our SFM model was tested before proceeding to other hypothetical scenarios.
Results showed fairly good agreement with the analyzed videos.

The investigation is organized as follows. A brief review of the basic “Social
Force Model” can be found in Section 2. In Section 3 we present the real-life
situations considered in our investigation, while in Section 4 we provide
details on the numerical simulations. Section 5 present the results from both
the empirical and numerical analyses. The corresponding conclusions are
summarized in Section 6.

##  2 Background

###  2.1 The social force model

This investigation handles the pedestrians dynamics in the context of the
“Social Force Model” (SFM) [14]. The SFM exploits the idea that human motion
depends on the people’s own desire to reach a certain destination at a given
velocity, as well as other environmental factors [26]. The former is modeled
by a force called the “desire force”, while the latter is represented by
“social forces” and “granular forces”. These forces enter the equation of
motion as follows

$m_{i}\,\displaystyle\frac{d\mathbf{v}^{(i)}}{dt}=\mathbf{f}_{d}^{(i)}+\displaystyle\sum_{j=1}^{N}\displaystyle\mathbf{f}_{s}^{(ij)}+\displaystyle\sum_{j=1}^{N}\mathbf{f}_{g}^{(ij)}$
(1)

where the $i,j$ subscripts correspond to any two pedestrians in the crowd.
$\mathbf{v}^{(i)}(t)$ means the current velocity of the pedestrian $(i)$,
while $\mathbf{f}_{d}$ and $\mathbf{f}_{s}$ correspond to the “desired force”
and the “social force”, respectively. $\mathbf{f}_{g}$ is the friction or
granular force.

The $\mathbf{f}_{d}$ describes the pedestrians own desire to reach a specific
target position at the desired velocity $v_{d}$. But, due to environmental
factors (i.e. obstacles, visibility), he (she) actually moves at the current
velocity $\mathbf{v}^{(i)}(t)$. Thus, he (she) will accelerate (or decelerate)
to reach any desired velocity $v_{d}$ that will make him (her) feel more
comfortable. Thus, in the Social Force Model, the desired force reads [14]

$\mathbf{f}_{d}^{(i)}(t)=m_{i}\,\displaystyle\frac{v_{d}^{(i)}\,\mathbf{e}_{d}^{(i)}(t)-\mathbf{v}^{(i)}(t)}{\tau}$
(2)

where $m_{i}$ is the mass of the pedestrian $i$ and $\tau$ represents the
relaxation time needed to reach the desired velocity. $\mathbf{e}_{d}$ is the
unit vector pointing to the target position. Detailed values for $m_{i}$ and
$\tau$ can be found in Refs. [14, 21, 27, 28].

The “social force” $\mathbf{f}_{s}(t)$ represents the socio-psychological
tendency of the pedestrians to preserve their _private sphere_. The spatial
preservation means that a repulsive feeling exists between two neighboring
pedestrians, or, between the pedestrian and the walls [14, 26]. This repulsive
feeling becomes stronger as people get closer to each other (or to the walls).
Thus, in the context of the Social Force Model, this tendency is expressed as

$\mathbf{f}_{s}^{(ij)}=A_{i}\,e^{(r_{ij}-d_{ij})/B_{i}}\mathbf{n}_{ij}$ (3)

where $(ij)$ corresponds to any two pedestrians, or to the pedestrian-wall
interaction. $A_{i}$ and $B_{i}$ are two fixed parameters (see Ref. [10]). The
distance $r_{ij}=r_{i}+r_{j}$ is the sum of the pedestrians radius, while
$d_{ij}$ is the distance between the center of mass of the pedestrians $i$ and
$j$. $\mathbf{n}_{ij}$ means the unit vector in the $\vec{ji}$ direction. For
the case of repulsive feelings with the walls, $d_{ij}$ corresponds to the
shortest distance between the pedestrian and the wall, while $r_{ij}=r_{i}$
(see Refs. [14, 26]).

It is worth mentioning that the Eq. (3) is also valid if two pedestrians are
in contact (i.e. $r_{ij}>d_{ij}$), but its meaning is somehow different. In
this case, $\mathbf{f}_{s}$ represents a body repulsion, as explained in Ref.
[22].

The granular force $\mathbf{f}_{g}$ included in Eq. (1) corresponds to the
sliding friction between pedestrians in contact, or, between pedestrians in
contact with the walls. The expression for this force is

$\mathbf{f}_{g}^{(ij)}=\kappa\,(r_{ij}-d_{ij})\,\Theta(r_{ij}-d_{ij})\,\Delta\mathbf{v}^{(ij)}\cdot\mathbf{t}_{ij}$
(4)

where $\kappa$ is a fixed parameter. The function $\Theta(r_{ij}-d_{ij})$ is
zero when its argument is negative (that is, $r_{ij}<d_{ij}$) and equals unity
for any other case (Heaviside function).
$\Delta\mathbf{v}^{(ij)}\cdot\mathbf{t}_{ij}$ represents the difference
between the tangential velocities of the sliding bodies (or between the
individual and the walls). Detailed values for all the parameters ($m_{i}$,
$\tau$, $A_{i}$, $B_{i}$, $\kappa$) can be found in Refs. [14, 27, 28].

###  2.2 Clusters

Many previous works showed that clusters of pedestrians play a fundamental
role during the evacuation process [10, 11, 13]. In this sense, researchers
demonstrated that there exist a relation between them and the clogging up of
people. Clusters of pedestrians can be defined as the set of individuals that
for any member of the group (say, $i$) there exists at least another member
belonging to the same group ($j$) in contact with the former. Thus, we define
a human cluster ($C_{g}$) following the mathematical formula given in Ref.
[29].

$C_{g}:p_{i}~{}\epsilon~{}C_{g}\Leftrightarrow\exists~{}p_{j}~{}\epsilon~{}C_{g}/r_{ij}<(R_{i}+R_{j})$
(5)

where ($p_{i}$) indicate the ith pedestrian and $R_{i}$ is his (her) radius
(shoulder width). This means that $C_{g}$ is a set of pedestrians that
interact not only with the social force, but also with physical forces (i.e.
friction force).

##  3 Experimental data

This investigation focuses on real life videos of human rush incidents. We
fully analyzed three specific episodes of the “Running of the Bulls” festival
in San Fermín (Spain), as occurred in 2013. The videos are freely available in
YouTube [23, 24, 25] and show a crowd of pedestrians running away from bulls.

The bull-run takes place along the streets of the city of Pamplona (Spain).
Participants escape from the bulls and some of them stumble and fall during
the rush. Fig. 1 captures three instances of the running when, at least, one
pedestrian stumbles and falls. Each row in Fig. 1 corresponds to a falling
episode. The snapshots on each row capture the time sequence of the falling
event (see captions for details).

For convenience, we will refer to each episode in Fig. 1 as the “pre-corridor
event” [23], the “corridor event” [24] and the “post-corridor event” [25]. The
latter occurs at the entrance of the bull-ring, but still in the corridor
limits.

Pre-corridor scenario
---
|  |
a (t=0 s) | b (t=1.5 s) | c (t=2.5 s)
Corridor scenario
|  |
d (t=5 s) | e (t=20 s) | f (t=30 s)
Post-Corridor scenario
|  |
g (t=10 s) | h (t=25 s) | i (t=30 s)

Figure 1: (Color on-line only) (Upper) Layout of the street and locations
where the incidents took place. Pre-corridor event (first row), corridor event
(second row) and post-corridor event (third row). The snapshots address the
time line of the event. The corresponding time-stamp is indicated below the
snapshots. The yellow and red circles indicate the region where the first
pedestrian stumbles and where the stumbling pedestrians block the exit,
respectively. In the last picture (lower right) many pedestrians pile-up (out
of the scene).

The following are the feature actions taking place in the events:

  * 1.

Pre-corridor: One pedestrian stumbles and falls during the rush. Few seconds
after, he (she) manages to get up and continues his (her) way out. Neighboring
pedestrians run without major incidents.

  * 2.

Corridor: Three pedestrians stumble and fall during the event. Notice that,
unlike the pre-corridor event, they stumble in a higher density scenario.
Fortunately, they get up quickly and continue their escaping route.

  * 3.

Post-corridor: A cluster of more than forty pedestrians stumble and fall
during the rush. The video shows that the majority of falls are correlated in
both space and time. Unlike the other two events, the first fallen pedestrian
is unable to stand up quickly, triggering a massive stumble. Consequently, the
crowd evolved towards a completely block scenario (see red circle in Fig. 1).

The very first inspection of the events brings out the fact that the local
density around the first falling pedestrian is a crucial feature to the
forthgoing events. That is, the higher the local density, the greater tendency
towards a massive stumble. We will leave the above matter to Section 5.1.1 and
for now we will focus on the data acquisition procedure (see Section 3.1). We
will further introduce some definitions in Section 3.2.

###  3.1 Data acquisition

In order to quantify the three incidents exhibited in Fig. 1, we sampled the
movies at a rate of 8 frames per second (attaining a time interval between
successive images of 0.125 seconds). The criteria for choosing this sampling
period was: 1) The period should be large enough to include (at least) one
fallen individual and to achieve a reasonable amount of measurements, 2)
during this period, the camera must be fixed to perform the right perspective
correction.

The total number of acquired images for the pre-corridor event was 32, while
the other two events attained 240 images each. This allowed a semi-manual
tracking of the pedestrians by means of the ImageJ software [30]. We made sure
that any pedestrian trajectory included at least 10 data points.

###  3.2 Definitions of relevant magnitudes

The following are the most meaningful magnitudes in the context of our
investigation.

  * 1.

The local density at any position $\vec{r}=(x,y)$ and time $t$ was computed as
follows

$\rho_{i}(\vec{r},t)=\frac{N_{i}(\vec{r},t)}{\pi R^{2}}$ (6)

where $N_{i}(\vec{r},t)$ represents the number of neighbors of pedestrian $i$
within a fix radius $R=1\,$m (see Fig. 2).

  * 2.

The instantaneous speed (modulus) for each runner was computed as the
symmetric incremental ratio

$v_{i}(t)=\frac{|\vec{r_{i}}(t+\Delta t)-\vec{r_{i}}(t-\Delta t)|}{2\Delta t}$
(7)

for $\Delta t=0.125\,$s,

  * 3.

The local density gradient around a pedestrian $i$ (modulus) was defined as
follows

$\nabla\rho_{i}(t)=|N_{f}(t)-N_{b}(t)|$ (8)

where $N_{f}$ and $N_{b}$ represent the number of moving pedestrians located
in the forward and backward directions, respectively (see Fig. 2 for details).
We only considered moving pedestrians for the gradient computation. That is,
fallen individuals were excluded from the computation. The $N_{f}$ and $N_{b}$
counts only considered pedestrians within the distance $R=1\,$m to pedestrian
$i$ (see Fig. 2).

  * 4.

We defined the product

$f_{s,i}=v_{i}\nabla\rho_{i}$ (9)

as a “susceptibility” to the fall. This “susceptibility” accounts for the
compound nature of the falling: the pedestrian’s own speed ($v_{i}$) and the
pushing environment ($\nabla\rho_{i}$).

The sampling area was located in the middle of the pathway, as depicted in
Fig. 4a. As already mentioned, each magnitude was recorded every $0.125\,$s.

(a)

(b)

Figure 2:  (Color on-line only) Schematic diagram for the individuals in the
corridor. The circles represent pedestrians moving from left to right. The
upper and lower lines represent the walls of the corridor. The dashed circle
corresponds to the measurement area around pedestrian $i$. In this case, he
(she) is surrounded by five individuals ($N_{i}=5$). Therefore, the local
density is $\rho=5/\pi$ (p/m${}^{2})$ (see Eq. 6). Also, notice that three of
these neighbors are located backward with respect to pedestrian $i$ (orange
circles), while the other two are placed forward (blue circles). Therefore,
the local gradient density around pedestrian $i$ is $\nabla\rho_{i}=1$ (see
Eq. 8).

##  4 Numerical simulations

###  4.1 Boundary and initial conditions

In order to study this process, we recreated the “Running of the bulls” event
(see Section 3) by placing from 100 up to 700 pedestrians inside a straight
corridor of width $w=5\,$m (similar to the width at the entrance of the bull-
ring) and length $L=50\,$m. The global density ranged from $\rho=0.5$ to 3
p/m2. This is similar to the local density values measured in the analyzed
scenes of the San Fermín festival (see Fig. 4).

The individuals were initially distributed at random positions along the
simulation box, with random initial velocities according to a Gaussian
distribution with null mean value. If certain conditions were met (see Section
4.2), any moving pedestrian could switch his (her) behavior to the “fallen”
pedestrian behavior. Fallen pedestrians were those that remained at a fix
position until the end of the simulation process.

The desired velocity $v_{d}$ was the same for all the individuals, meaning
that all of them had the same anxiety level. The desired direction
$\mathbf{\hat{e}_{d}}$ was updated at each time step in order to point to the
exit (say, the entrance of the bull-ring) as depicted in Fig. 3.

Figure 3: (Color on-line only) Snapshot of simulated evacuation process
through a $5\,$m $\times\,50\,$m straight corridor. $\mathbf{\hat{e}_{d}}$
points directly to the exit (say, the entrance of the bull-ring at the bottom
of this scheme). All the individuals correspond to “moving pedestrians”. The
left and right yellow lines represent the walls of the corridor.

###  4.2 Moving and fallen pedestrians

We assumed two behavioral patterns, as already mentioned: “moving” individuals
and “fallen” individuals. The former are those that move according to Eq. (1).
The latter are those that are not able to move at all until the end of the
evacuation process. Moving pedestrians, however, are able to switch to the
fallen behavior, but fallen pedestrians always remain in the same condition.

The falling transition was implemented as follows. First, we computed the
“fallen susceptibility” (say, $f_{s}^{(i)}=v^{(i)}\nabla\rho^{(i)}$) for each
moving pedestrian, at instance $i$ (see Eq. 9). Secondly, we switched the
state of each “moving” individual to the “fallen” category with the following
probabilities

$p(f_{s}^{(i)})=\left\\{\begin{matrix}p_{F}(f_{s}^{(i)})&\mathrm{if~{}there~{}is~{}a~{}fallen~{}individual~{}nearby}\\\
p_{NF}(f_{s}^{(i)})&\mathrm{if~{}there~{}is~{}not~{}a~{}fallen~{}individual~{}nearby}\end{matrix}\right.$
(10)

where fallen neighbors are considered to be within a radius of 1 m. That is,
we considered those fallen neighbors located backward and forward the
pedestrian $i$. Notice that $p_{F}$ and $p_{NF}$ depend on the susceptibility
$f_{s}^{(i)}$, and are obtained from the fitting of experimental data (see
Section. 5.1.2 for details on the computation of these probabilities). The
time-step between instances was 0.5 s.

The moving pedestrians were allowed to pass through the fallen ones. In order
to make this possible, we assumed that no social repulsion was present between
the moving and fallen individuals. See Ref. [22] for details.

###  4.3 Simulation software

The simulations were implemented on the Lammps molecular dynamics simulator
[31]. Lammps was set to run on multiple processors. The chosen time
integration scheme was the velocity Verlet algorithm with a time step of
$10^{-4}\,$s. Any other parameter was the same as in previous works (see Refs.
[21, 32]).

We performed between 30 and 100 simulations for each scenario in order to get
enough data for statistical analyses (see figures caption for details). Data
was sampled at time intervals of 0.05 s along 100 s. The recorded magnitudes
were the pedestrian’s positions and velocities for each evacuation process.

##  5 Results

We present in this section the main results of our investigation. The section
is divided into two parts. In the first part (Section 5.1), we present the
empirical results extracted from each incident. In the second part (Section
5.2), we compare the empirical results with numerical simulations.

###  5.1 Empirical results

The analysis of the videos is divided into two separate Sections. The first
one examines the crowd behavior previous to the first fallen pedestrian
(within the sight of the film), while the second one focuses on the dynamics
following the fall. These are two naturally different stages, since the former
corresponds to an homogeneous crowds, while the latter deals with two
behavioral patterns (i.e. with “moving” and “fallen” pedestrians).

####  5.1.1 Before the first fall

The analysis started by slicing the scenes shown in Fig. 1. We made the
corresponding perspective corrections and computed the magnitudes defined
previously in Section 3.2. Fig. 4a schemes the procedure for the computation
of each observable.

(a) Measuring diagram

(b) Pre-corridor

(c) Corridor

(d) Post-Corridor

Figure 4: (Color on-line only) (a) Schematic diagram for individuals during
the running. In order to minimize boundary effects, the measurements were
taken from the middle of each scenario (gray box). This box was binned in
cells of size $3\,\mathrm{m}\times 0.5\,\mathrm{m}$. (b-c-d) Density contour
lines along time for each scenario. The maps were splined to get smoother
curves. Each cell of size $0.5\,\mathrm{s}\times 0.5\,\mathrm{m}$ corresponds
to the occupancy density at time $t$ in position $x$ (see the measuring
diagram). The scale bar on the right is expressed in pedestrians/m2. The
vertical red dashed line represents the time of the first fall. The letters
above each plot mean the corresponding snapshots in Fig. 1. The horizontal and
vertical axis represent the time and the pedestrian’s $x-$location in each
corridors, respectively. The walls are located at $x=0$ and $x=5$ (bottom and
top of each figure).

Figs. 4b-4d show the local density ($\rho$) pattern along time for each
scenario (see caption for details). We can notice a common behavior from the
contour maps in Fig. 4: the first fall occurs _after_ an increase in the local
density. Or, in other words, an increase in the density somehow anticipates a
forthcoming fall. This does not mean that _any_ increment in the density
naturally yields a falling incident, but sets an environmental condition for
expecting it (in the context of the San Fermín festival).

A closer inspection of the scenes _before_ the falls shows that the natural
tendency of the pedestrians is to preserve some distance between them while
running, presumably because of trying to dodge other runners. However, the
upcoming bulls disturb this tendency towards a more chaotic scenario.
Consequently, there is no time left for trying to dodge others, and the
density increases.

In order to quantify the above observation, we measured the pedestrians’
velocity ($v_{i}$) and the local density gradient ($\nabla\rho$), as defined
in Section 3.2. Both magnitudes and some comments on its time evolution can be
found in A. However, we actually prefer to focus on the susceptibility $f_{s}$
as defined in Section 3.2. This is, from our point of view, a more meaningful
magnitude to express the disturbed scenario.

Fig. 5 shows the susceptibility histograms as a function of time for each
scenario (see caption for details). Likewise the local density patterns, we
can notice a common behavior from the histograms: the first fall occurs
_after_ an increase of the number of pedestrians exposed to high
susceptibility values (see white circles in Fig. 5).

Notice that the white circles in Fig. 5 do not enclose a large amount of
events (i.e. many counts) but those that are significant to the falling
process. This means that only a few pedestrians attain the susceptibility
level to (probably) produce a falling incident.

The susceptibility appears as a better indicator for an imminent fall with
respect to the local density. Fig. 5 provides a rough threshold (say,
$f_{s}\approx 4$) pointing to an imminent fall. The increase in the local
density stands for a likely falling condition, although we can not set any
clear threshold for the San Fermín context.

(a) Pre-corridor

(b) Corridor

(c) Post-Corridor

Figure 5:  (Color on-line only) 2-D histograms for the falling susceptibility
($v\nabla\rho$) along time for each scenario. The scale bar on the right
corresponds to the number of pedestrians exposed to $v\nabla\rho$ ($y$-axis)
at time $t$ ($x$-axis). The bin size was $0.125\,\mathrm{s}\times
1\,\mathrm{people~{}m/s}$. The vertical red dashed line represents the time-
stamp of the first fall. The horizontal and vertical axis represent the time
and the falling susceptibility, respectively. For clarity reasons, the upper
limit of each color bar is different. The letters above each plot mean the
corresponding snapshots in Fig. 1.

####  5.1.2 After the first fall

We will now focus on the “fallen” pedestrians. We first computed the number of
stumbles in presence (or not) of fallen neighbors as a function of time. This
is shown in Fig. 6 for the post-corridor scenario only. The pre-corridor and
corridor scenes are not shown because the scarcity of data did not allow a
reliable analysis.

As can be seen in Fig. 6, near 40 pedestrians stumble and fall in the vicinity
of, at least, one fallen neighbor at the end of the process. But this number
barely reaches 10 individuals in the absence of a neighboring fallen
pedestrian. Thus, it becomes clear that the presence of at least one fallen
neighbor increases dramatically the ratio of casualties along time.

Figure 6:  (Color on-line only) Empirical number of fallen pedestrians with
(red circles) and without (green squares) fallen neighbors as a function of
time in the post-corridor scene.

We further examined the number of fallen neighbors around any new falling
pedestrian. The sample probabilities are exhibited in Fig. 7. We estimate a
sampling error of $\sigma\approx 0.2$ for each bin (see caption for details).
This means that the case of only one fallen neighbor is actually the
significant one. More than one fallen neighbors may occur, but its relevancy
is still in question according to our data.

Figure 7:  (Color on-line only) Normalized cumulative number of fallen
pedestrians as a function of the number of fallen neighbors in the post-
corridor scenario. The count of fallen neighbors was done within a fix radius
$R=1\,$m. The diagrams above the bins are schematic representations for each
case. The dashed circle corresponds to the measurement area around pedestrian
$i$. The plot is normalized with respect to the total number of fallen
pedestrians. The mean number of fallen individuals in this sample distribution
is $\mu=1.83$. The bin error is expected to be $\sigma=\sqrt{p\,(1-p)/6}$ for
$p\approx\mu/6$.

The fact that fallen neighbors increase dramatically the possibility of new
fallings turned our investigation to the following working hypothesis: the
presence of “fallen neighbors” increases the chance of falling, in addition to
the expectation for susceptible (moving) individuals alone. This means that
those environment attaining “fallen individuals” should be studied separately
from the “non-fallen” ones.

We first computed separately the falling probabilities for individuals that
belong to an environment with, at least, one fallen neighbor, and with no
fallen neighbors at all. We studied both situations as a function of the
susceptibility $f_{s}$. The following is the expression used for computing
both falling probabilities ($p_{F}$ and $p_{NF}$, see Eq. 10):

$p(f_{s})=\frac{\displaystyle\sum_{t=0}^{T}N_{F}(f_{s})}{\displaystyle\sum_{t=0}^{T}N_{T}(f_{s})}\\\
$ (11)

where $N_{F}$ and $N_{T}$ represent the number of falling pedestrians and the
total number of pedestrians (either fallen and moving ones) attaining a
falling susceptibility $f_{s}$, respectively. Both sums run over all the
discrete time samples (see Section 3.1 for details). Notice that the value of
$f_{s}=v\nabla\rho$ was always sampled just before the fall occurs.

Figure 8:  (Color on-line only) Falling probability as a function of the
pedestrian’s susceptibility ($f_{s}=v\nabla\rho$) with (red circles) and
without (green squares) neighboring fallen pedestrians (see Eq. 11). Solid
lines correspond to the quadratic fitting of each data set:
$p_{F}(s)=0.33\,f_{s}^{2}-0.025\,f_{s}+0.229$ (red curve) and
$p_{NF}(s)=0.006\,f_{s}^{2}-0.011\,f_{s}+0.001$ (green curve).

Fig. 8 shows the falling probability as a function of the susceptibility
($f_{s}$) in the post-corridor scene (see caption for details). It can be seen
that the falling probability fits accurately a quadratic function with respect
to $f_{s}$ in both environmental conditions (say, with and without fallen
neighbors). Moreover, Fig. 8 is in agreement with the rough threshold
($f_{s}\approx 4$) observed in Fig. 5 for an imminent falling. We can now
confirm that the probability of falling increases rapidly beyond this value.

Interestingly, both curves exhibit a similar qualitative shape regardless of
the environmental condition. The presence of fallen neighbors, however,
introduces, at least, an additional $20\%$ of chances for the new fall to
occur. This additional probability due to a fallen neighbor increases as
$f_{s}$ becomes larger, as can be noticed from the comparison of the slopes in
Fig. 8.

#### Concluding remarks from the empirical results

Our major conclusions from this Section are as follows. First, two different
causes of stumbles can be noticed during the rush: a few individuals fall as a
consequence of their interaction between other moving pedestrians only.
However, the majority stumble in presence of, at least, another fallen
pedestrian.

As a second conclusion, we noticed that the falling probability increases as
the susceptibility $f_{s}=v\nabla\rho$ (say, the product between the local
density gradient and the velocity of each pedestrian) increases. Thus,
fallings are more likely to occur whenever people rush away from danger (_i.e_
the bulls) or embody a striking situation (_i.e_ high density gradient). These
are the commonly observed situations in the San Fermín festival.

###  5.2 Numerical results

In this Section we widen the empirical picture of the San Fermín festival by
means of numerical simulations. Our first aim is to mimic this situation
within the context of the SFM (see Section 2 for details). We introduce
falling probabilities to the basic SFM, according to estimates from Section
5.1. We then investigate possible scenarios according to the SFM.

####  5.2.1 The morphology in the bulk of the crowd

Fig. 9 exhibits a few snapshots of the corridor simulations for different
crowd densities. The moving direction is from from top to bottom. The corridor
width is similar to the one in San Fermín (see caption for details). The
overlapping individuals are actually the ones passing through fallen
pedestrians.

$v_{d}=3\,$m/s
---
|
$\rho~{}=~{}1\,$p/m2 |  $\rho~{}=~{}2\,$p/m2

$v_{d}=5\,$m/s
---
|
$\rho~{}=~{}1\,$p/m2 |  $\rho~{}=~{}2\,$p/m2

Figure 9: (Color on-line only) Snapshots of different evacuation processes for
two desired velocities and two densities at the time-stamp of 15 s. Moving and
fallen pedestrians are represented in green and red circles, respectively. The
moving direction is from top to bottom. The yellow lines represent the walls
of the pathway. The corridor width is $5\,$m.

The first two columns in Fig. 9 correspond to the lowest value of the explored
anxiety levels ($v_{d}=3\,$m/s). We can see that no fallings occur for
$\rho=~{}1~{}\,$p/m2, but a unique cluster of fallen pedestrians appears for
$\rho~{}=~{}2\,$p/m2. An inspection of the animations shows that the first
fallen runner triggers the falling of neighboring runners. This is in
agreement with the falling events reported in the videos, and confirms once
more, the relevancy of the first falling.

Interesting, there is an increase in the number of isolated (i.e not in
contact) human clusters of fallen pedestrians for the more stressing scenario
(last two columns of Fig. 9). This is also in agreement with the empirical
patterns exhibited in Fig. 1. Furthermore, the complete video recordings show
that stressed pedestrians tend to rush chaotically, and thus, the probability
of fallings increases (see Fig. 8).

The fallings shown in Fig. 9 appear somehow related to the overall density of
the crowd and the degree of clusterization during the run. Both issues will be
studied separately in the next sections.

####  5.2.2 Number of fallen pedestrians versus the density

Fig 10 shows the number of stumbles occurring during the first 35 s of the
escaping process as a function of the global density (see caption for
details). We analyzed two desired velocities ($v_{d}=3\,$m/s and
$v_{d}=5\,$m/s) since these appear as quite reasonable values according to
data in A.

Figure 10: (Color on-line only) Normalized number of fallen pedestrians as a
function of the initial density for $v_{d}=3\,$m/s (blue circles) and
$v_{d}=5\,$m/s (red squares) during the first 35 s (same corridor as in Fig.
9). Initially, pedestrians were randomly distributed along the simulated
corridor. Mean values were computed along 50 realizations. The plot is
normalized with respect to the total number of simulated pedestrians. The
error bars corresponds to $\pm\sigma$ (one standard deviation).

We can see in Fig. 10 that the number of fallen pedestrians increases
monotonically with density. This is an expected result since the more crowded
environment, the more strikes between runners, and consequently, the more
chances to fall down. These chances also increase with the desired velocity,
that is, with the anxiety level of the individuals. Notice that these results
are in agreement with in situ videos (see Section 3). Specifically, the
snapshots in Fig. 1 show that the stumbles commonly occur in the high density
scenario (i.e. in the post-corridor scenario).

We should call the attention on the fact that Fig. 10 computes the fallings
regardless of the clusterization context. This opens the question on whether
the clusters formation tend to increase the number of fallings or not. We turn
to this point in the following Section.

####  5.2.3 Clusters

We now turn to study the clustering structures, as defined in Section 2.2. We
classified the human clusters (of fallen pedestrians) into three categories
(according to their size $n$) as follows: small ($1\leq n\leq 5$), medium
($5<n<25$) and big ($n\geq 25$). Fig. 11 shows the mean multiplicity of human
clusters for each category (see caption for details).

A very first inspection of Fig. 11 shows that small human clusters have a
major role along the whole process (for $v_{d}=5$), while its role at the
early stage of the processes appears as essential.

A closer inspection of Fig. 11 shows the following:

  * 1.

Small human clusters ($1\leq n\leq 5$): The occurrence of small human clusters
is greater in the higher stress scenario ($v_{d}=5\,$m/s) than in the lower
one ($v_{d}=3\,$m/s), as depicted previously in Fig. 9. This means that, the
higher the anxiety level, the most likely that (non contacting) pedestrians
can initiate a falling process. This is a very dangerous environmental
condition since any pedestrian becomes a potential instance of falling (for
himself/herself and others).

  * 2.

Medium human clusters ($5<n<25$): The higher the anxiety level, the more
likely to occur. This is somehow engaged to the existence of small clusters
since the latter may increase its size (due to new fallings) as the process
evolves. In other words, an examination of the process animations shows that
small clusters can merge into a medium cluster.

  * 3.

Big human clusters ($n\geq 25$): A single big cluster can be observed in the
low stress scenario, while six of them occur in the high stress one. As in the
medium human clusters, an inspection of the animations shows that many medium
clusters can merge into bigger one.

Figure 11: (Color on-line only) Mean multiplicity of human clusters of fallen
pedestrians ($n$) for three different size categories (see legend for details)
as a function of time. The size of each human cluster was obtained every
$1\,$s. Data was acquired from 100 simulation processes. Solid and dashed
curves correspond to $v_{d}=3\,$m/s and $v_{d}=5\,$m/s, respectively. The
density was $\rho~{}=~{}2\,$p/m2.

We resume the highly stressing scenario as the one where fallings occur in
many places and fallen clusters of any size are likely to occur. The low
stressing scenario is substantially different since people fall around a
single cluster.

As a concluding remark, the massive stumble in the post-corridor event shown
in Fig. 1 is similar to the above low stressing scenario, attaining a big
fallen cluster. We presume that the runners at this instance were quite tired,
resembling a low anxiety behavior (in the context of the SFM).

####  5.2.4 Size of the largest avalanche

Besides the size of the human clusters, we further examined the size of the
largest avalanche of fallen pedestrians. That is, the maximum number of new
fallen pedestrians (regardless of the falling location). This is, from our
point of view, a meaningful magnitude to express the cluster formation rate.
Fig. 12 shows the histograms of the largest avalanche distribution for two
density levels (see caption for details). We omitted the case
$\rho~{}=~{}1\,$p/m2 because it does not provide enough falling samples for a
proper statistic. We considered in Fig. 12 the densities $\rho~{}=~{}2\,$p/m2
and $\rho~{}=~{}3\,$p/m2.

(a) $\rho~{}=~{}2\,$p/m2

(b) $\rho~{}=~{}3\,$p/m2

Figure 12:  (Color on-line only) Normalized distribution of the largest
avalanche size for two stress levels and two densities (see legend for
details). The bin size is 5 pedestrians. The plot is normalized with respect
to the total number of events. Data was recorded from 100 escaping processes.

We first notice that the mean value of the distribution depends on the
pedestrian’s anxiety level in the low-density situation
($\rho~{}=~{}2\,$p/m2). As can be seen in Fig. 12a, the (maximum) avalanche
events do not involve more than 5 pedestrians for $v_{d}=3\,$m/s, yielding a
narrow distribution. But, in the highly anxious situation of $v_{d}=5\,$m/s,
the corresponding distribution widens to allow up to 15 or 17 new fallings.
This phenomenon appears reasonable since anxious pedestrians rush up to
$v\approx v_{d}$, and thus, they become more susceptible to fallings
($f_{s}\approx v_{d}\,\nabla\rho$).

Besides, Fig. 12b shows that the avalanche events, mainly, involve 10 to 15
pedestrians for $\rho=3\,$p/m2, regardless of the anxiety level. These are
massive events that occur in a quite crowded environment, somewhat similar to
the one shown in Fig. 1 for the post-corridor scene. As already observed in
Section 5.1, these fallings occur because pedestrians embody striking
situations (i.e. high density gradients), regardless of the rushing level.
Recall that high density gradients yield highly susceptible conditions for the
falling.

We stress the fact that varying the anxiety level does not significantly
change the behavior of the largest avalanche distribution. Or, in other words,
the avalanches are mainly controlled by density regardless the anxiety level
in high-density situation.

#### Concluding remarks from the numerical results

The results from the numerical simulations show that highly dense environments
tend to increase the number of fallings. The final number of fallings,
however, is a complex interplay between the crowd density and the pedestrians
anxiety level:

  * 1.

For a moderate environmental density and low anxiety levels
($\rho~{}=~{}2\,$p/m2 and $v_{d}=3\,$m/s, respectively), the fallings are
expected to occur in small avalanches at a single location. Thus, the picture
is of a single cluster of fallen pedestrians.

  * 2.

For a moderate environmental density but high anxiety levels
($\rho~{}=~{}2\,$p/m2 and $v_{d}=5\,$m/s, respectively), the fallings occur at
many locations and can involve a varying number of falling people each.

  * 3.

For more crowded environments (say, $\rho~{}=~{}3\,$p/m2), the anxiety level
looses significance since the pedestrians strike among others, no matter their
escaping desire.

##  6 Conclusions

Our research focused on the microscopic analysis of the stumbling phenomena
during an emergency situation (the Running of th Bulls’ Festival at San
Fermín). We first analyzed three real life videos of the rush and proposed a
stumbling mechanism in the context of the SFM. We then performed ex post
simulations for evaluating the accuracy of the model improvements.

In order to properly understand the stumbling mechanism, we examined the crowd
behavior before and after the first falling event. This allowed us to
distinguish between an homogeneous scenario (with “moving” pedestrians) and an
heterogeneous one (with “moving” and “fallen” pedestrians).

We identified the product between the local density gradient and the velocity
of each pedestrian (named here as susceptibility $f_{s}$) as a good indicator
for an imminent fall. We noticed that $f_{s}$ is monotonically related to the
falling probability. This relation can be found in Section 5.1.

We further noticed through the experimental data that the presence of fallen
pedestrians dramatically increases the falling probability of neighboring
pedestrians. Our computations show that the latter have, at least, an
additional $20\%$ of chances for falling down. Thus, the presence or not of a
fallen neighbor is a crucial feature of pedestrians (local) environment. This
is actually the main achievement of our investigation.

The pedestrian’s own susceptibility and the presence (or not) of fallen
neighbors are the only features allowing a stumble within the context of the
Social Force Model. Although the simplicity of this model, we attained fairly
good agreement with empirical data from the San Fermín Festival.

We performed ex-post simulations of the “Running of the Bulls”. We used
different sets of densities ($\rho$) and anxiety levels ($v_{d}$) for
mimicking three instances of the race (see Section 5.2). Our results
reproduced satisfactorily the corresponding video recordings for adequate
values of $\rho$ and $v_{d}$. We further focused on $v_{d}=3\,$m/s and
$v_{d}=5\,$m/s for a detailed analysis of the stumbling process (see Section
5.2). The conclusions are as follows:

  * 1.

The moderately dense scenarios ($\rho~{}=~{}2\,$p/m2) may evolve quite
differently, according to the anxiety level. If $v_{d}=3\,$m/s (low anxiety
level) a small stumbling process can takes place at a single location. In this
case, we will observe single cluster of fallen pedestrians. However, if
$v_{d}=5\,$m/s (high anxiety level) the stumbling process may take place in
multiple places within the crowd. Indeed, we observe a varying number of
people at falling and clusters of fallen pedestrians among the crowd.

  * 2.

The highly dense scenario ($\rho~{}=~{}3\,$p/m2) may evolve in the same way,
no matter the anxiety level (within our explored range). This way will show up
as a massive avalanche within the crowd.

## Acknowledgments

C.O. Dorso is a main researcher of the National Scientific and Technical
Research Council (spanish: Consejo Nacional de Investigaciones Científicas y
Técnicas - CONICET), Argentina and full professor at Departamento de Física,
Universidad de Buenos Aires. G.A. Frank is a researcher of the CONICET,
Argentina. F.E. Cornes is a PhD Student in Physics. This work was supported by
the Fondo para la investigación científica y tecnológica (FONCYT) grant
Proyecto de investigación científica y tecnológica Number
PICT-2019-2019-01994. G.A. Frank thanks Universidad Tecnológica Nacional (UTN)
for partial support through Grant PID SIUTNBA0006595.

##  Appendix A Velocity and local density gradient histograms

Fig. 13 shows the velocity histograms as a function of time for each scenario
(see caption for details). We observe that, likewise the local density pattern
(see Figs. 4b-4d), the three scenarios present a common behavior: the first
fall occurs _after_ an increase in the number of pedestrians attaining high
velocity (say, $v>2\,$m/s).

A careful inspection of the scenes _before_ the falls shows that some
pedestrians moving with high velocity trigger the run of many other relaxed
individuals. Consequently, this “fear spreading” among the crowd makes the
maximum of the velocity distribution to move towards higher values.

The local density gradient histograms along time for each scenario can be
found in Fig. 13 (see caption for details). Notice that this magnitude carries
information on the local density profile around each pedestrian (see Eq. 8).
That is, the higher the local density gradient, the greater the local density
asymmetry in the front–back direction. And therefore, the higher the force
unbalance on the corresponding pedestrian.

It can be noticed that there is an increment in the occurrence of high local
density gradients (say, $\nabla\rho>1$) few seconds _before_ the first
falling. In this sense, in the pre-corridor and post-corridor scenarios, a
sizable portion of the pedestrians experienced a significant amount of
unbalance in the front-back forces (say, $\nabla\rho\geq 3$). Notice that this
is not significant in the corridor scenario.

Velocity
---
|  |
Pre-corridor | Corridor | Post Corridor
Local density gradient
|  |
Pre-corridor | Corridor | Post Corridor
Figure 13: (Color on-line only) 2-D histograms for the velocity (upper) and
local density gradient (lower) for each scenario. The horizontal and vertical
axis represent the time and the measured value of each magnitude,
respectively. The bin size was $0.125\,\mathrm{s}\times 0.33\,\mathrm{m/s}$
and $0.125\,\mathrm{s}\times 1\,\mathrm{people}$ for the velocity and the
local density gradient, respectively. The vertical red dashed line represents
the time of the first fall. Also, for clarity reasons, the upper limit of each
color bar is different. The crowd states at different moments (see letters
above figures) can be seen in Fig. 1.

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