# A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods

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## Abstract

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## 1. Introduction, Motivations, and Plan of the Paper

## 2. Modelling Requirements and Strategy

**What is a crowd?**Agglomeration, in the same venue, of many walkers whose collective dynamics are determined by interactions with other individuals. The assessment of a critical threshold, somewhat analogous to the Knudsen number, suitable to separate individual motion from collectively driven motion appears to be an open problem.**Which is the most unsafe output of emotional states?**The term “panic”, which is occasionally criticized by experts in the field, can induce a breakdown of ordered, cooperative behaviour. Alternatively, we use the term “stress” and observe that, in many cases, stress conditions are induced by a perceived struggle for survival which drives walkers towards not safe situations rather than their survival. The concept of panic/stress should be related, as we shall see in the item below, to the concept of collective rationality/irrationality.**How we can define a collective intelligence?**It is a collective strategy which induces an overall collective emergent behavior in a large population of walkers due to nonlinearly additive interactions which modify individual strategies and might lead to a commonly shared consensus. Consensus towards a common strategy does not imply rationality towards safe conditions, on the contrary crisis situations observe consensus towards irrational behaviors.

- Mathematical structures to support the modelling approach: The modelling approach refers, as we shall see in the next section, to a strategic selection of one of the three scales, i.e., micro-scale (individual based), meso-scale (kinetic), and macro-scale (hydrodynamic). The selection of the scale should be developed by a preliminary analysis of the reproduction by models of the specific features of living systems. Further modelling requirements concern the validity of models and of related computational tools.
- Derivation of a mathematical structure and modelling: All requirements reported in Item 1 should be included in a general differentials structure specialized for each scale. These structures provides the conceptual basis for the derivation of models based on a detailed analysis and modelling of walkers (pedestrians) among themselves and with walls and obstacle. In addition, the quality and main physical features of the venue, where the dynamics occur, should be taken into account.
- Validation: Validation should be addressed to verify how far models can reproduce, quantitatively, empirical data and, qualitatively, expected emerging behaviors. The former specifically refers to the velocity diagram, representing mean velocity versus density. The latter is based on the empirical observation that collective motions exhibit a self-organization ability leading to patterns which are reproduced qualitatively, but it might be subject to large quantitative deviations for small variations of the flow conditions.
- Computing: After validation, computational codes should be developed. As it is known [13], different mathematical structures, hence different computational tools, correspond to each scale.
- Simulations to support crisis managers: Managers can use simulations with various specific purposes, namely “training”, to support decision making. In addition, simulations can be used to improve the design of venues by comparing venues which, with equal transport ability, induce situations of minor overcrowding.

**Strategy:**Individual walkers can develop strategies which take into account the geometry of the venue and the interactions with the surrounding walkers.**Heterogeneity:**The ability to express a strategy is not the same for all walkers. This strategy includes different walking targets and rules in the crowd, for instance, in evacuation it includes the possible presence of leaders.**Learning ability:**Living systems receive inputs from their environments and have the ability to learn from past experience. Therefore their strategic ability and the rules of interactions can evolve in time. Stress conditions can induce important modifications to collective behaviors.**Nonlinear Interactions:**Interactions are nonlinearly additive and involve neighbors in the sensitivity zone of each walker. In some cases, also distant walkers manage to communicate. Walkers perceive, at distance, walls and obstacles and modify their walking strategy accordingly.**Quality of the venue:**The walking strategy, and hence the overall dynamics, is affected by the quality of the venue, where they move, for instance, environment, weather conditions, geometry, and specific features of the venue.

## 3. On the Selection of the Mesoscopic (Kinetic) Scale

- Micro-scale (individual based) corresponding to a system with finite number of degrees of freedom, where pedestrians are individually identified by their position and velocity, while rotational motion is generally neglected.
- Meso-scale (kinetic), where the micro-scale state is identified, as in individual based models, by position and velocity, but the dependent variable the probability distribution function over the micro-scale state.
- Macro-scale (hydrodynamic), where the dependent variable is defined by locally averages quantities, typically density and momentum.

- (i)
- Assessment of the mathematical structures;
- (ii)
- Rationale towards the validation of models;
- (iii)
- Overview of the existing literature;
- (iv)
- Selection of the kinetic theory approach;
- (v)
- Critical analysis.

#### 3.1. Mathematical Structures

**Micro-scale:**The state of the system is given by position ${\mathbf{x}}_{i}={\mathbf{x}}_{i}\left(t\right)=({x}_{i}\left(t\right),{y}_{i}\left(t\right))$ and velocity ${\mathbf{v}}_{i}={\mathbf{v}}_{i}\left(t\right)=({v}_{x}^{i}\left(t\right),{v}_{y}^{i}\left(t\right))$ of all i-th walkers with $i=1,\dots ,N$. The derivation of models requires the specific modellling the acceleration ${\mathbf{F}}_{i}(\xb7)$ impressed to each walker by the action of the other interacting walkers. The equation modelling this dynamics is as follows:

**Macro-scale:**The state of the is represented by the local density $\rho =\rho (t,\mathbf{x})$ and the mean velocity $\xi =\xi (t,\mathbf{x})$. The derivation of models requires the specific modelling the acceleration $\mathcal{A}(\xb7)$ impressed the walkers in the elementary volume $\mathbf{x}$ the action the surrounding walkers. The equation modelling this dynamics is a follows:

**Meso-scale:**The state of system is delivered by a statistical distribution function over the micro-scale state of walkers, namely their position $\mathbf{x}$ and velocity $\mathbf{v}$ which are the micro-scale variable. As in the classical kinetic theory, macroscopic quantities are obtained by velocity weighted moments, see [15].

#### 3.2. Rationale towards Validation

- Capturing the complexity features of human crowds.
- Depicting, as an emerging behavior, the velocity diagrams of crowd traffic depending on environmental conditions can determine different observable dynamics.
- Reproduce “qualitatively” emerging behaviors which are observed in experiments.

#### 3.3. Overview of the Existing Literature

#### 3.4. Selection of the Modelling Scale

## 4. Towards Modelling Perspectives and Applications

- Computational tools;
- Analytic problems;
- Support to crisis managing.

#### 4.1. Computational Tools

#### 4.2. Analytic Problems

#### 4.3. Safety Problems to Support to Crisis Managers

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Elaiw, A.; Al-Turki, Y.; Alghamdi, M.
A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods. *Symmetry* **2019**, *11*, 851.
https://doi.org/10.3390/sym11070851

**AMA Style**

Elaiw A, Al-Turki Y, Alghamdi M.
A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods. *Symmetry*. 2019; 11(7):851.
https://doi.org/10.3390/sym11070851

**Chicago/Turabian Style**

Elaiw, Ahmed, Yusuf Al-Turki, and Mohamed Alghamdi.
2019. "A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods" *Symmetry* 11, no. 7: 851.
https://doi.org/10.3390/sym11070851