1. Introduction
The scientific management of crowd-gathering scenarios is a significant issue regarding public safety. Using physics, transportation, and mathematical tools to describe the characteristics of a crowd flow, crowd flow dynamics has been a well-known academic hotspot for several decades. It comprises the processes of establishing a mathematical model of a crowd flow and then carrying out numerical simulations and solving them.
There were some similarities between crowd flows and continuous fluids. Since Swiss mathematician and mechanics scientist Euler proposed a classical assumption for fluid continuity [
1], the irregular thermal motion of molecules in a fluid were simplified as the regular motion of the molecules; i.e., the fluid was considered as a continuous medium. Then, fluids were mainly studied based on a macro-mechanical motion theory of a continuous medium.
In general, a crowd flow consists of pedestrians, with physical gaps between them. From a micro point of view, crowd flows are not continuously distributed, but if the distance between pedestrians and the surrounding environment is defined as a pedestrian area, it is feasible to assume a crowd flow as a continuous pedestrian fluid. Therefore, like the continuum hypothesis in fluid dynamics, the macroscopic quantities of crowd flows are the velocities, densities, and flows, which also conform to the physical laws of mass conservation, momentum conservation, and energy conservation. Henderson first proposed a macroscopic model for pedestrian flow [
2,
3].
.It was shown that the distribution functions for crowd densities and velocities were consistent with the results from the Maxwell–Boltzmann theory, except for significant deviations near the frequency mode of each distribution [
3]. Henderson further proposed a hydrodynamic model of a pedestrian crowd, in which a particle collision was used to describe the interactions between macroscopic pedestrian groups [
3]. Hughes considered pedestrian flow as a continuous fluid medium and used fluid dynamics equations to describe the relationships between the speed, density, and flow rate of pedestrians
. According to references [
4,
5], a typical macro dynamic model for the field of crowd flows is established in the form of Equation (1).
where
represents the horizontal velocity, and
represents the lateral velocity.
Most studies on macro crowd flow used this crowd flow model in Equation (1) as their primary theory base. This model assumed that the behaviors of the crowds were homogeneous, i.e., that crowds make approximately the same responses under the same stimulus; however, this was not sufficiently realistic in practice.
In real pedestrian-gathering scenarios, the panic behavior characteristics of crowd flows were relatively complex. Due to the subjective initiatives of the crowds and external environments, once internal panic disturbances due to emergencies such as pedestrian falls, stampedes, shootings, terrorist attacks, explosions, and fires occur, the stop-and-go phenomena of crowds can cause a chaotic phenomenon of the sudden acceleration and deceleration of pedestrians. In this context, more complex crowd trampling and other accidents often easily occur [
5].
It was significant to keep a pedestrian crowd stable considering that a large-scale crowd (including more than 2000 pedestrians) system consisted of a large number of autonomous individual pedestrians, movement rules, and the scene environment [
6]. As a continuous flow, a large-scale crowd flow also had three typical system states: stability, critical stability, and instability.
Large-scale crowd trample disasters were typical phenomena of crowd instability owing to internal disturbances. Most of the existing literature about crowd stability is based on traffic flow theory and lacks systematic consideration. The panic characteristics of pedestrian self-organization behaviors and crowd-merging flow layouts make crowd flows more complex, while few studies have been conducted on the propagation characteristics of sudden panic disturbances in crowd flows, which motivate us to study this issue further.
The main contributions of this study are the following: (1) we propose a dynamic disturbance model of crowd panic behaviors, considering the anisotropy of crowd movements based on the conservation law of fluid dynamics; (2) the anisotropy property of the disturbance propagation is proven with theoretical derivations and simulation experiments; (3) a crowd stability criterion is put forward under internal disturbances based on Lyapunov theory.
The outline of this paper is as follows: in
Section 2, we investigate the related work with a literature survey; in
Section 3, we propose a dynamic disturbance model of crowd panic behaviors; in
Section 4, we analyze the stability of crowd flow considering disturbances due to panic behaviors; in
Section 5, simulation experiments conducted in the waiting hall of Shanghai Hongqiao Railway Station to validate the proposed DPM are presented.
3. Dynamic Disturbance Model of Crowd Panic Behaviors
In fact, a point in the crowd movement area will be disturbed when sudden changes in some physical environment or sudden changes in pedestrian behavior occur. These cause the crowd flow to gradually change from a stable and orderly state to a chaotic unstable state, and these key points are referred to as perturbation points. According to the formation principle of a crowd-disturbing point, such points can be divided into mutation-disturbing points and release-disturbing points [
30,
31]. Among them, a mutation-disturbing point is a sudden abnormal behavior inside a crowd of pedestrians; it mainly manifests as abnormal postures, expressions, and voices.
Abnormal postures are the most conductive and harmful and include pedestrian speed mutations, pedestrian U-turns, pedestrian falls, disease assistance, group fights, and violent terrorist attacks. These types of abnormalities can easily cause massive disturbing forces in the crowd, thereby disturbing an originally stable crowd flow. A release disturbance point corresponds to external disturbance factors such as insufficient numbers of entrances and exits in crowded places, insufficient numbers of passageways, or evacuation passageways and stairs in inappropriate places, leading to limited crowd flow movements. Release disturbances are generally discussed before the establishment of the scenarios. For public places already in use, sudden pedestrian abnormal behavior disturbances are worthier of attention. Moreover, based on statistics, mutant crowd disturbances are more likely to lead to serious crowd accidents. The disturbances discussed in this paper are mutation disturbance points, that is, they represent disturbance forces suddenly generated in a crowd.
3.1. Dynamic Model of Internal Crowd Disturbance
In time domains, the dynamic behavior of a sudden internal disturbance of a crowd is similar to a sudden white noise disturbance in the crowd movement. Its randomness cannot be completely expressed by deterministic variables but can be expressed by a stochastic process. However, the numerical solutions of stochastic equations and the inherent mathematical complexity of stochastic calculus are challenging problems. Based on a probability density function, a Langevin equation can be used to describe the turbulent velocity of fluid particles, which is modeled as Brownian motion. Initially, this equation was applied to homogeneous and isotropic turbulence. The Brownian motion model was widely used in internal random disturbance modeling.
If the random process with value R satisfies the following properties:
(1) The orbit
is almost necessarily continuous:
(2) For any
and Borel set
where if
, then
is called the standard Brownian motion. The standard Brownian motion satisfies a Gaussian distribution. For the sake of simplicity, this study only considers the disturbances of single pedestrian behaviors, so the probability distribution of the random disturbance intensity can be expressed by a single Gaussian function.
Figure 1a shows the two-dimensional Gaussian distributions of four groups of different expectation values (mu) and standard deviations (sigma).
Figure 1b shows the standard three-dimensional Gaussian distribution when the expectation value is 0 and the standard deviation is 5.
When a crowd moves forward in a crowded state, it will produce a certain “pressure” among the crowd. In this study, the crowd is regarded as fluid, and a crowd flow model is established based on fluid dynamics. Therefore, the pressure among a crowd can be established according to the characteristics of fluid dynamics, as shown in Equation (4) as follows:
Here,
is a pressure factor. In the case of sudden abnormal situations occurring in crowds, such as pedestrian falls and terrorist attacks, a disturbing force will be produced and disturb the balanced crowd flow; this can be directly assumed as the change in “pressure” among the crowd. To construct the dynamic model of a disturbance caused by an abnormal behavior in a pedestrian crowd, we assumed that the disturbance burst point is
, then the crowd pressure at this point is the largest, and the crowd pressure around it decays exponentially in the form of
e, with same character of a Gaussian distribution in
Figure 1. This result showed a single disturbance without damping from the neighboring pedestrians. Therefore, under the influence of random disturbances in the crowd, an equivalent pressure
at
can be defined as follows:
Here is the pedestrian density, is the power of density, is the pressure intensity coefficient of the random disturbance, is the position of the disturbance center, and is the duration of a disturbance burst. Therefore, the equivalent pressure is a comprehensive and dimensionless definition considering the pedestrian density, scenario-based pressure intensity, disturbance propagation time, and two-dimensional layout of observed measurement points.
3.2. Dynamic Disturbance Model of Crowd Panic Behaviors
The Hughes crowd flow model, as shown in Equation (1), is widely used in the study of crowd macro flows. However, the crowd behavior represented by the Hughes model is homogeneous; that is, it assumes that pedestrians make the same responses under the same stimuli. In reality, as affected by individual subjective consciousness, pedestrian movements exhibit evident acceleration inconsistencies. For example, the rear crowd can directly observe the movement changes of the front crowd, and then change their movement characteristics. However, the front crowd cannot see the rear crowd, so the front crowd is hardly disturbed by the rear crowd, reflecting the anisotropy of crowd disturbance propagation. Therefore, in addition to the hyperbolic conservation equation shown in Equation (1), another equation is needed to reflect the anisotropy of crowd disturbance propagation.
Pedestrian acceleration, velocity, and density, as the three basic motion variables, have complex nonlinear couplings, but meet the basic momentum conservation. Therefore, many scholars in fluid mechanics have proposed second equation models based on momentum conservation. Aw and Rascle used a convection derivative to replace the spatial derivative of “pressure” in a second-order model, and developed the Aw–Rascle prediction (AR) model, thereby improving the non-physical characteristics pointed out by Daganzo [
32]. The AR model is a typical anisotropic traffic flow model.
In this study, based on the AR model and the proposed crowd disturbance pressure as shown in Equation (5), a propagation dynamic model for the internal disturbance of a crowd is established, as shown in Equations (6)–(8).
where
represent the horizontal and lateral equilibrium velocities, respectively; these are represented by the steady-state relationship between velocity and density (fundamental diagrams).
is a relaxation factor.
,
are the horizontal and lateral crowd disturbance pressures, as shown in Equations (9) and (10).
3.3. Proof of Anisotropy of Dynamic Disturbance Model
As mentioned above, the anisotropy of pedestrians determines that the propagation of internal disturbances in crowd movement is anisotropic. The proof of the anisotropy of the model is given below.
The propagation dynamic model of an internal disturbance of a crowd can be expressed in a conservation form as shown in Equation (11).
here, the conserved vector,
,
,
and
.
Therefore, the eigenvalues in the horizontal and lateral directions of Equations (6)–(8) are shown in Equation (12).
It is evident that the characteristic velocity of the propagation dynamic model of the internal disturbance of the crowd is less than or equal to the macroscopic fluid velocity . Daganzo indicated that a continuum model with a characteristic velocity greater than the macroscopic fluid velocity is not heterogeneous. Moreover, it is difficult to demonstrate the nonphysical effects of vehicle flows in some cases. Crowd flows based on hydrodynamics have the same properties. The characteristic velocity of the model proposed in this study is not greater than the macroscopic fluid velocity; therefore, the model is anisotropic.
4. Stability Analysis of Crowd Flow Considering Internal Disturbances
4.1. Lyapunov-Based Crowd Stability Analysis
The definition of crowd stability based on Lyapunov is as follows. Assume that the equilibrium flow of the macro crowd flow model is . The disturbance flow is . If the spatial gradient of the disturbance flow is bounded, then the propagation of the crowd flows is stable under the disturbance of , i.e., and at , . is the whole region of the crowd movement scenario. In addition, when and , the propagation of the equilibrium flow is asymptotically stable.
Based on the above definition of crowd stability, this study uses the wavefront theory in fluid dynamics to express the crowd disturbance propagation, as shown in
Figure 2 [
33]. The equilibrium state of the crowd flow
is disturbed suddenly at position
. The wavefront propagation curve of crowd flow is shown in
Figure 2.
If the crowd flow system is stable, then the initial disturbance will not increase in the process of its propagation, and will gradually disappear; that is, the wavefront curve of the disturbance flow is bounded. However, if the crowd flow system is unstable, the disturbed flow wave may increase in the propagation process and eventually form a shock wave or bottleneck in the crowd flow scenario.
4.2. Stability Analysis of Crowd Flow with Internal Disturbance
Based on the wavefront theory, the nonlinear dynamic model of crowd disturbance propagation, that is, Equations (6)–(8), are linearized, and then the stability of crowd flow is analyzed based on Lyapunov.
It is assumed that internal disturbances occur in the equilibrium state
. The initial solutions of Equations (6)–(8) are
and
. First, the solutions of the equations near the wavefront are expanded by the power series, as follows:
Here,
is the location of the wavefront at time
. Because the wavefront is the boundary of the disturbance influence range under the crowd equilibrium state, the eigenvalue method remains valid near the wavefront. Thus, the derivatives of
and Y
can be obtained using the eigenvalues, as follows:
According to Equation (13), we can obtain the power series expansions of
,
, and
as follows:
where
,
,
,
. Therefore, the partial derivatives of
, and
are as follows:
Similarly, the partial derivatives of the pressure term
and equilibrium velocities
are obtained as follows:
here,
,
,
,
,
,
,
,
, and
,
.
By substituting Equations (16)–(29) into Equations (6)–(8) and retaining the previous two terms
and
, we can obtain the following formulas:
In the above equations,
. Substituting Equation (30) into Equation (32) yields the following:
We can see that the coefficients of
and
are linearly independent, and can be estimated by Equations (31) and (33). It can be concluded from Equation (30) that
, which is substituted into Equations (31) and (33) to obtain the Bernoulli equation, as follows:
here,
and
.
Assuming that
in
Figure 2,
is the slope of the wavefront at position P. The solution of the Bernoulli equation is
. Therefore, the Bernoulli equation can show the evolution of the wavefront slope. In the case of a disturbance, generally, the crowd density increases sharply and the crowd velocity decreases sharply,
, and
,
, so the stable range is
. Then, we can determine the followings:
here, “-” indicates the direction, and
is the relaxation factor. Let
.
is the critical acceleration function, and is related to the disturbance pressure in the crowd. Substituting
into
yields the following:
Therefore, the stability criterion of the horizontal crowd flow can be obtained as follows:
Similarly, the stability criterion of the lateral crowd flow can be obtained as follows:
6. Conclusions and Future Work
In this study, a dynamic disturbance model of crowd panic behaviors based on fluid dynamics was proposed to quantitatively investigate the disturbance dynamics of pedestrian panic behaviors. Further, the anisotropy property of this model was proven to keep consistent with the ground truth of real pedestrian movement.
In addition, based on the Lyapunov stability theory, a criterion of crowd stability for sudden internal disturbances was proposed; Experimental results in Hongqiao railway station showed that the equivalent pressure was related to the pedestrian density, duration of the disturbance, and distance from the center of the disturbance. In the Lyapunov-based crowd stability analysis, two main conclusions have been drawn:
(1) The dynamic propagation of panic disturbance pressure in a crowd was mainly affected by the density, direction of flow, and obstacles, indicating the heterogeneous characteristics of the propagation.
(2) Regardless of the limitations on crowd movement directions, the dynamic disturbance of crowd behaviors changed in crowd stability, revealing the characteristics of wave diffusion.
Further work on model calibration based on measured data with more complex scenarios is suggested to be conducted, which can support more pedestrian flow control in public places.