Effect of Changing Path on Pedestrian Traffic under the Cumulative Effect of Delay Time

In this paper, a lattice hydrodynamic model of two-dimensional bidirectional pedestrian traffic is proposed with consideration of altering path under the cumulative effect of delay time. ,e stability condition is acquired by linear analysis, and the mKdV equation to describe congestion evolution is derived by reductive perturbation technique. According to the result from the stability analysis, the stability region of pedestrian flow can be divided into stable region, unstable region, and metastable region. On the basis of stable condition, the unstable region is narrowed with the increase of delay time td and the path changing rate c. It indicates that changing path can effectively improve the stability of the pedestrian flow under the cumulative effect of delay time td. For numerical simulation and analysis of density wave, it is found that the increase of path changing rate c and the cumulative effect of delay time td are conducive to alleviate pedestrian congestion.


Introduction
With the increase of urbanization process, the study of city traffic system has become the hot spot which the scientific and social pays attention. As an important part of the city traffic system, pedestrian transportation system brings great convenience for people to travel. But in recent years, the pedestrian stampedes events happened frequently, which has brought serious disasters to the society and brought pain to the victims' families. On December 31, 2014, 36 people were killed and 49 injured in a stampede at Chen Yi Square on the Bund in Shanghai [1]. Saudi authorities issue the Hajj stampede in Mecca on September 26, 2015, have killed 769 people, and wounded 934 others who are from different countries [2]. e stampede caused 22 deaths and 27 injured in Mumbai, India, on September 29, 2017 [3]. How to keep evacuate people safely and effectively in limited space has attracted major concern.
In order to better solve the problem of pedestrian traffic congestion, researchers have made a lot of efforts. From about 1980 onwards, research on pedestrian traffic was often divided into two branches: pedestrian simulation and pedestrian model. Pedestrian simulation focuses on reproducing the natural behavior of pedestrian flow. And with the development of computer graphics, virtual pedestrians can be realized.
ere are many similar behaviors and different characteristics between the vehicular traffic and the pedestrian traffic. First of all, on the macro level, pedestrian traffic presents the same flow characteristics as vehicle traffic. Similarly, the average velocity and flux can be introduced to describe flow characteristics. However, pedestrian traffic composed of the different individuals shows stochastic properties and multidirection movement due to fewer constraints. us, pedestrian traffic is more complex and flexible than vehicular traffic. Henderson firstly put forward the macroscopic model of pedestrian flow in 1971 by applying gas dynamics model and fluid dynamics model to the crowd experience data. He argued that the crowd's behavior is similar to gas under the freely moving phase with low density and similar to fluids under congested phase with high density. However, the energy conservation in the interaction of pedestrians cannot be satisfied in case of pedestrian conflict [4]. In 1992, Helbing proposed a better fluid dynamics model on the basis of Henderson's model by taking into account the intention, the desired velocity, and interaction between pedestrians [5]. Hughes' model of crowd gives the following three assumptions: (1) pedestrian velocity is only determined by the pedestrian density; (2) pedestrian has about the destination of the common sense; and (3) pedestrians seek the shortest travel time [6]. Existing pedestrian models mainly include cellular automata model, lattice gas model, agent-based model, social force model, and hydrodynamic model [4,[7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Helbing and Molnár presented the social force model to describe pedestrian dynamics in 1995 and assumed that the interaction between pedestrians and pedestrians and between pedestrians and environment is expressed by social forces [21]. But it will cause unexpected effects when the superposition principle of force is satisfied under high density. Cellular automata model (CA model) of pedestrian traffic flow is a micromodel which discretizes time and space. Introducing the cellular automata model of traffic flow into pedestrian flow, it is necessary to refine the discrete space. Each pedestrian to occupy the multiple cells selects the moving direction and speed according to the simple rules [22]. At present, these models have been used for pedestrian facility design, calculation of channel capacity, and emergency evacuation. And many characteristics of pedestrian traffic can be extensively applied to many occasions such as bidirection pedestrian, fire evacuation, aircraft evacuation, T-shaped intersection, bottleneck road section, and slope evacuation [23][24][25][26][27].
Comparing with micro-CA model, the macroscopic model is suitable to real-time online simulate pedestrian flow and carry out prediction due to the neglect of pedestrian movement details resulting in rapid calculation.
e first microlattice hydrodynamic model was put forward by Nagatani in 1998. He simplified Kerner and Konhäuser's high-order continuous model and referenced to the idea of the car-following model obtained by flow dynamics equation and conservation equation [28]. And for simulating more actual status of traffic flow, Nagatani proposed a two-lane traffic model to consider the problem of vehicle lane change in 1999 [29]. When the motion of a crowd is seen as fluid flow at a macrolevel, the lattice dynamic equations describing the flow can be applied to pedestrian traffic. Based on Nagatani's lattice hydrodynamic model, Tian et al. extended the two-lane lattice hydrodynamic model of traffic flow to the two-dimensional lattice hydrodynamic model considering path change in bidirectional pedestrian traffic flow and studied the impact of path changing rate on the stability of pedestrian traffic flow through discussing the value of critical point a c [30,31]. Kuang et al. presented a new lattice hydrodynamic model for two-way pedestrian flow considering pedestrian visual field effect. e phase transitions between the freely moving phase, the coexistence phase, and the jamming phase below which the critical point a c has been found [32]. Zhou et al. studied the effect of memory effect on pedestrian flow and assumed pedestrians may walk by his memory when it is difficult for pedestrians to accurately capture the position [33]. Li et al. extended the original hydrodynamic model of traffic flow to single row pedestrian movement with medium and high density and considered asymmetric interaction [34]. Yu et al. solve the general solution of the mKdV (modified Korteweg-de Vries) equation [35]. In 2015, Redhu and Gupta present a new lattice hydrodynamic model based on delayed feedback control considering the flow difference effect [36]. In 2016, Redhu and Gupta propose a multiphase lattice hydrodynamic model by considering the forward looking effect and discuss the impact of forward leading lattices on multiphase traffic dynamics [37]. Peng et al. consider the lateral effects and extended the lattice hydrodynamic model to the nonlane-based lattice hydrodynamic model [38]. Wang et al. study the impact of the multiple density difference effect on the stability of traffic [39].
Until now, the lattice hydrodynamic model of pedestrian flow with consideration of path change has been little developed. Also, the cumulative effect of delay time is important to adjust current pedestrian flow reaching the optimal status. e movement of pedestrians will be affected by the transverse effect and the longitudinal effect between them. Assumed that if there is more space to allocate pedestrian in the transverse direction based on the optimal effect and comfort level, one can select to continue along the way forward or also select to change path with a faster rate. In the longitudinal direction, Tian et al. have studied that the current grid point flow is adjusted by the optimized flow of the grid point ahead. e pedestrian flow reaches the optimal flow within the delay time t d , while the downstream optimal flow can continuously adjust the current flow under the cumulative action of the delay time. Considering both the transverse and longitudinal effects of pedestrian traffic, the analysis of pedestrian traffic evolution characteristics will be more real.
In this paper, a lattice hydrodynamic model of change path pedestrian traffic under the cumulative action of delay time is presented in Section 2. Section 3 carries out the linear stability analysis to obtain the stability condition. In Section 4, the mKdV equation of congestion evolution is derived by applying perturbation theory to perform nonlinear analysis. Section 5 executes the numerical simulation to analyze the evolution of the density wave to discuss the impact of the cumulative effect of delay time. Finally, the main conclusion is simply summarized.

Models
In 1999, Nagatani extended the one-lane lattice model to two-lane traffic lattice model and assumed the changing lane happens when the density ρ 2,j− 1 (t) at site j − 1 on the second lane is greater than the density ρ 1,j (t) at site j on the first lane, and the changing lane rate is c|ρ 2 , where ρ 2,j (t) and ρ 1,j (t), respectively, denote the local density at site j on the second lane and the local density at site j on the first lane. e changing path frequency is denoted as c, which reflects the strength of the changing path. e total average density is ρ 0 and the e velocity V(ρ 0 ) is called the optimal velocity function of the density ρ 0 , which is defined as follows [28]: where v max denotes the maximal velocity of traffic flow and ρ c represents the critical density which is the inflection point of the function of optimization velocity V(ρ). Similarly, if the density at site j on the first lane is greater than that at site j + 1 on the second lane, the changing lane rate from the first lane to the second lane is e conservation equation of one-dimensional lattice hydrodynamic model is expressed as follows: We will extend equation (2) to two-dimensional pedestrian flow with consideration of pedestrian changing path. Pedestrians usually change travelling path along the transverse direction in Public Square and so on. As shown in Figure 1, four-way pedestrian traffic is considered: eastbound, westbound, northbound, and southbound pedestrian. Assumed the fraction of pedestrians moving along each direction is different. e ratio of eastbound and westbound pedestrians to total pedestrians is C, in which the ratio of westbound pedestrians to eastbound is C 1 and the westbound pedestrians correspond to (1 − C 1 ). e ratio of northbound and southbound pedestrians to total pedestrians is (1 − C). e ratio of southbound pedestrians to southbound is C 2 and the northbound pedestrian is e eastbound, westbound, northbound, and southbound pedestrian local densities at site (j, m) at time t are represented by ρ x+ (j, m, t), ρ x− (j, m, t), ρ y+ (j, m, t) and ρ y− (j, m, t), respectively. e average density can be obtained as follows: e conservation equation of the two-dimensional bidirectional pedestrian traffic flow can be written as follows: where z t � z/z t . e eastbound, westbound, northbound, and southbound pedestrian flows are denoted as q x+ (j, m, t), q x− (j, m, t), q y+ (j, m, t) and q y− (j, m, t), respectively.
To simulate more realistic pedestrian traffic conditions with consideration of the impact of downstream pedestrian flow on current pedestrians, the evolution equation of onedimensional pedestrian traffic flow is obtained as follows: where q j is the pedestrian flow at site j at time t and a is the sensitivity coefficient of pedestrian flow (the inverse of the response time τ). In effect, the response time is a relaxation time when the pedestrian flow reaches a steady state from a nonequilibrium state, which corresponds to the time interval in response to a stimulus. λ denotes the strength coefficient, and the integral term represents the cumulative effect of downstream optimization flow and current pedestrian flow differences within delay time t d . e delay time t d is the time pedestrian needs to make decisions about what to travel next and its value is larger than the response time. We employ the following trapezoidal integral formula to approximately estimate integral in equation (5) and extend to four-way pedestrian flow: e evolution equation of four-way pedestrians flow can be written as follows: Mathematical Problems in Engineering By eliminating velocity in equations (4) and (7), the density evolution equation of four-way pedestrian flow is derived as follows:

Linear Stability Analysis
To obtain the stability condition of equation (8), the linear stability analysis of the lattice hydrodynamic model is performed as follows. At first, we consider the uniform pedestrian flow which has constant density ρ 0 and optimization velocity V(ρ 0 ). e solution of the uniform steady state is obtained as ρ j,m (t) � ρ 0 , v j,m (t) � V(ρ 0 ). Under disturbance y(j, m, t), the density of pedestrian flow is expressed as ρ j,m (t) � ρ 0 + y j,m (t). Substituting it into equation (8), the linearized equation is derived as follows: Letting the disturbance y(j, m, t) be expressed as a Fourier series y(j, m, t) ∼ exp(ik(j + m) + zt) and substituting it into equation (9), we can gain the following equation:

Mathematical Problems in Engineering
By performing Taylor expansion z � z 1 ik + z 2 (ik) 2 + · · · for z and keeping the first-and second-order terms of (ik), we can abstract the following equation from equation (10): where g � ( e uniform steady-state flow becomes unstable pedestrian flow for long-wavelength modes in the condition if z 2 is a negative value. When z 2 is a positive value, the state of the steady flow maintains. e neutral stability condition can be obtained as follows: When c � 0.0, the following stability condition is obtained: e stability condition is identical with the solution of the lattice hydrodynamic model of bidirectional pedestrian traffic presented by Cen et al. [40]. By comparing the stability condition, we discovered that the critical value of the sensitivity coefficient without changing path frequency is larger than the critical value of the sensitivity coefficient considering changing path frequency. It shows the importance of considering changing lane to alleviate traffic congestion. Moreover, if the cumulative effect of delay time t d is not taken into account, the stability condition equation (12a) become as follows: It is the same as the research results in [31]. us, it indicates the cumulative effect of delay time t d is conducive to suppress traffic jamming.

Nonlinear Analysis
Using the reductive perturbation technique to equation (8), we carry out nonlinear analysis by considering longwave patterns in pedestrian flow on coarse-grained scales. In order to describe the long-wave modes, the slowly varying behavior of long wavelengths near the critical point (ρ c , a c ) is studied. Introducing a small parameter ε (0 < ε ≪ 1), we defined the slow space variables X and slow time variable T: where b is an undetermined constant. e instantaneous density is obtained as follows: using equations (13) and (14) and expanding each term of equation (8) to the fifth-order of ε, we gain Mathematical Problems in Engineering where the first derivative and third derivative of the optimal velocity V(ρ) at the critical point (ρ c , a c ) are and introducing a � a c (1 − ε 2 ), the second-order term and the third-order term of ε are eliminated. We derived the following equation: where Mathematical Problems in Engineering 7 For obtaining the high-order regularized equation, the transformations are introduced: Consequently, the mKdV equation with O(ε) term is obtained as follows: where X R ′ }. e solution of the mKdV equation (19) is given by where us, the solution of kink-antikink wave of the mKdV equation is yielded as follows: where ε 2 � a c /a − 1, and the amplitude A of kink-antikink wave is expressed as follows: e solution of kink-antikink wave indicates the coexisting phase which is composed of the free flow phase (low density) and the congested phase (high density). e density of the free flow phase isρ j,m � ρ c − A, and the density of the congested phase is ρ j,m � ρ c + A. ereafter, the coexisting curve can be shown as the dotted curve in Figure 2. Figure 2 reveals the neutral stability lines with solid lines and the coexisting curves with dotted curves for a fixed value of the ratio C 1 , C 2 , and C, the delay time t d , the cumulative coefficient λ, and the changing path rate c. e phase diagram can be divided into three regions: the stable region above the neutral line, the metastable region between the neutral stability line and the coexisting curve, and the unstable region below the neutral line. In the metastable region, if the disturbance strength exceeds its critical amplitude, the disturbance will become larger with the time. e pedestrian flow is in the unsteady state, which causes pedestrian congestion. When the disturbance is less than the critical amplitude, pedestrian flow is still in the steady state. In the stable region, pedestrian flow remains the steady state due to the attenuation of disturbance. On the contrary, the disturbance in the unstable region gradually increases with time and eventually develops into kink-antikink wave.

Simulation and Analysis
In order to verify the results in the stability analysis and nonlinear analysis, we carry out simulation and comparison with the consequence of the theoretic analysis. e periodic boundary condition is adopted in the process of the numerical simulation. e initial density and the critical density are chosen as ρ(j, m, 0) � ρ c � 0.25 and the sensitivity of driver a = 1.8. e four-way traffic on square is divided into L × L lattice with L = 100. e small perturbations are added to a two-dimensional four-way pedestrian traffic: ρ((L/2), (L/2), 1) � 0.1 and ρ(L/2 − 1, L/2 − 1, 1) � 0.3. e unit time step is equal to the inverse of the sensitivity a. e evolution pattern of density wave is depicted in Figure 3 at 2000 time steps for C 1 = C 2 = 0.1 without delay time t d . As we can see that the fluctuation of density wave in Figure 3(a) is great without changing path frequency. It is easy to cause congestion and is not conducive to evacuation. In Figures 3(b) and 3(c), the fluctuation of density wave is gradually weakened under the increase of path change frequency c, it indicates that changing path can improve the stability of the traffic system on certain level, and it is consistent with the stability condition equation (12c). When the ratio C is less than or equal to 0.5, Figures 3(b) and 3(d) exhibit that there is no congestion of pedestrian traffic with the increase of the ratio C, which is consistent with the theoretical analysis of system stability. Figure 4 shows the spatial distribution of density at 2000 time steps for C 1 � C 2 � C � 0.1. It is clearly observed that the cumulative effect of delay time t d can significantly improve the pedestrian traffic stability and eliminate jams of 8 Mathematical Problems in Engineering  (5) is suitable to use as a feedback control one.

Conclusions
In this paper, a two-dimensional lattice hydrodynamic model of pedestrian traffic flow is proposed considering path change under the cumulative effect of delay time.
Using the linear stability analysis, the neutral stability condition is obtained. By performing the nonlinear analysis, the mKdV equation describing the density wave of pedestrian flow is derived and the coexistence curve is gained. e phase diagram is divided into three regions: stable region, unstable region, and metastable region. e theoretical analysis shows that changing path c and the cumulative effect of delay time t d can reduce the value of critical point a c and significantly improve the stability of pedestrian flow. e unstable region can be shrunken by the increase of path change frequency, which may be conducive to crowd evacuation. Numerical simulations further illustrate the improvement of changing path rate and delay time t d on pedestrian traffic flow. Moreover, increasing ratio C (C < 0.5) of eastbound pedestrians and westbound pedestrians to total pedestrians can also improve the stability of pedestrian traffic flow. It indicates pedestrian density must be properly distributed in order to improve the stability of the traffic system.

Data Availability
e data used to support the findings of this study are included within the article's references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.