Passenger flow analysis and emergency response simulation in a metro network using virus transmission model

Abstract

Objectives

The potential virus in transportation facilities poses a serious risk to travelers. This research focus on the commuting by metro on the risk of the coronavirus disease 2019 (COVID-19). The main purpose is to explore the trajectory of virus transmission and the effectiveness of various control measures.

Methods

A transmission model was established on the basis of the susceptible-infected-recovered (SIR) model, combined with the spatial and temporal characteristics of the metro passenger flow. The implementation effects of the emergency strategies were analyzed through a series of simulation experiments. The changes in passenger flow affected by the virus transmission were analyzed both under the single intervention condition of the disinfection or off-peak travel policy and their double interventions.

Results

The results of the experiments show that disinfection and off-peak travel can effectively reduce the number of the infected people. To promote the disinfection is better than the off-peak travel strategy. The optimal solution is the combination of these two strategies, thereby reducing the infection rate in the stations effectively. In particular, it can reduce the number of potential infected people in high-traffic stations by 50%.

Conclusions

This study provides a scientific basis for the prevention of COVID-19 in the urban transportation system and the formulation of public emergency strategies. It can also be applied to other epidemic diseases such as the seasonal flu, for public health prevention.

1. Introduction

The current coronavirus disease 2019 (COVID-19) epidemic has originated from the novel coronavirus, which was popular since the end of 2019. It is highly contagious that poses a danger to people's safety. At present, the known transmission routes include the droplets and the close contact. A positive correlation exists between the virus spread and the population density. After a long vacation or the house quarantine, peoples from all walks of life gradually resume their work offline. Thus, the metro in the city carries a large number of commuting passengers. Meanwhile, unfortunately, the carriage has a relatively close space, providing favorable condition for the virus spread. Following the travel route of the susceptible group of the newly diagnosed patients on the metro is necessary. Metro network is one of the key points for epidemic prevention and emergency management in the urban transportation network.

The analysis of the passenger flow in the metro network is important to both public transportation and public health safety. When a station in the metro fails (Gu et al., 2020), the travelling becomes inconvenient to people. Therefore, the virus spread should be controlled while maintaining the connectivity of the metro network. However, in the metro network, the passenger flow has a high-density aggregation in the carriage, thereby spreading to each station. The spatial and temporal trends of the metro passenger flow should be analyzed to predict the early warning to the stations with high passenger flow. Under the constraints of the network connectivity and the power guarantee, reasonable measures for effective prevention and control should be formulated.

The contributions of this study are highlighted as follows: (1) the virus transmission process in the metro network caused by the passenger flow movement is modeled based on the susceptible-infected-recovered (SIR) model. (2) Multi-source data were fused to prepare for the transmission simulation. The metro passenger flow data were extracted from the automatic fare collection (AFC) to analyze the spatial and temporal trend of passenger flow. The location of the district of confirmed cases was obtained from Tencent bulletin board, where the data were collected from the Municipal Health Commission as the initial value of virus input. (3) Through the multi scenario experiments, the virus transmission after the disinfection and off-peak travel policy was simulated. The changes in the virus transmission are discussed for the transmission speed and the number of potential infected people to analyze the implementation effect in the metro network, which can provide a decision basis for emergency management and guarantee the travel safety.

2. Related works and research gap

2.1. Metro network and ridership assignment

The study on the metro network structure and the passenger flow characteristics are the fundamental of the emergency control policy in the metro network. In terms of the structure, Sienkiewicz and Holyst (2005) built the public transport networks of 22 cities in Poland through the L space method. That is, two adjacent points can be linked. This method can be used in the construction of the metro network. In terms of the passenger flow, Li et al. (2016) classified the metro transit passengers by the mobile phone signaling data. Guo et al. (2017) focused on passenger demand and studied the optimization problem of metro network schedule. Li et al. (2017) established an optimization model through the passenger flow prediction and traffic demand analysis to achieve the optimal operation efficiency of the metro. Zhou et al. (2016) studied the relationship between the transfer passenger flow and the walking time. The movement of the passenger flow creates a risk of the virus spreading. Some researches focus on the movement of passenger flow inside the metro. The passenger flow from the origin to the destination was allocated to each link by all-in allocation (Sun and Guan, 2016). Li et al. (2019) established a spatio–temporal prediction model of the urban metro transit passenger flow under emergency conditions.

2.2. Transmission model

The network transmission model originated from the study of infectious virus in the biomedical field. Infectious disease is a great challenge in public health. In nature, the common routes of the virus transmission include the air, droplets, contact, fecal, mouth, blood source, and mother-to-child transmission. The infected individuals and infectious carriers transmit the virus to their close contacts through these patterns, leading to the virus widespread. Since 1760s, researchers have attempted to explore the human smallpox epidemic model (Kermack and Mcendrick, 1927). The studies on the spread of infectious diseases and the forecast model are in constant progress. Nowadays, the susceptible-infected (SI) model, susceptible-infected-susceptible (SIS) model, and susceptible-infected-recovered (SIR) model are commonly used. They are not only used to predict the spread of infectious diseases, but also applied widely in various network transmission phenomena, such as the information flow in the social network (He and Liu, 2020; Lai et al., 2020; Liu et al., 2018, Zhang et al., 2014; Zhang et al., 2014) and computer network (López et al., 2019, Yin, 2016), the passenger flow in the transportation network (Yang et al., 2012; Zhang et al., 2015).

The study of virus transmission is based on mathematical models. The classical SI model is a relatively simple system dynamics model of virus transmission. It divides the population into the susceptible (S) and the infective (I). Through the close contact or the other transmission pattern with the infected persons (I), at a certain rate, the susceptible persons (S) become newly infected. This category is suitable to describe the spread process of the sudden outbreak of the virus before effectively controlled, such as the early period of SARS. Another transmission model is SIS, in which the transmission process has one more step than SI. The infected persons (I) will return to the state of susceptible (S) with a certain probability. It can be used to describe the spread of a secondary infection, such as influenza and hand-foot-and-mouth disease.

Subsequently, Kermack and Mcendrick (1927) proposed SIR model. In this model, the population is divided into three categories, the susceptible (S), the infective (I), and the recovered (R). R stands for healing without being re-infected. The SIR model has also been applied to measure the crowdedness in public in public transportation. Dai et al. (2015) established an air traffic congestion propagation model on the basis of SIR model. Saberi et al. (2020) studied the congestion propagation in cities by SIR model and identified the congested links.

Besides the similarity of the traffic flow and the virus transmission, the movement of the humans plays a key role in the virus transmission (Noland, 2021, Pestre et al., 2011). Ronchi et al. (2020) proposed an exposure assessment model to analyze personnel movements from a micro perspective for risk assessment, which can explain different types of disease transmission. A significant correlation was found between the number of COVID-19 infections in Chinese cities and the frequency of the high-speed metro services and flights departing from Wuhan in May, 2020 (Zhang et al., 2020). Freire De Souza et al. (2021) pointed out that airports and highways were responsible for the interiorization of the COVID-19. If an infected person travels in the vehicle without the mask, the virus particles can be deposited directly on the surface of the vehicle or suspended in the air (Horve et al., 2020). Qian and Ukkusuri (2021) described the dynamics of urban migration as the process of leaving home, getting to and from places of activity and participating in activities. They embed the susceptible-exposed-infectious-recovered process over the mobility dynamics which explained how infections can occur during travel and daily activities. In this study, we combined the SIR model with the movement of metro passenger flow to predict risks.

The knowledge about the complex network and system dynamics has been combined to describe the dynamic process of virus transmission (Duan et al., 2019, Krause et al., 2018, Li and Wei, 2019) to better control and predict the epidemic situation. Brockmann and Helbing (2013) proposed SIR-based model of the virus spreading in the transportation network with data of H1N1 and 2003 SARS epidemic. They also introduced the effective distance between cities. The travelling from one city to another directly without passing through other cities was depicted. Some scholars analyzed the relationship between the spread of diseases and that of the relevant information among individuals (Liu et al., 2016, Nian and Yao, 2018, Zhan et al., 2018). It was found that high information transmission can effectively slow down the spread of the epidemic.

2.3. The outbreak of COVID-19

In response to the sudden outbreak of COVID-19, many researchers have conducted follow-up studies. Some studies have given the safe distances in confined spaces (Megahed and Ghoneim, 2020). WHO suggested the safe interpersonal distance for 1.5 or 2 m to minimize the risk of infection. However, the published study in April, 2020 found that the virus could spread more than 2 m from an infected person (Setti et al., 2020).

Another spotlight focuses on the corresponding measures and their effects to prevent the virus spreading. Roosa et al. (2020) did a short-term forecast of the cumulative number of cases in Hubei province, where COVID-19 was firstly discovered and reported, by using phenomenological models. It is confirmed that the control strategies implemented in China have successfully reduced the transmission. Using a fuzzy set qualitative comparative analysis of 323 cities in China, Fan et al. (2021) identified four categories for blocking COVID-19 transmission: social reassurance, proactive defense, decisive resiliency, and strengthened coercion. Lan et al. (2020) found that some of the recovered patients were still carriers of 2019-nCoV. It indicates that it is important to track the passengers’ travel route for public health prevention and emergency control. From a population antibody test of 3330 samples in Bay Area of Northern California, USA, Bendavid et al. (2021) found that the actual number of the infected patients was 50–85 times more than the confirmed cases. The findings illuminate a certain proportion of the asymptomatic virus carriers in the community. The asymptomatic infections (A) are not as easily detected as normal patients. It is the possible to infect other passengers when the asymptomatic use the public transportation network. The previous study stated that asymptomatic infections accounted for about 60% of all infections (Qiu, 2020). Wu studied the outbreak in Wuhan and found out that about 59% of the infected people were asymptomatic. Comparing the results of several studies, Chowell indicated that the asymptomatic account for about 40–50% of all infections (Qiu, 2020) with the data in Hubei, China and Diamond Princess cruise ship, Japan. Al-Sadeq and Nasrallah (2020) reviewed the studies on asymptomatic infections of COVID-19. They pointed out that the percentage of asymptomatic infections in the total population was between 1.2% and 12.9% in a large sample size (>1000), while in the smaller sample size that could be up to 87.9%.

Above all, most studies primarily focused on the transmission model to discover the virus spread from city to city. In fact, the connection with the travel by the vehicles within the city and the spread of COVID-19 should be taken into account. However, the joint study on asymptomatic infections and their communicate routes are relatively rare. This paper aims to fill up the research gap in these aspects.

3. Problem formulation

3.1. Notation and definition

Notation 1. Origin-Destination (OD) The origin and the destination of the traveller, are used to describe the trip distribution of the travellers. In general, the travel prediction and calculation model consist of four basic steps: trip generation, trip distribution, mode choice and travel assignment. For the passengers travelling by the metro, their trip distribution, the pair of ODs, can be obtained from AFC data.

Notation 2. Ridership Assignment (RA) The process that the volumes of persons on the route are estimated in the transit system. It is the last step in the ridership calculation model. In the metro network, though OD stations can be recorded by AFC system, the specific route of the trip is a lack in the data. Usually, there may be more than one feasible route between a pair of ODs. Ridership assignment is to allocate the ridership to the route according to a certain rule, such as the shortest path, stochastic user equilibrium, etc.

3.1.1. Definition: Time stamp

The study period is discretized into a series of different time stamps with the mode of departure interval $\delta$. Then, a set of obtained time stamps can be expressed as $T=\left\{t_1,t_2,\cdots ,t_m\right\}$. For example, if the study period is 1 h, then $m$ is $60/3=20$ with a departure interval $\delta$ of 3min.

3.1.2. Problem formulation

When a person is infected, if the symptoms have appeared, he or she will be quarantined, as shown in the bottom part of Fig. 1 . However, since the asymptomatic one is hard to be found, he or she will continue their daily travel, such as to work, for leisure, etc., until one or some symptoms appear. When the asymptomatic infected person takes the metro, as shown in the top part of Fig. 1, the ridership S in the metro system can be transformed with the probability of $\alpha$, while the infected people will leave the metro system with the probability of $\beta$. The constant movement of people through the system will continue to expand the group of infected people. The problems to solve in this paper are (1) to study the mechanism of the virus transmission in the metro system, and (2) to provide different emergency management measures and analyze their effectiveness by the simulation tests.

Fig. 1.

Flow diagram of virus transmission.

3.2. Network construction and passenger flow distribution

The metro network is abstracted into a directed graph $G=\left(V,A,W\right)$ by the L space method, where $V$ represents the station set,$A$ represents the directed arcs between two stations and $W$ is the passenger flow matrix. The value of the element in $W$ is obtained by RA.

The virus spread in the metro system can be divided into two sections, in the station and in the vehicle carriage. If an asymptomatic is in the station, the other passengers may get infected when they enter/exit the station or during the waiting for the train on the platform. After the asymptomatic person takes on the train, the virus spread in the carriage of the train becomes dynamics along the railway lines, moving from one station to another. It should be notice that the virus does not spread only directly between OD stations, all the stations, and all the ridership at these stations in the route are involved. OD ridership assignment is a key but difficult point in the modeling of virus transmission.

A mature metro network usually has stable check-in ridership. The travellers, especially the commuters in the peak hour have a certain route selection pattern. The minimum number of stations is consistent with the metro passengers’ perception of the route choice during the morning peak hours. OD passenger flow was allocated by the shortest path with the minimum number of stations, forming the passenger flow matrix $W$.

3.3. Confirmed cases

In epidemiological investigations, the potential “contacts” of the virus should be calculated on the basis of the confirmed cases. To better track the close contacts during the epidemic and carry out the follow-up epidemic prevention management, the location of the initial epidemic patient (referred to the confirmed area) is of great importance. The information of COVID-19 confirmed areas in Beijing, China are updated by the health commission every day. The data on February 21st, 2020 were taken as the initialization of the model to find out the latent spread link.

The product of infections and the proportion of asymptomatic infections are taken as the number of input cases, which can be regarded as traffic generation. Considering that not all the asymptomatic infections choose the metro when travelling, the input case number should be multiplied by the share of metro travel, which is 16.2% according to the previous study (Beijing Transport Institute, 2019). This step can be regarded as a mode choice.

The nearest metro station is selected by the asymptomatic infections to travel, carrying the virus into the metro system, as shown in Fig. 2 .

Fig. 2.

Distribution of epidemic area.

4. Virus transmission and emergency management measures simulation

4.1. The SIR model based on traffic

To predict the virus spread in the metro system, an intercity virus transmission model (Brockmann and Helbing, 2013), as shown in Equations (1), (2), (3), is applied to the urban metro network. Different from the transmission process of direct travel between cities, the passing stations along the railway lines will also be affected. The model is revised to redefine some variables. The number of potential infected people in the metro network can be estimated. The passenger flow granularity is accurately calculated through two considerations, the information of asymptomatic infections obtained from Section 3.3 and people in close contact with the infected persons in the same carriage. Then, the revised number of the infected persons is set as the input value of the differential equation.

$$\frac{\partial s_n\left(t\right)}{\partial t}=-\frac{\alpha }{K}{s}_n\left(t\right)j_n\left(t\right)+\gamma \sum _{m\ne n}{p}_{mn}\left(s_m\right(t)-s_n(t))$$ (1)

$$\frac{\partial j_n\left(t\right)}{\partial t}=\frac{\alpha }{K}{s}_n\left(t\right)j_n\left(t\right)-\beta j_n\left(t\right)+\gamma \sum _{m\ne n}{p}_{mn}\left(j_m\right(t)-j_n\left(t\right))$$ (2)

$$s_n\left(t\right)+j_n\left(t\right)+r_n\left(t\right)=1$$ (3)

where $s_n\left(t\right)$ and$j_n\left(t\right)$ respectively represent the proportion of susceptible people and that of infected people at the node $n$ at the time $t$ . The product of the two can applied to detect the number of passengers who co-exist in the same station at time $t$. $\alpha$ represents the probability of infection. It is a prior value, which is related to the characteristics of the virus itself. $K$ represents the multiplier relative to the smallest area station. $\beta$ represents the exit ratio, which is equivalent to the reciprocal of the average time in the metro during the study period; $\gamma$ is the average moving population ratio within the time stamp, which is equal to the moving population divided by the total population in the system, the total population contains all arrivals and departures; $r_n\left(t\right)$ represents the proportion of people leaving the station $n$ at the time $t$. $p_{mn}$ is the element in the $m^{th}$ row and $n^{th}$ column of the passenger flow transfer probability matrix $P$, which is defined as the probability of moving from node $n$ to node $m$. It is obtained by the ratio of the value from node $n$ to node $m$ in passenger flow matrix $W$ to the sum of all the flows from node $n$, as shown in Equation (4).

$$p_{mn}=\frac{f_{n\to m}}{f_{n\to •}}$$ (4)

4.2. Emergency management measures

The disinfection or the off-peak travel can prevent and control the virus spread in the metro stations. Among them, the disinfection measure is a rectangular window display function $\varpi \left(t\right)$, as shown in Equation (5). It is multiplied into the interaction item of Equations (1), (2)), as shown in Equations (6), (7)). The function $\varpi \left(t\right)$ assumes that the interaction at the disinfection time $t_x$ is 0.

$$\varpi \left(t\right)=\{\begin{array}{l}0，t_x\le t<t_x+\delta \\ 1，others\end{array}$$ (5)

$$\frac{\partial s_n\left(t\right)}{\partial t}=-\frac{\alpha }{K}\varpi \left(t\right)s_n\left(t\right)j_n\left(t\right)+\gamma \sum _{m\ne n}{p}_{mn}\left(s_m\right(t)-s_n(t))$$ (6)

$$\frac{\partial j_n\left(t\right)}{\partial t}=\frac{\alpha }{K}\varpi \left(t\right)s_n\left(t\right)j_n\left(t\right)-\beta j_n\left(t\right)+\gamma \sum _{m\ne n}{p}_{mn}\left(j_m\right(t)-j_n\left(\mathrm{t}\right))$$ (7)

Two common methods of disinfection are proposed, to spray in the carriage and to send out a disinfected vehicle. As shown in Fig. 3 , at the time of disinfection, the station can be disinfected by spraying disinfectant. Or a disinfected train departs from each of the first stations, as shown in the blue point on the right of the figure. It takes approximately 1 h to disinfect a train by the second method. However, it can be considered as that a disinfected standby vehicle is dispatched at the time of disinfection. The accumulated virus in the vehicle is eliminated. The reset interval $\Delta t$ is set as the departure interval. After passengers aboard the vehicle in the next time stamp, the transmission coefficient returns to the transmission process between I and S in Fig. 1.

Fig. 3.

Disinfection methods.

The infected passenger flow is a continuous function prior to considering the emergency disinfection measures. Thus, the time node corresponding to the maximum infection speed ratio is selected as the disinfection moment to contain the outbreak of the peak epidemic effectively.

Another single interference measure is to promote off-peak travel policy. The flow matrix $W$ can be modified to transfer part of the passenger flow in the peak period to other adjacent periods. It is equivalent to calculating the rotation axis of the passenger flow transfer matrix through Gaussian transformation without affecting the flow balance. In the simulation experiment, the new passenger flow transfer probability matrix $P$ after transformation is used while the values of other variables remain unchanged.

5. Experiment

5.1. Data sources

The weekday passenger flow without regulation was selected to analyze the actual travel demand to compare the spread of the epidemic before and after emergency management measures. The following experiment calculation was carried on through certain conversion calibration. The preprocessing of AFC data was shown in Fig. 4 .

Fig. 4.

Flow diagram of data preprocessing.

The data of this study were selected through the historical data of the Beijing metro system. Without loss of generality, the AFC data of the week from April 11, 2016 (Monday) to April 15, 2016 (Friday) were adopted for subsequent calculation. During this period, there were neither mega events nor metro accidents to cause an outburst passenger flow. The average daily data of the week were used to represent the common situation on weekdays.

The virus spread requires a certain population density. Therefore, the peak of the ridership is the key duration of the emergency response. According to the change of metro passenger flow, the centralized travel time was 7:00 to 9:00 in the morning and 17:00 to 19:00 in the evening. Without loss of generality, the morning peak hours were selected for this study. To better prevent the spread of the epidemic, an additional hour before and after the morning peak was also taken into account. The peak time period was chosen as 6:00 a.m. to 10:00 a.m., 4 h in sum, as marked in the red box in Fig. 5 . The AFC data contains card ID, arrival time, arrival station name, departure time, departure station name, travel time, station longitude and latitude information. Matching with the entrance station name and the exit station name, the pairs of records with the same OD were assigned into groups. The daily average number of OD pairs in the study period was about 60 thousand.

Fig. 5.

Spatial and temporal statistics of metro passenger flow in Beijing on weekdays.

Two factors were taken into account in the data conversion, the natural growth rate $g$ of passenger flow and the resumption rate $r$ by the influence of COVID-19. By these two operators, the passenger flow on weekdays in 2016 was converted into the passenger flow on weekdays after the Spring Festival in 2020. The specific steps were as follows. First, in accordance with the metro passenger flow statistics of the China Urban Rail Transit Association from 2016 to 2018 (China Urban Rail Transit Association, 2016–2018), the average annual growth rate $g$ was set as 0.026 to calculate the passenger flow in 2020. Secondly, the travel intensity in the city during weekdays from February 17 to February 28 in 2019 and 2020 was obtained through the migration data of the Baidu Map (2020). After that, the resumption rate $r$, the ratio of the two years’ average travel intensity (Urban Data Cluster, 2020), was obtained as 0.466. Finally, the ridership of each OD pair in the morning peak during weekdays in 2016 was converted into that in 2020, as shown in Equation (1).

$$Q^{2020}=Q^{2016}\times {\left(1+g\right)}^{2020-2016}\times r$$ (8)

where $Q^{year}$ represents the ridership of the study period in a given year.

5.2. Simulation experiment design

The simulation experiment was designed to better observe the trend of epidemic spread and the change of the passenger flow of virus transmission under emergency management measures. Equations (1), (2), (3) in the traffic-based SIR model and Equations (5), (6), (7) after regulation were applied to each station in the network.

The departure intervals of each metro line during the morning peak hour in Beijing are listed in Table 1 . The length of the time stamp was set to 3 min according to the average departure time in the peak period.

Table 1.

The departure intervals of Beijing metro during morning peak hour.

Line	Departure intervals (min)	Line	Departure intervals (min)
1	2	13	3
2	3	14	8
4	2	14	5
5	2	15	5
6	3	Changping Line	4
7	3	Batong Line	3
8	3	Yizhuang Line	4
9	3	Daxing Line	3
10	2	Fangshan Line	4

The parameter $\alpha$ was obtained by the product of the basic reproductive number $R_0$ and $\beta$, where $R_0$ was selected as 3.28 (Liu et al., 2020), and $\beta$ was the reciprocal of the average time in the train during the survey period. The average time in the train was 40 min, and $\beta$ was $\frac{3}{40}$. Assuming all the stations were of the same size, $K$ was setting as 1. When allocating paths, some OD's had more than one shortest path, and at most 4 shortest paths were considered. $\gamma$ and $p_{mn}$ were calculated according to the definition. The proportion of the asymptomatic infections is set as 50%. For various scenarios in Table 2 , simulation experiment parameters were set, and iterative operations were started through Python.

Table 2.

Simulation experiment scheme.

Scenario	Measures	Figure
a	No regulation + Consider OD	6
b	No regulation + Consider Ridership Assignment	7
c	Disinfection + Consider Ridership Assignment	9,10
d	Off-peak travel + Consider Ridership Assignment	11
e	Disinfection + Off-peak travel + Consider Ridership Assignment	12

6. Results and discussion

6.1. Comparison of virus transmission before and after ridership assignment

The virus transmission is compared between considering OD and ridership assignment or not. As shown in Fig. 6, Fig. 7 , the horizontal axis jumps with the time stamp length (departure interval $\delta$ of 3min). The starting scale of 0 is 6:00 a.m. The time stamps 20, 40, 60, and 80 correspond to 7:00 a.m., 8:00 a.m., 9:00 a.m., and 10:00 a.m., respectively. The vertical axis is the number of potential infected passengers in a station. Each line in the figure represents a station. The peak number of infected passengers is marked with a gray dotted line and marked with ${\mathrm{W}}_{peak}^{scenario}$. Typical stations, which are near the large residential areas, business districts, and high-tech industrial zones are shown in different colors, while other stations are in gray. The position of the typical stations is shown in Fig. 8 . Huilong Guan, Tiantongyuan Bei, Tiantongyuan and Dongzhi Men are close to the outbreak site. Xi'erqi, Dongzhi Men, Guomao and Dawang Lu are transfer stations, carrying large number of passengers, where need attention to the virus transmission. Xitucheng and Caofang are common stations.

Fig. 6.

Virus spread in the metro station considering only OD data (Scenario a).

Fig. 7.

Virus spread in the metro station considering Ridership Assignment (Scenario b).

Fig. 8.

Position of the typical stations.

There are two considerations for the identification of the peak: (1) the number of stations with peak infection at a given time stamp is higher than the number of nearby time stamps; (2) the average number of infections is not less than 5.

Three outbreak peaks is found in scenario a, as shown in Fig. 6 (a). Peak 1 appears at approximately the 43rd time interval, that is, at the early hour of 8:00. It can be seen clearly from Fig. 6 (b) that 7 stations reach the peak at the 43rd time stamp with an average of 14 infections. Tiantong Yuan and Huilong Guan are the stations close to the confirmed area. Peak 2, between 8:30 and 9:30, has the most stations break out. There are 9 stations that reach the peak at the 55th time stamp with an average of 17 infections. At this time, in Xi'erqi, Guomao, and Dawang Lu stations, the passenger flow is the highest during the morning peak. The infected number of people potentially vulnerable to infection is also the highest, reaching more than 40 persons within this time interval. The numbers in Xitucheng, Tiantongyuan Bei, and Caofang are slightly low, at approximately 10. Peak 3 appears approximately at the 64th time stamp. It can be seen clearly from Fig. 6 (b) that 18 stations reach the peak. The average number of potential infected people at each station is 8. The concentration areas in the figure show that most stations are susceptible to less than 25 people during peak outbreak time.

Considering that once an infected person enters the metro, there is a probability that the virus spreads to the station on the route. The assignment of OD passenger flow data is carried out in scenario b (Fig. 7 (a), (b)), which is more in line with the actual commuting situation. By the simulation test, two peaks of the virus spread is found.

Compared with the peaks in scenario a, the first and the second peak barely moved in scenario b. Taken Xi'erqi as a representation of peak 2, it is more violent than the curve of scenario a, with nearly 45 passengers infected. The reason is that Xi'erqi is a transfer station of Changping Line and Line 13, carrying the most passenger flow in the network. However, this phenomenon cannot be observed through only OD observation, because the transfer ridership is much more than the passengers enter or exit at this station.

Some stations like Guomao and Dawang Lu break out relatively late at 9:00 a.m. with approximately 25 infected passengers. These stations are nearby the business districts. They are usually the destination of the communute trip in the morning peak. The results inferred that the improved model with the metro passenger flow assignment is more suitable for the simulation and prediction of urban public transportation than the traditional virus transmission model.

In general, the outbreak time of high passenger flow stations is concentrated in the time stamp range of 40–60, that is, between 8:00 a.m. and 9:00 a.m., which is consistent with the nine-to-five working time stipulated by most occupations and enterprises. The number of potential infections is mostly concentrated below 20.

6.2. Virus transmission after regulation

6.2.1. Disinfection

In accordance with the virus outbreak trend predicted by this experiment, the disinfection interference strategy is adopted when the number of virus infections increases the fastest, that is, the position with the maximum curvature of line changes in the figure.

The first is to spray and sterilize all vehicles at one time stamp. As shown in the blue auxiliary line ①② in Fig. 9 (a), the disinfected time is at the 41st and the 52nd time stamp. Compared with scenario b, three outbreak peaks are found in scenario c. No significant change is found in the position of peak 1. Peak 2 moves from 8:45 a.m. to 8:36 a.m. A new peak appears at 9:12 a.m. The number of potential infections is decreased by more than 20% in peaks 2 and 3, namely 5 to 10 people in each station. The simulation test tells that after the disinfection, the peak of most stations appeared at nearly 10 o ‘clock, with only about one potential infected person, which can be seen from Fig. 9 (b). After the disinfection, many stations are less likely to be spread by the virus, and do not appear during peak hours. It indicates that the disinfection can effectively reduce the number of potential infections.

Fig. 9.

Virus spread in the metro station after Disinfection (Scenario c).

The second measure is to send a disinfected vehicle from each of the first stations at the disinfected time. In this scenario, the disinfected vehicles are placed at the 41st and the 52nd time stamp. The result is shown in Fig. 10 . Between 40 and 60 steps, there are still stations (gray lines) that remain high. It means this approach has little effect on the stations without disinfected.

Fig. 10.

Virus spread in the metro station (with disinfected vehicle departure).

6.2.2. Off-peak travel

According to the analysis, the outbreak time of high passenger flow stations is concentrated in the 40th to 60th time stamps, that is, at 8:00 a.m. to 9:00 a.m. The travel time intervention is applied to this interval (yellow area in Fig. 11 (a)). In scenario d, through the policy guidance such as the feasible working time and travel fare benefit, some passengers are adjusted to travel during the period with less passenger flow, i.e. from 6:00 a.m. to 7:00 a.m.

Fig. 11.

Virus spread in the metro station with Off-peak Travel policy (Scenario d).

To verify the prevention and control effect of virus transmission, the passenger acceptance of the travel transition policy is set by 25% and 50% for two tests. The results have the similar trend. The test with 50% transmission results in an even greater decrease in the number of travellers to the outbreak stations, which is shown in Fig. 11 (a) (b) as an example. Compared with the situation prior to the regulation, the number of potential infected passengers in each station decreased by approximately five. The average of potential infected passengers in many stations was reduced to one, and the time was concentrated around 10 o ‘clock, as shown in Fig. 11 (b).

The results of regulating and controlling the number of potential infected passengers in the morning peak hour indicate that the off-peak commute policy can evidently affect the control of the virus outbreak. Therefore, it can be performed in advance to prevent the virus and reduce the infected population by transferring the peak passenger flow to the off peak. The peak of passenger flow also appears between 7:00 a.m. and 8:00 a.m., and no peak of infected people is observed, indicating that the virus outbreak needs a certain time to accumulate.

6.2.3. Disinfection and off-peak travel

Metro disinfection and off-peak travel measures are implemented simultaneously in Scenario e. Fig. 12 (a) (b) show the implementation effect by the two measures jointly together. Compared with scenario b of the control group without any measures, peak 1 and peak 2 moves forward and peak 3 appears backward. In addition, the number of potential infected people at the peak is significantly reduced, especially in the high passenger flow stations. The number of potential infected people under the worst conditions can be reduced by half to approximately 20 people. The experiments show that the disinfection and off-peak travel, while prolonging the possible duration of virus transmission, can significantly reduce the number of potential infections. For the stations with general passenger flow intensity, the measures can reduce the maximum number of potential infected people below 10 people. The peak of 56 stations is around 10 o ‘clock, with about only one potential infected person, as shown in Fig. 12 (b).

Fig. 12.

Virus spread in the metro station (Disinfection + Off-peak Travel) (Scenario e).

The series of simulation results show that the effect is best when the two interventions are implemented simultaneously. For low passenger flow periods, such as the stations with the maximum number of potential infected people below 10 in the figure, the three measures slightly affect the number of potential infected people. In many stations, the number of potential infected people has dropped to 1 with the infection time around 10 o ‘clock. This means that during the peak period of 7–9 o ‘clock, these stations can hardly be infected and the virus spread is effectively controlled. This study starts with the emergency management measures under the high passenger flow risk of the metro. The infection rate can be reduced by 50%, indicating that the intervention effect is remarkable.

6.3. Parameter analysis

In this experiment, when the departure interval is reduced to 2 min, the numbers of the infected passengers at the stations with high passenger flow are decreased by one to two, while the overall trend is slightly forward. When the departure interval is increased to 4 min, the number of the infected passengers at high passenger flow stations is increased by one to two. Because there is a certain relationship between the ridership and the departure interval. The excessive growth of the departure interval may lead to an increase in the passenger flow density on the platform and in the carriage. Therefore, from 8:00 a.m. to 9:00 a.m., the passenger flow can be restricted at the entrance station. The situation of the high-density passenger flow gathering in the platform can be avoided by a shorter departure interval.

The proportion of asymptomatic infections may have an influence on the results. In order to explore the spread and outbreak of the virus with different inputs, the parameter analysis on the proportion of asymptomatic infections is carried out, as shown in Fig. 13 .

Fig. 13.

Analysis of the proportion of asymptomatic infections.

The average number of potential infected people at each station calculated with a ratio of 0.5 is taken as the benchmark. As the number of asymptomatic infections decreases, the number of potential infected people also decreases. 0.5 is a critical value. When the proportion of asymptomatic infection is greater than 0.5, no more one person will be infected in each time stamp of each station. This result is based on the worst case of the severe outbreak in Beijing in February, 2020. Fewer people would have been infected in the post-outbreak world than the result.

6.4. Further discussion

According to the simulation test results, to do the disinfection at the most profitable time, to promote the travel time transition policy, have a notable effect to control the virus breakout in the metro system. For the cities with large ridership, the dual intervention controls of disinfection and off-peak travel should be implemented, especially for the stations with high ridership. Passengers are suggested to take the metro during the off peak hours to reduce the passenger flow density in the carriage at peak hours. Therefore, the probability of infection caused by the cumulative effect of virus transmission is reduced.

In the actual emergency response, for the travel choice transition to reduce the ridership peak, the relevant policies contain changing the working hours, adjusting the time-sharing electricity price of enterprises, time-sharing metro ticket price, ticket save in designed station etc. For example, Hong Kong Mass Transit Railway (MTR) company launches a serial of ticket discount schemes. One of them is the fare saver in specific stations to persuade people to use the stations with less ridership instead of the hot stations. The passenger can wave the Adult Octopus card (smart card for transit in Hong Kong) over the reader on the MTR Fare Saver and enjoy $2 discount on the next MTR ride from designated stations. The intervention from the government is also needed in the high-risk area of the virus spread. The station skip, load factor control, strong ventilation are all recommended measures for the public transit including metro, bus, ferry, train, etc. For other cities with a general passenger flow intensity, appropriate single-intervention emergency management and control policies can be formulated by referring to the period of low passenger flow.

Besides the management scheme in the transit system, other valid actions for the individuals should be carried out at the same time to prevent and control the epidemic. For the personal protective measures, the mask wearing and frequent hand washing are very easy but effective. The hierarchical isolation control measures are also valid. It is worth noting that masks and vaccines can reduce the rate of infection $\alpha$ in the transmission model respectively, which can reduce the impact of human-to-human transmission.

Compared to the previous studies on the virus spread from city to city, the study focuses on the virus transmission when the passengers travel in the metro system. Unlike the other studies that focus on how to prevent the transmission from person to person, it aims to discover the influence of implementing counter-measures such as disinfection or off-peak travel management on the transmission speed of the virus. Both of these two strategies and the superposition of them can effectively reduce the infected people. The proposed model and simulation method can also provide ideas for the emergency response of other viruses, such as high-incidence seasonal influenza in the metro network.

The epidemic prevention intensity should be determined according to the severity of the local epidemic. When the epidemic prevention intensity is too high, it takes additional operation schedule to guarantee the safety social space in the carriage, which cost lower operation efficiency for the transit system and longer waiting time for the passengers. The operation cost will increase by the disinfection and the ridership control in the carriage. On the other hand, while the epidemic prevention intensity is insufficient, it may cost a higher risk of the virus spread. The policy maker, the operator, and each passenger of the public transit must realize that the life and health are above any cost in the public transportation system.

7. Conclusions

On the basis of the analysis of the network and the mechanism of public health epidemic prevention, a virus transmission prediction model for emergency management in the metro network is established. The simulation analysis on the effect of disinfection and off-peak travel measures is conducted. The main contributions are as follows.

• (1)Considering the travel of asymptomatic infections, the mechanism of virus transmission and diffusion in the process of metro passengers during commuting is depicted.
• (2)The spatial and temporal characteristics of metro passenger flow and network transmission characteristics are analyzed. The passenger flow movement path on the space is established through RA based on the shortest path. It provides the basis of simulation to explore the trend of virus transmission.
• (3)The virus transmission simulation experiment under various prevention and control strategies is designed. Disinfection and off-peak travel measures have been verified to effectively reduce the number of potential infections. The combination of the two approaches is optimal, especially in high passenger flow stations, can reduce the infection rate by 50%.
• (4)The relationship of the emergency response measures in the metro network and the virus spread factors are deduced based on the experimental results. Some suggestions on emergency intervention and control for the metro stations and cities in different scales are put forward to ensure personal safety during travel. It provides scientific decision-making basis for the control of the COVID-19 epidemic and other similar diseases in the urban transportation network.

The limitation of this work is that the ridership assignment is obtained through the shortest path. In the real life, people's route choice will be affected by other factors such as travel time and the prior knowledge of the route. Moreover, the current approach has limitations due either to uncertainly of infection transmission mechanism or to other potential transmission sources. In addition, due to the unavailability of data, the area of all stations is considered consistent in the text, which can be improved in the future. At last, it should be notice that in this research the virus spread is discussed from the perspective of traffic flow but not its biological characteristics.

Funding

National Key Research and Development Program of China (No. 2018YFB1600900) and Beijing Natural Science Foundation Program (No. L181002).

Author statement

The authors confirm contribution to the paper as follows: study conception and design: Yuyang Zhou, Shuyan Zheng, Feng Feng and Yanyan Chen; data collection: Shuyan Zheng, Yanyan Chen; analysis and interpretation of results: Yuyang Zhou, Shuyan Zheng; draft manuscript preparation: Yuyang Zhou, Shuyan Zheng, Feng Feng. All authors reviewed the results and approved the final version of the manuscript.

Declaration of competing interest

There are no financial conflicts of interest to disclose.