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Graphical Abstract
A Queue-epidemic Model: Revealing the Transmission Mechanism of Epidemics in Metro Systems from a
Meso Level
Aoping Wu, Lu Hu, Pan Shang, Juanxiu Zhu
Train
Train
Outside()
Hall
Platform
Station
boarding
leaving
Up-direction
Down-direction
Highlights
A Queue-epidemic Model: Revealing the Transmission Mechanism of Epidemics in Metro Systems from a
Meso Level
Aoping Wu, Lu Hu, Pan Shang, Juanxiu Zhu
•A meso-level Queue-Epidemic Model (QEM) is proposed for metro systems.
•The Macro-level Epidemic Model (MEM) is proved to underestimate the epidemic transmission compared with
QEM.
•A recursive algorithm is proposed to solve the QEM.
•The proposed QEM is validated versus the micro-level agent-based simulation.
•The effectiveness of four prevention measures is evaluated by QEM and MEM.
A Queue-epidemic Model: Revealing the Transmission Mechanism of
Epidemics in Metro Systems from a Meso Level
Aoping Wua,b, Lu Hua,b,∗, Pan Shangc, Juanxiu Zhub
aSchool of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China
bNational Engineering Laboratory of Integrated Transportation Big Data Application Technology, Chengdu, 610031, China
cSchool of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China
Abstract
The COVID-19 pandemic has affected communities worldwide. The metro system, an essential means of public
transportation in many cities, is particularly vulnerable to the spread of the virus due to its limited space and com-
plex passenger flow structure. As the basis of quick and effective management decision-making, it is very important
but intractable to accurately and quickly capture the transmission mechanism of epidemics. This study addresses
this challenge by proposing a meso-level Queue-Epidemic model (QEM). The QEM integrates a feedback queuing
network model, which captures the nonlinear stochastic effect of the congestion propagation on passenger mobility
dynamics in the metro system, with an extended SEIAR (Susceptible, Exposed, Infected, Asymptomatic, and Recov-
ered) epidemic model in multiple-subgroups situation. The risk of metro systems is measured using the Total Number
of Newly Exposed Travellers (TNNET). The Macro-level Epidemic Model (MEM) is proved to underestimate the
TNNET compared with the proposed QEM. A recursive algorithm is proposed to solve the QEM, with time complex-
ity that is independent of passenger volumes and station and train capacities, making it suitable for the analysis and
decision-making of large-scale metro networks. The QEM is validated versus the micro-level agent-based simulation.
Numerical experiments reveal some interesting findings: (1) The gap between QEM and MEM in terms of TNNET
will become larger with congestion; (2) From QEM, low-demand and low-risk stations may become high-risk because
of congestion propagation, while MEM does not think so; (3) There exists a moderate social distance to minimize the
risk, and the optimal value increases with the growing travel demand; and (4) The allowed entering proportion control
is more effective than social distancing.
Keywords: metro system, COVID-19, transmission mechanism, queuing model, epidemic model
PACS: 0000, 1111
2000 MSC: 0000, 1111
1. Introduction
The COVID-19 pandemic continues to ravage the global community, with transmission primarily occurring through
respiratory droplets and close contact (Galbadage et al., 2020). Despite widespread vaccination efforts, the situation
remains challenging. The emergence of new strains of the virus, such as the highly transmissible Delta (Yu et al.,
2021) and Omicron variants (Tian et al., 2022), has added further complications to the containment of the epidemic.
As of October 30, 2022, the World Health Organization reports over 620 million confirmed cases and over 6.5 million
deaths worldwide.
In recent years, numerous studies have been conducted on the COVID-19 pandemic, covering topics such as the
allocation of epidemic prevention resources (Abdin et al., 2023), the evaluation of epidemic prevention measures
(Prem et al., 2020; Ku et al., 2021), and the spread of the epidemic through public transport (Mo et al., 2021).
Particularly, some studies (Chen et al., 2020; Qian and Ukkusuri, 2021; Liu et al., 2022) have emphasized the critical
∗Corresponding author
Email address: hulu@swjtu.edu.cn (Lu Hu)
Preprint submitted to Transportation Research Part C April 19, 2023
link between the movement of people and the outbreak of epidemics. Most of these studies are based on either
micro-level or macro-level epidemic models. The micro-level models consider each individual an agent and simulate
interactions between agents and the environment. This approach allows for the consideration of heterogeneity among
individuals and the modelling of complex interactions in real-world systems, particularly in closed systems where the
spread of the infectious disease can be modelled in detail. However, this method may not be computationally feasible
when dealing with large-scale problems. On the other hand, the macro-level models are population-based and ignore
individual heterogeneity and behaviours. This method is computationally efficient as the number of variables is not
dependent on the number of individuals. It can be expressed mathematically to derive the basic reproduction number
R0(Diekmann et al., 2010) to quickly estimate the long-term trajectory of the outbreak. However, this approach
may not accurately capture the diffusion patterns of COVID-19, particularly in complex networks. To the best of our
knowledge, there is a lack of studies that focus on meso-level epidemic models, which combine the advantages of
both micro-level and macro-level models. The meso-level models have the potential to accurately and quickly capture
the transmission mechanism of epidemics. However, developing such models is a challenging task, given the complex
traffic flows and the dynamic random interaction between passengers with different infection statuses.
In this research, the transmission mechanism of COVID-19 in a metro system is analyzed from a meso-level
perspective. As the backbone of public transport in the metropolis, the metro system often faces pressure from over-
loading passengers and accompanying queues. More and more citizens choose the metro as their daily commuting
means of transportation in the context of urbanization. In China, there are four cities’ metro systems with a daily
average passenger ridership of more than 5 million: Beijing, Shanghai, Guangzhou, and Shenzhen. Heavy ridership
makes the transmission mechanism of COVID-19 in a metro system complex (nonlinear) and unpredictable (stochas-
tic). Firstly, with the time-varying and stochastic travel demand, the system may face a sudden surge of passengers
during peak hours. Secondly, in the same facility, inbound and outbound passengers interleave with each other, which
intensifies the interaction and cross-infection of passengers. Thirdly, in facilities with limited capacity (such as halls
and platforms), the travel time is stochastic and depends on the density (a nonlinear relationship) as pointed out by
Smith (1991), and thus congestion often occurs. It not only sharply reduces the service rate of this facility (Hu et al.,
2019), but also makes the passengers in the upstream facilities have to queue, that is, congestion propagation. This
phenomenon of congestion propagation not only appears in continuous facilities in the same station but also spreads
between different stations with the movement of trains. Fourthly, the congestion propagation lengthens the travel
time of passengers and makes they stay in a hotbed of epidemic diseases (the metro with complicated passenger flow
structure and poor air circulation) for a long time. Therefore, describing these realistic features for revealing the
transmission mechanism in a metro system is very important but intractable. Hereafter, these features are referred to
as the nonlinear stochastic impact of congestion propagation unless otherwise specified.
To fight COVID-19 and avoid large crowd gatherings, many countries require their citizens to wear masks in public
places and keep a certain social distance. The difficulty in maintaining social distancing within the metro system is a
significant challenge, as passengers face a trade-offbetween leaving the system quickly and keeping a safe distance.
In China, the metro system has reduced the risk of the crowd gathering by controlling the allowed entering proportion
of passengers and mandating social distancing at entrance gates. Other measures, such as adjusting the train dwell
time and increasing departure frequency, may also be effective in reducing congestion (Chen, 2022) and (Li et al.,
2021b).
In this study, we aim to evaluate the nonlinear stochastic impact of congestion propagation on the spread of
COVID-19 in metro systems using a meso-level mathematical model. To achieve this, we propose a Queue-Epidemic
Model (QEM), which integrates an extended SEIAR (Susceptible, Exposed, Infected, Asymptomatic, and Recovered)
epidemic model with a multi-customer feedback queuing network model Mt/G(x)/C/C. Here, the SEIAR model
is extended to consider the interaction and cross-infection among multiple passenger subgroups (i.e., multi-class
customers including inbound and outbound passengers in different infection status). The arrival process of each
passenger, denoted as Mt, follows a non-homogeneous Poisson distribution with a time-varying rate. The service time
for a passenger is defined as the time taken to traverse the facility. This travel time follows a general distribution G. It is
also dependent on the state, represented by the number of passengers (x) that are simultaneously using the facility, and
is therefore state-dependent. When a facility becomes congested (xreaches its capacity C), congestion propagation
occurs in the upstream facility, causing passengers in the upstream facility to repeatedly visit the downstream facility.
This phenomenon is referred to as feedback queues. The proposed QEM enables us to assess the epidemic intensity
within the metro system accurately. To quantify the epidemic severity, we introduce the concept of the Total Number
2
of Newly Exposed Travellers (TNNET), which refers to the accumulated number of travellers who transition from
a susceptible status to an exposed status after arriving at the metro system. We also compare the TNNETs under
different mitigation measures, including the allowed entering proportion control, the train dwell time adjusting, social
distancing, and mandatory mask-wearing.
1.1. Literature Review
Our research intersects with several key areas of study: (1) metro passenger flow modelling, (2) congestion prop-
agation modelling, (3) epidemic modelling, and (4) the operation and management of public transport during the
COVID-19 pandemic. By bringing together insights from these fields, we aim to provide valuable insights into man-
aging public transportation systems during a pandemic.
The movement process of passengers in metro systems can be analyzed through the three commonly utilized
traffic flow models: the micro-level simulation model, the macro-level network flow model, and the meso-level queue
model. Micro-level simulation models, such as discrete event simulation (Liu et al., 2021) and agent-based simulation
(Wang et al., 2013; Zou et al., 2021), have been widely utilized to study passenger mobility process in metro systems.
The agent-based simulation method, in particular, considers a single passenger as the basic unit and can simulate
various complex behaviours in reality, but at the cost of increased complexity. Micro-level simulation models may
require a lengthy simulation time for multi-node and high-complexity networks, and the simulation results are prone to
randomness, hindering the stability of the results. These limitations make micro-level simulation methods unsuitable
for quick decision-making by managers. Macro-level network flow models (Shi et al., 2022; Niu and Zhou, 2013),
on the other hand, treat passengers as a homogeneous population behaving like a fluid or gas. These models are
computationally efficient but ignore the nonlinear stochastic impact of congestion propagation. Thus integrating them
into epidemic models may underestimate the risk of metro systems. In contrast, meso-level queue models could
better consider realistic features and meanwhile have a low computational complexity (Chen, 2022; Xu et al., 2014;
Smith, 1991). The Markov feedback queuing network models M/G(n)/C/C(Xu et al., 2014; Smith, 1991) consider
the stochastic demand, the stochastic and state-dependent walking speed of customers, and congestion propagation
among facilities. However, they are stationary and time-invariant models, and thus can not depict the dynamic (time-
varying) stochastic system. Based on the fluid queuing model Mt/G(x)/C/Cproposed by Hu et al. (2019), Chen
(2022) developed the feedback queuing network version to model the dynamic stochastic metro system, but he only
considered a single inbound passenger flow on side platforms. Given the transmission characteristic of COVID-19
and the universality of island platforms, a queuing network model with multiple passenger flows is more suitable
for metro systems with island platforms. In this paper, we extend the Mt/G(x)/C/Cqueuing models of Chen (2022)
and Hu et al. (2019) to multi-class feedback queuing network model by considering multi inbound-passenger flow
and multi outbound-passenger flow in different infection status. This feedback queuing network model can capture
nonlinear stochastic effects of congestion propagation in metro systems and be integrated into epidemic models, but
it brings enormous challenges due to the dynamic stochastic interaction among multiple flows.
Congestion propagation in the case of road or railway traffic (vehicles) is studied extensively (Bourrel and Henn,
2002). It can be divided into two main categories: data-driven artificial intelligence (AI) modelling and classic traffic
flow modelling. Each approach has its own strengths and weaknesses, and the choice of approach will depend on the
specific application and data availability. Data-driven AI modelling has become increasingly popular in recent years
due to the availability of large amounts of traffic data and advances in machine learning techniques. One advantage
of data-driven AI modelling is its ability to capture complex patterns and relationships in data that classic traffic
flow models may struggle to identify (Fan et al., 2019; Luan et al., 2022). However, data-driven AI models can be
difficult to interpret and may require large amounts of high-quality data to train effectively. Therefore, data-driven
AI modelling is suitable to manage and control the system after a long-term operation, and it may not be appropriate
for planning, design and operation decisions in the early stages of the system due to a lack of high-quality data. The
primary objective of classic traffic flow modelling is to uncover the correlation among system parameters, with much
less reliance on data. It’s highly applicable in system planning, design, and operation. These models can be broadly
categorized into three levels: micro, macro, and meso. Micro-level models (e.g., car following) focus on individual
vehicle behaviour (Treiber et al., 2000). Macro-level models describe the traffic flow by aggregation methods. For
instance, Daganzo (1994) proposed the cellular transmission model (CTM) to describe the congestion propagation
process in the road network. Long et al. (2011), Tao et al. (2016), and Zhang and Gao (2012) extended the CTM
to model the congestion propagation in different scenarios. Saberi et al. (2020) investigated the spreading of traffic
3
jams in urban networks by the famous Susceptible-Infected-Recovered (SIR) epidemic model. B¨
uchel et al. (2020)
observed empirical data and Dekker et al. (2022) built a diffusion-like model to research the propagating process
of the delays in railway networks. Meso-level models transform the congestion propagation process into a dynamic
queuing problem where the system randomness and heterogeneity could be considered in detail by aggregation or
disaggregation techniques (Luan et al., 2022; Helbing, 2003; Osorio and Yamani, 2017). Helbing (2003) proposed
a deterministic dynamic queuing model to describe the hysteresis of the traffic flow and congestion propagation.
They created a solid foundation in the dynamic queuing modeling field of congestion propagation, but they do not
consider the randomness and heterogeneity of traffic flow. Osorio and Yamani (2017) developed a transient tandem
Markovian feedback queuing network Mt/M/1/Cto describe urban traffic dynamics and congestion propagation. They
explicitly model the system randomness and derive the state probabilities at any time, but the queuing model ignores
the state-dependence of service time and the heterogeneity of vehicle traffic. Our proposed queuing network model
simultaneously considers the randomness, state-dependence, heterogeneity, and dynamic congestion propagation. It
could be easily applied for vehicle traffic flow because of the versatility of Mt/G(x)/C/C.
Studies on the transmission dynamics of epidemics can be broadly categorized into two groups: macro-level
models and micro-level models. The most well-known and classic macro-level model is the SIR model (Kermack
and McKendrick, 1927), which has been followed by several other population-based models such as the SIS model
(Yang et al., 2022), the SIRS model (Jin et al., 2007), and the SEI model (Sigdel and McCluskey, 2014). In particular,
the SEIAR model (Chen et al., 2020; Jia and Chen, 2021) is widely used to model the COVID-19 pandemic, as it
incorporates five statuses of individuals: susceptible (S), exposed (E), infected (I), asymptomatic (A), and recovered
(R). Qian and Ukkusuri (2021) proposed a novel modelling method that embed the SEIR model over the mobility
dynamics. Despite the simplicity of macro-level models, but the homogeneous nature of the population makes them
difficult to accurately reflect the complex traffic behaviours and contact patterns of individuals in reality. To address
these limitations, micro-level models have been proposed. For instance, the agent-based simulation model (Li et al.,
2021a) is capable of considering individual heterogeneity and is suitable for describing personal social networks
and contact activities. However, the need for large amounts of personal data and long computation times (Mao,
2014; Crooks and Hailegiorgis, 2014; Ku et al., 2021) limit its practical applications in effective and rapid epidemic
prevention and control. Therefore, a meso-level model that combines the advantages of both macro- and micro-level
models is desired to provide a more reliable representation of the transmission dynamics of epidemics. In this paper,
we first extend the SEIAR model by taking into account the interaction among multiple passenger subgroups and
their potential of infecting one another. Then, we incorporate the extended SEIAR model into the proposed queuing
network model to develop the meso-level epidemic model QEM.
According to Chen et al. (2020), the mobility of the population can exacerbate the spread of the epidemic. How-
ever, confining people to their homes for prolonged periods is unrealistic. Hence, some researchers have focused on
examining public transport operations and management during the COVID-19 pandemic. Ku et al. (2021) discov-
ered that the mandatory wearing of masks and social distancing during peak hours on public transportation resulted
in a reduction of COVID-19 transmission by 98.1%. Qian and Ukkusuri (2021) analyzed the trade-offbetween en-
trance screening and external resources (e.g. economic costs and manpower), and then investigated the allocation of
resources in urban transport systems. Sun and Zhai (2020) emphasized the importance of ventilation efficiency in pre-
venting COVID-19 transmission. Our paper contributes to the expanding literature on public transport operations and
management during the COVID-19 pandemic by examining the nonlinear stochastic effect of congestion propagation
on the metro epidemic diffusion. We provide insights into the risk of metro systems and how different anti-epidemic
measures affect it when considering or no considering the nonlinear stochastic effect of congestion propagation.
We list the major characteristics of some closely related studies in Table 1 to show this paper’s innovations.
Although these studies have laid a strong groundwork for studying the transmission mechanism of epidemic in the
metro system, there are still the following gaps:
(a) In the existing metro passenger flow models, micro-level models require the lengthy simulation time for a
complex network, and macro-level models are lack of accuracy. Although Chen (2022) built a meso-level
model describing the metro system with dynamic stochastic demand and walking speeds, his model can not be
applied to study the spread of infectious diseases because it focuses on simple inbound passenger flow on the
side platform. The more realistic metro passenger flow model simultaneously considering complex passenger
flow structure and good expansibility (with epidemic models) must be developed.
4
Table 1: Differences between the main relevant literature and this study.
Publications Model type Epidemic Consideration of traffic flow features
CTFS TD CP SD SR
Shi et al. (2022) Macro-level !
Zou et al. (2021) Micro-level1! ! ! !
Chen (2022) Meso-level2! ! ! !
Saberi et al. (2020) Macro-level ! !
Helbing (2003) Meso-level2! ! !
Osorio and Yamani (2017) Meso-level2! ! ! !
Mao (2014) Micro-level1! ! !
Chen et al. (2020) Macro-level ! !
Qian and Ukkusuri (2021) Macro-level ! !
Ku et al. (2021) Micro-level1! ! ! ! ! !
This study Meso-level2! ! ! ! ! !
1Agent-based Simulation; 2Queuing Theory; Complex Traffic Flow Structure (CTFS); Time-varying Demand (TD); Con-
gestion Propagation (CP); State Dependence (SD); System Randomness (SR)
(b) Congestion propagation is critical for depicting the traffic system dynamics. Data-driven AI modelling is not
suitable for planning, design, and operation decisions in the early stages of the system due to the shortage of
high-quality data. Meso-level dynamic queuing models could balance the accuracy and efficiency compared
with micro- and macro level models. However, the existing dynamic queuing models ignore several realistic
features, such as stochastic demand, stochastic and state-dependent speeds, and the heterogeneity of traffic flow.
A more thorough exposition of congestion propagation must be implemented.
(c) Most of the existing epidemic studies are based on either micro-level or macro-level models. Qian and Ukkusuri
(2021) proposed a novel modelling method that embeds the macro-level SEIR model over the mobility dy-
namics, but they ignored traffic behaviours and contact patterns of passengers. A more accurate and efficient
epidemic model that combine the advantages of both micro-level and macro-level models must be investigated.
1.2. The main contributions of this study
In summary, we present the first meso-level model QEM to comprehensively consider epidemic transmission in a
metro system. Precisely, the main contributions of our study are as follows.
(a) We develop a multi-class Mt/G(x)/C/Cfeedback queuing network model for metro systems by considering
various realistic features, such as mixed passenger flows, dynamic stochastic demand, interaction among mul-
tiple subgroups of passengers, stochastic and state-dependent speed of passengers, and congestion propagation
among metro facilities. This provides an accurate and efficient passenger flow model of metro systems. As a
differential and probabilistic model, it has a good expansibility with epidemic models. It can be used for the
analysis and decision-making of metro systems under normal circumstances. It could be also applied for vehicle
traffic flow modelling due to the universality of Mt/G(x)/C/Cin traffic circulation systems (such as roads and
corridors).
(b) We enhance the existing respiratory infectious disease diffusion model, the SEIAR model (Chen et al., 2020),
to accommodate the complexities of passenger flow in metro systems. This extension takes into account the
interaction between multiple passenger subgroups and their potential for infecting one another. To fully capture
the dynamics of the epidemic spread within the metro system, we incorporate the extended SEIAR model into
the multi-customer feedback queuing network model, resulting in the establishment of the QEM. Although
passengers with the same departure time, origin, and destination are aggregated to reduce the complexity of
the model, the heterogeneity and traffic behaviour of these aggregated subgroups are modelled by considering
the aforementioned various realistic features. As a result, our proposed QEM is highly efficient and offers a
relatively accurate analysis of epidemic transmission in metro systems.
5
(c) The proposed QEM is proved to have a unique solution, providing a solid foundation for further analysis and
interpretation of the results. We demonstrate that the Macro-level Epidemic Model (MEM) underestimate
epidemic transmission compared with our proposed QEM. A recursive algorithm is proposed to solve the QEM.
Its time complexity is independent of passenger volumes and capacities of stations and trains. Thus, it is suitable
for analysing and optimizing large-scale metro networks during epidemics. We validate the proposed QEM
versus the micro-level agent-based simulation.
(d) In a small-scale case and a real-world scenario in the Chengdu Metro of China, our numerical experiments
implemented by QEM and MEM provide insights into the risk of metro systems and how different anti-epidemic
measures affect it. For instance, from QEM, low-demand and low-risk stations may become high-risk because
of congestion propagation, while MEM does not think so. Unexpectedly, it is found that the allowed entering
proportion control is more effective than social distancing, according to QEM, but MEM is insensitive to these
measures.
The structure of the present study is outlined as follows: Section 2 outlines the problem and the underlying as-
sumptions. The formulation of the metro system feedback queuing network model, which integrates the extended
SEIAR model, is described in Section 3, resulting in the development of the QEM. In Section 4, we discuss two
propositions of the QEM. Section 5 presents a recursive algorithm to solve the QEM. The proposed approach’s per-
formance is demonstrated by examining a small-scale case and a real-world case in Section 6. Finally, we provide
some concluding remarks and suggest avenues for future research in Section 7.
2. Problem description
This study considers a bi-directional metro line system consisting of |N| stations, as illustrated in Fig. 1. Each
station is indexed by i∈ N ={1,2, ...i, ..., |N |}, where Nrepresents the set of stations. For simplicity, each station is
assumed to have one hall and platform, and passengers move through the hall and wait at the platform for the service
trains. Once they reach their destination, they pass through the platform and hall again to exit. The service trains are
numbered consecutively as k∈ K ={1,2, ..., |K |}, where Krepresents the set of service trains. The direction of the
operating service trains is indicated by u∈ U ={1,2}, where u=1 (down-direction) indicates the service train is
moving from station ito station i+1, and u=2 (up-direction) represents the opposite direction.
Station 1 Station 2 ...... Station 𝒾...... Station 𝒩 − 1 Station 𝒩
Down direction（𝒰 = 1）
Up direction（𝒰 = 2）
Turn-
back
tracks
Turn-
back
tracks
Fig. 1. Structure of the metro line.
We consider all platforms to be island platforms, as depicted in Fig. 2, so the high risk of intermingling passenger
flows in the metro system can not be ignored. This type of platform is shared by passengers waiting for up-direction
and down-direction trains. And it is also shared by all disembarking passengers. The close contacts between these
passengers increase the risk of infection, especially during peak hours. The complex and dense passenger flow struc-
ture, combined with high passenger volume, creates a challenging environment for epidemic control and prevention
measures. The situation is similar in the hall. This scenario highlights the need for a comprehensive and effective
solution to explain the spread of epidemics in metro systems.
Trapped by this, we devote ourselves to exploring a meso-level epidemic model to understand the transmission
mechanism of the epidemic in a metro system with a bi-directional metro line. Our goal is to understand how the
6
Island platform
Fig. 2. Structure of the island platform
number of passengers and newly exposed travellers in different facilities of the metro system change over time. Using
the proposed method, we also intend to evaluate the effectiveness of various epidemic prevention measures, such as
the allowed entering proportion control, train dwell time adjusting, social distancing, and mandatory mask-wearing in
different travel demand scenarios. The ultimate goal is to provide a basis for large-scale decision-making in epidemic
control and prevention in metro systems.
The notation xi(t) represents the number of overall passengers in the hall iat time t, while yi(t) represents the
number of overall passengers on the platform iat time t. Notion tis a moment in the timeline T, namely t∈ T .
The subgroups of these passengers are defined as xd
i,u(t) and yd
i,u(t), respectively, where the purpose of the subgroups
is indexed by d∈ D ={1,2}. If d=1, the passengers in the subgroup intend to board the train, while if d=2, they
intend to leave the station through an exit. The direction of the train is indexed by u∈ U ={1,2}. The number of
passengers in the service train is represented by zk,u(t). e∈ P ={S,E,I,A,R}is the index for the epidemiological
status of the passengers. The number of passengers with each status in each subgroup is represented as xd,e
i,u(t), yd,e
i,u(t)
or ze
k,u(t) for the hall, platform, and train subgroups, respectively.
In this paper, some assumptions have been made to establish the meso-level epidemic model:
Assumption 1. The time-varying travel demand is known and we do not consider the elasticity of travel demand.
This information can be obtained from historical data or short-term ridership forecasting.
Assumption 2. Given the poor ventilation and relatively closed spaces of the metro system, it is assumed that virus
carriers can transmit to susceptible individuals within the same facility. The relationship between transmissibility and
population density is linear, meaning the higher the density, the faster the transmission.
Assumption 3. Passengers with the same origin station but different destinations arrive at the origin station in
random orders, and the spread of the epidemic is also random. Therefore, it is reasonable to assume that passengers
of different epidemiological statuses with different destinations are evenly mixed, which makes it easier to express
transitions between different subgroups within the population. A similar assumption in the study of passenger flow
control has been given by Shi et al. (2022).
3. Model
In Section 3.1, we present the SEIAR model as a tool to capture infectious disease dynamics, taking into account
the characteristics of COVID-19. In Section 3.2, we address the complex passenger flow structure and the nonlin-
ear stochastic impact of congestion propagation on mobility dynamics by developing a feedback queuing network
model of metro traffic flow. This model partitions passengers into multiple subgroups. In Section 3.3, the SEIAR
model is extended to accommodate the situation of multiple subgroups, and then it is integrated into our proposed
7
Susceptible Exposed
Infected
Asymptomatic
Recovered
1
2
)1( −
Fig. 3. Schematic diagram of the disease dynamics for the COVID-19 pandemic.
feedback queuing network model to construct the QEM for metro systems. The QEM sheds light on the transmission
mechanisms of infectious diseases within metro systems from a meso-level perspective.
3.1. The SEIAR model
The SIR model, first introduced by Kermack and McKendrick (1927), is a classic macro-level epidemic model
that has been widely used to analyze pandemic dynamics in long term, such as plague and cholera. The SIR model
divides the population into three statuses: Susceptible (S), Infected (I), and Recovered (R). In this study, we use the
SEIAR model (Chen et al., 2020) to capture the dynamics of COVID-19, which includes two additional populations:
the Exposed (E) and the Asymptomatic (A). The symbols used in this section are listed in Table A1.
•The susceptible status refers to individuals who have not developed immunity to the virus. The susceptible
individuals will switch to the exposed status when encountering virus carriers, namely, infected, asymptomatic,
and exposed individuals.
•The exposed status denotes the population who are lurked by the virus. People in this status, eventually switch-
ing to the infected or asymptomatic statuses, are also infective to the susceptible individuals.
•The infected status represents the individuals who have been infected and have symptoms. They can expose
susceptible people.
•The asymptomatic status refers to individuals who have been infected but do not display symptoms.
•The recovered status represents individuals who have either been cured or developed immunity to the virus.
While they are immune in this status, they will return to being susceptible in the future.
The dynamics of the SEIAR model can be depicted through a flowchart, Fig. 3, and can be described mathemat-
ically using a set of ordinary differential equations (ODEs) Eq. 1-Eq. 5. The variables S,E,I,A, and Rin the ODEs
represent the number of individuals in the susceptible, exposed, infected, asymptomatic, and recovered statuses, re-
spectively. Over time t, the populations transition from one status to another, forming a complex interplay of disease
spread and recovery.
dS /dt =−β1S(I+kA)/N−β2S E/N+θR(1)
dE/dt =β1S(I+kA)/N+β2S E/N−ωE(2)
dI/dt =(1 −η)ωE−γI(3)
dA/dt =ηωE−γA(4)
dR/dt =γ(I+A)−θR(5)
Specifically, the fundamental rules (process) of the SEIAR model are as follows:
(1) The system is closed, meaning the total number of individuals Nis constant and equal to S+E+I+A+R.
8
(2) The population is evenly mixed and homogeneous, and the parameters β1and β2represent the average number
of susceptible individuals exposed by infected and exposed individuals, respectively. Symbol kis the contagion
intensity of the asymptomatic population relative to the infected population. These factors are reflected in the first and
second terms on the right side of Eq. 1 and Eq. 2.
(3) The latent period of the virus in exposed individuals is represented by an average length of 1/ω. The exposed
individuals turn into either asymptomatic or infected individuals with probability ηor (1 −η), respectively. These
processes are shown in the first term on the right side of Eq. 3 and Eq. 4.
(4) The average lengths of the recovery and immunity periods are 1/γ and 1/θ, respectively. Infected and asymp-
tomatic individuals will eventually turn into recovered individuals and then become susceptible again, as indicated in
Eq. 5.
3.2. A feedback queuing network model
In the context of modelling the spread of epidemics in metro systems, the mobility dynamics of travellers play a
crucial role. Thus, prior to constructing the QEM, the metro traffic flow is modelled in this section. The M/G(n)/C/C
state-dependent queuing models proposed by Smith (1991) and Yuhaski and Smith (1989) are appropriate for mod-
elling the metro traffic flow as they are initially proposed to model the unidirectional or multidirectional traffic flows in
circulation systems, including corridors, stairways, and other physical paths of movement in metro systems. However,
M/G(n)/C/Cis a stationary queuing model and cannot capture the passenger dynamics. To address this limitation, Hu
et al. (2019) extended the stationary model to a fluid version (Mt/G(x)/C/C) that models the passenger dynamics of
the unidirectional traffic flow of a single facility. In this paper, we extend the Mt/G(x)/C/Cmodel to a network version
that can capture the nonlinear stochastic effect of congestion propagation and is more suitable for modelling the metro
traffic flow with multidirectional passengers and feedback queues (repeat visits). The proposed metro feedback queu-
ing network model has several advantages: (1) The model considers both inbound and outbound passengers, making
it more consistent with the reality of infectious disease transmission compared to existing traffic flow models that only
consider inbound passengers (Shi et al., 2018; Niu and Zhou, 2013); (2) The model is expressed as a series of differ-
ential equations, making it more suitable for integration with the SEIAR model; and (3) The time complexity of the
algorithm is independent of the passenger volume and capacities of stations and trains, making the model applicable
to large-scale networks.
In the following, we introduce the proposed feedback queuing network model of metro traffic flow. We begin by
presenting the framework in section 3.2.1. Next, in sections 3.2.2, 3.2.3, and 3.2.4, we describe the queuing models
for passengers in the hall, on the platform, and boarding and alighting on the train, respectively. A summary of the
symbols used in the feedback fluid queuing network model can be found in Table A2.
3.2.1. A feedback queuing network framework
According to the theory of fluid queues (Wang et al., 1996), the average number of passengers in a queuing system
at time tcan be represented as the system state, denoted by x(t). The flow conservation principle suggests that the
change rate of the system state is equal to the difference between the average input flow rate, represented by f in(t),
and the average output flow rate, represented by f out(t). The change rate of the system state with respect to time can
be mathematically expressed as Eq. 6, which is commonly referred to as the fluid equation.
x(t)′=f in(t)−f out(t) (6)
In the Mt/G(x)/C/Cqueuing system, let λ(t) represent the time-varying arrival rate and µ(x(t)) denote the service
rate. The idle probability of the system with finite capacity is represented by PE(x(t)) and the blocking probability by
PB(x(t)). These parameters are dependent on the system state x(t), and their calibration procedures are described in
Appendix B. The term ”feedback” refers to the fact that customers may visit the system repeatedly if it is congested.
Therefore, we use λ(t) to denote the current arrival rate, which is equal to the difference between the total numbers of
customers who have arrived and entered the system as described in Eq. 7. The input and output flow rates at time t
9
can be formulated by Eq. 8 and Eq. 9, respectively.
λ(t)=Zt
0
λ(τ)dτ−Zt
0
λ(τ)1−PB(x(τ))dτ(7)
f in(t)=λ(t)1−PB(x(t))(8)
f out(t)=µ(x(t))1−PE(x(t))(9)
Now consider two consecutive Mt/G(x)/C/Cqueuing systems, the states of which can be represented by x1(t) and
x2(t), as depicted in Fig. 4. The blocking probabilities, idle probabilities, and service rates of the first and second
𝒙𝟏(𝒕) 𝒙𝟐(𝒕)
𝒇𝒊𝒏𝟏𝒕 𝒇𝒐𝒖𝒕𝟏𝒕 = 𝒇𝒊𝒏𝟐𝒕𝒇𝒐𝒖𝒕𝟐𝒕
Fig. 4. Schematic diagram of two consecutive queuing systems
systems are represented by PB1(x1(t)), PB2(x2(t)), PE1(x1(t)), PE2(x2(t)), µ1(x1(t)), and µ2(x2(t)), respectively. The
current arrival rate λ1(t) and input flow rate f in1(t) of the first system can be calculated using Eq. 10 and Eq. 11. The
output flow rate f out1(t) depends not only on its own idle probability but also the blocking probability of the down-
stream system as described by Eq. 12. This is known as the congestion propagation phenomenon, where customers in
the upstream system have to queue if the downstream system is blocked.
λ1(t)=Zt
0
λ1(τ)dτ−Zt
0
λ1(τ)1−PB1(x1(τ))dτ(10)
f in1(t)=λ1(t)1−PB1(x1(t))(11)
f out1(t)=µ1(x1(t))1−PE1(x1(t))1−PB2(x2(t))(12)
The input flow rate for the second queuing system is equal to the output flow rate of the first system, as expressed
in equation Eq. 13. The output flow rate for the second system can be calculated as shown in equation Eq. 14, which
is similar to the calculation for a single queuing system.
f in2(t)=f out1(t) (13)
f out2(t)=µ2(x2(t))1−PE2(x2(t))(14)
In the metro system, passengers are categorized into four categories based on their queuing locations: outside,
hall, platform, and train. The system states for each of these categories are denoted by o,x,y, and z, respectively. The
capacities of the hall, platform, and train are defined as Cx
i,Cy
i, and Cz
k,u, respectively. Note that the outside queuing
area is considered to have an infinite capacity. For passengers in the hall at station i, they are further divided into four
subgroups. The number of passengers in each subgroup is represented by xd
i,u(t), where i,u, and dstand for the station
number, the train direction, and the passenger’s purpose, respectively. For example, x2
i,1(t) represents the number of
passengers in the hall of station iwho alight from the down-direction (u=1) train and plan to exit (d=2) the station.
Similarly, the number of passengers on the platform of station iis divided into four subgroups, with each subgroup
represented by yd
i,u(t). The states of the hall and platform of station ican be obtained by summing up the number of
passengers of all subgroups, as defined in Eq. 15 and Eq. 16. Regarding the number of passengers in the k-th service
train, only the operating direction is considered, represented by zk,u(t).
xi(t)=X
d∈D X
u∈K
xd
i,u(t),∀i∈ N (15)
yi(t)=X
d∈D X
u∈K
yd
i,u(t),∀i∈ N (16)
10
The passengers mobility pattern in the metro system is shown as Fig. 5. The state of each subgroup (represented by
yellow rectangles in Fig. 5) is associated with two corresponding flow rates, represented as f in and f out (represented
by blue rectangles in Fig. 5). The subscript and superscript notations used in the flow rates allow for clear indexing of
the specific subgroup. Note that the relationship between some flows is established by the system topology, such as the
connection between f outx,1
i,1(t) and f inty,1
i,1(t). These fluid equations are used to analyze and understand the dynamics
of the metro system.
xd
i,u(t)′=f inx,d
i,u(t)−f outx,d
i,u(t),∀i∈ N,u∈ U,d∈ D (17)
yd
i,u(t)′=f iny,d
i,u(t)−f outy,d
i,u(t),∀i∈ N,u∈ U,d∈ D (18)
zk,u(t)′=f inz
k,u(t)−f outz
k,u(t),∀k∈ K,u∈ U (19)
Train
Train
Outside()
Hall
Platform
Station
boarding
leaving
Up-direction
Down-direction
Fig. 5. System passengers mobility pattern diagram.
3.2.2. Queue model for passengers in the hall
In this section, we will build the queue model for passengers in the hall. The rectangular entity labelled as Hall
in Fig. 5 includes four subgroups, which have four input flows, denoted by f inx,1
i,1,f inx,1
i,2,f inx,2
i,1, and f inx,2
i,2, and four
output flows, denoted by f outx,1
i,1,f outx,1
i,2,f outx,2
i,1, and f outx,2
i,2. The dotted line in Fig. 5 distinguishes between the
boarding process (d=1) on the left and the leaving process (d=2) on the right. Our model development will start
with the boarding process (d=1) and then proceed to the leaving process (d=2).
According to Niu and Zhou (2013); Yin et al. (2016), the dynamic passenger demand can be represented as Eq. 20,
a time-varying original-destination matrix (TOD). The matrix element, odi,j(t), represents the number of passengers
11
arriving at station iwith destination station jat time t. The time-varying passenger demand data can be accurately
predicted from the historical automatic fare collection (AFC) system.
T OD(t)=
0od1,2(t). . . od1,N(t)
od2,1(t) 0 . . . . . .
.
.
..
.
....odN−1,N(t)
odN,1(t). . . odN−1,N(t) 0
(20)
In order to model the boarding passenger flow, it is necessary to consider the direction to take the train (u) and at
which station to get off(j). Therefore, we further mark the direction of odi,j(t) and denote it as λi,j,u(t) using Eq. 21.
Due to the limited capacity of the hall, if the hall is blocked, passengers may not be able to enter the hall immediately
and would have to wait outside and visit the hall repeatedly. The blocking probability of hall iis denoted by PBx
i(xi(t)),
which changes based on the state of the hall i xi(t). Hence, it is a state-dependent function. The number of current
arrival passengers for each OD denoted as λi,j,u(t), is equal to the difference between the total numbers of passengers
who arrived and entered hall i, as shown in Eq. 22.
λi,j,u(t)=
odi,j(t),j≥i(u=1)
odi,j(t),j<i(u=2) ,∀i,j∈ N,∀u∈ U (21)
λi,j,u(t)=
total passengers
actually arrived
z }| {
Zt
0
λi,j,u(τ)dτ−
total passengers actually entered
z }| {
Zt
0
λi,j.u(τ)·1−PBx
i(xi(τ)
| {z }
unblocking probability
of the hall
dτ, ∀i,j∈ N,u∈ U (22)
The input flow rate of boarding passengers for each origin-destination (OD) pair at hall iis denoted by f inx,1
i,j,u(t).
This rate is determined by multiplying the corresponding current arrival rate, λi,j,u(t), with the unblocking probability
of hall i,1−PBx
i(xi(t)). This relationship can be represented mathematically as Eq. 23. The input flow rate of
boarding passengers subgroup x1
i,u(t) can be obtained by summing up f inx,1
i,j,u(t) over corresponding OD pairs, as
Eq. 24.
f inx,1
i,j,u(t)=λi,j,u(t)·1−PBx
i(xi(t)),∀i,j∈ N,u∈ U (23)
f inx,1
i,u(t)=
PN
j=i+1f inx,1
i,j,u(t),u=1
Pi−1
j=1f inx,1
i,j,u(t),u=2,∀i∈ N,u∈ U (24)
The output flow rate of boarding passengers for each OD pair f outx,1
i,j,u(t) can be calculated by Eq. 25. In this
equation, x1
i,j,u(t)/xi(t)represents the ratio of the corresponding passengers to the total number of individuals in
hall i, and µx
i(xi(t)) is the mean service rate of hall i. Additionally, PEx
i(xi(t)) and PBy
i(yi(t)) represent the state-
dependent idle probability of hall iand the blocking probability of platform i, respectively. Therefore, µx
i(xi(t))1−
PEx
i(xi(t))1−PBy
i(yi(t))can denote the flow rate from hall ito platform iwhen and only when the hall is non-empty
and the platform is unblocked. The output flow rate of boarding passengers subgroups of hall ican be expressed as
the summation of the output flow rates for the corresponding OD pair, which is given by Eq. 26.
12
f outx,1
i,j,u(t)=
corresponding
ratio of the passengers
z }| {
x1
i,j,u(t)/xi(t)·
the maximal flow rate from the hall to the platform
z }| {
µx
i(xi(t))
| {z }
service rate
of the hall
1−PEx
i(xi(t))
| {z }
nonempty probability
of the hall
1−PBy
i(yi(t))
| {z }
unblocking probability
of the platform
,∀i,j∈ N,u∈ U (25)
f outx,1
i,u(t)=
PN
j=i+1f outx,1
i,j,u(t),u=1
Pi−1
j=1f outx,1
i,j,u(t),u=2,∀i∈ N,u∈ U (26)
In pedestrian flow theory (Smith, 1991), the calculation of µx
i(xi(t)) is described in Eq. 27, where vx
i(xi(t)) repre-
sents the state-dependent velocity of customers in hall iand Lx
idenotes the walking distance in hall i. The procedure
for calibrating state-dependent variables (vx
i(xi(t)),PBy
i(yi(t)) and PEx
i(xi(t))) is outlined in Appendix B.
µx
i(xi(t)) =xi(t)·vx
i(xi(t))/Lx
i,∀i∈ N (27)
In Eq. 25, x1
i,j,u(t) represents the decomposed form of x1
i,u(t) by destination j. Eq. 28 is its fluid equation.
x1
i,j,u(t)′=f inx,1
i,j,u(t)−f outx,1
i,j,u(t),∀i,j∈ N,u∈ U (28)
Now, let’s consider the leaving process (d=2), represented by the right portion of the Hall rectangle in Fig. 5.
It is important to note that for departing passengers, there is no need to differentiate by destination j. And it is noted
that, for leaving subgroups of the hall, the input flow rates come from the process of alighting passengers passing
through the platform and entering the halls.
Similar to Eq. 25 the input flow rate f inx,2
i,u(t) can be calculated as in Eq. 29. It is equal to the product of the
corresponding ratio, represented by x2
i,u(t)/xi(t), and the output flow rate from platform ito hall i, represented by
µy
i(yi(t))1−PEy
i(yi(t))1−PBx
i(xi(t)).
f inx,2
i,u(t)=y2
i,u(t)/yi(t)·µy
i(yi(t))1−PEy
i(yi(t))1−PBx
i(xi(t)),∀i∈ N,u∈ U (29)
To calculate the output flow rate, in Eq. 30, x2
i,u(t)/xi(t)denotes the ratio of the corresponding passengers to total
passengers in hall iand µx
i(xi(t))1−PEx
i(xi(t))represents the output flow rate from hall ito outside when and only
when the hall is non-empty. Because the capacity of the outside is considered to be infinite, the blocking probability
is always assumed to be zero.
f outx,2
i,u(t)=x2
i,u(t)/xi(t)·µx
i(xi(t))1−PEx
i(xi(t)),∀i∈ N,u∈ U (30)
3.2.3. Queue model for passengers on the platform
The relationships and structure of the fluid flows of the platform are very similar to those of the hall. We will again
start with the boarding process.
Due to the serial relationship between the hall and the platform in the same station, the output flow of the hall
and the input flow of the platform refer to the same group of boarding passengers. This relationship is shown in
Eq. 31-Eq. 32.
f iny,1
i,j,u(t)=f outx,1
i,j,u(t),∀i,j∈ N,u∈ U (31)
f iny,1
i,u(t)=f outx,1
i,u(t),∀i∈ N,u∈ U (32)
To illustrate the connection relationship between platforms and trains, we introduce the binary variable PTi,k,u(t).
This variable equals to 1 if the k-th service train in direction uis dwelling at platform i, and 0 otherwise. The output
flow rate of the platform for each OD f outy,1
i,j,u(t), can be calculated using Eq. 33. Here, y1
i,j,u(t)/y1
i(t)represents the
13
ratio of the corresponding passengers to the total number of individuals on platform i, and µy
i(yi(t))1−PEy
i(yi(t))1−
PBz
k,u(zk,u(t))represents the flow rate from platform ito train kwhen the platform is non-empty and the train is not
blocked, similar to Eq. 25. Finally, the output flow rate for the boarding subgroups of platform ican be obtained using
Eq. 34.
f outy,1
i,j,u(t)=X
k∈K
PTi,k,u(t)·y1
i,j,u(t)/y1
i(t)·µy
i(yi(t))1−PEy
i(yi(t))1−PBz
k,u(zk,u(t)),∀i,j∈ N,u∈ U (33)
f outy,1
i,u(t)=
PN
j=i+1f outy,1
i,j,u(t),u=1
Pi−1
j=1f outy,1
i,j,u(t),u=2,∀i∈ N,u∈ U (34)
As the decomposed form of y1
i,u(t) by destination j,y1
i,j,u(t) represents the number of passengers going to station j
on the platform iat time t. Eq. 35 is its fluid equation.
y1
i,j,u(t)′=f iny,1
i,j,u(t)−f outy,1
i,j,u(t),∀i,j∈ N,u∈ U (35)
For leaving subgroups of platform i, the input flow rate f iny,2
i,u(t) is equivalent to the output flow rate of stopped
trains f outz
k,u(t), as represented in Eq. 36. This is because the leaving passengers will enter the platform only when
the train is dwelling at the platform.
f iny,2
i,u(t)=X
k∈K
PTi,k,u(t)·f outz
k,u(t),∀i∈ N,u∈ U (36)
Because of the topological relationships mentioned earlier, the output flow rate f outy,2
i,u(t) equals to the input flow
rate of hall f inx,2
i,u(t), as expressed in Eq. 37.
f outy,2
i,u(t)=f inx,2
i,u(t),∀i∈ N,u∈ U (37)
3.2.4. Queue model for passengers boarding and alighting on the train
The passenger mobility pattern through trains is depicted in the bottom part of Fig. 5. The up-and-down direction
trains are identical, and we provide an illustration of the down-direction trains, where u=1.
The input flow rate of the service train kdenoted as f inz
k,u(t), equals to the output flow rate of the platform at which
the train dwells, as expressed in Eq. 38.
f inz
k,u(t)=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t),∀k∈ K,u∈ U (38)
The output (alighting) flow rate of train kin direction uexpressed as f outz
k,u(t), can be calculated as Eq. 39. The
service rate of train kwhen it dwells at platform jis given by Pj∈N PT j,k,u(t)·Pi∈N Rtd
i,k,u
t′=td
i,k−1,u
f outy,1
i,j,u(t′)dt′/tw
j,k,u.
The departure time of train kin direction ufrom platform iis denoted by td
i,k,u. During the time window between
the departure of train k−1 and train k, passengers can only board train k. The cumulative number of passengers
with destination jon train kis represented by Pi∈N Rtd
i,k,u
t′=td
i,k−1,u
f outy,1
i,j,u(t′)dt′. When train karrives at platform j, only
passengers with destination jintend to get offthe train. The service rate of train kwhen it dwells at platform jis
obtained by multiplying Pi∈N Rtd
i,k,u
t′=td
i,k−1,u
f outy,1
i,j,u(t′)dt′by Pj∈N PT j,k,u(t) and dividing by tw
j,k,u, which means the dwell
time of train kin direction uat platform j. The unblocking probability of platform jis denoted by 1−PBy
j(yj(t))
14
and the non-empty probability of train kis denoted by 1−PEz
k,u(zk,u(t)), as previously defined.
f outz
k,u(t)=
service rate of train kwhen it dwells at platform j
z }| {
X
j∈N
PT j,k,u(t)·X
i∈N Ztd
i,k,u
t′=td
i,k−1,u
f outy,1
i,j,u(t′)dt′
| {z }
the cumulative passengers
with destination jon train k
/tw
j,k,u
·1−PEz
k,u(zk,u(t))1−PBy
j(yj(t)),∀k∈ K,u∈ U (39)
3.3. Queue-Epidemic Model (QEM)
In this section, we propose the QEM, a meso-level epidemic model that incorporates an extended SEIAR model
into the proposed feedback fluid queuing network model. The framework of the QEM is depicted in Fig. 6. The
figure is inspired by Fig.1 in Qian and Ukkusuri (2021). The model requires three inputs: metro system network,
Metro system network
Commuting data
Disease parameters
Input
System Dynamics
S E
I
A
R
Hall TrainPlatform
Mobility Dynamics
Infectious Disease Dynamics
Output
Boarding Boarding
Leaving Leaving
Fig. 6. The framework of the QEM
commuting data, and disease parameters of interest. In the metro system network, dividing the passengers into mul-
tiple subgroups based on location (x,y,z), operating direction (u) and purpose (d), helps to overcome the limitations
of the homogeneity of population in traditional macro-level epidemic models. By combining the commuting data and
disease parameters, the QEM can provide insights into the spread of the disease and estimate the risk in the metro
system. The notations used in the QEM are listed in Table A3 to avoid symbol confusion.
The gap between the existing benchmark model MEM and our proposed QEM in describing the transmission of
infectious diseases in metro systems can be attributed to two main factors: (1) the MEM model fails to account for the
traffic behaviour of passengers, such as congestion propagation, state-dependence, and stochasticity of the system, and
(2) it ignores the contagion caused by the complex passenger flow structure, including the interactions among multiple
subgroups. To address these limitations, we propose considering both passengers’ mobility dynamics and infectious
disease dynamics simultaneously, as visualized in Fig. 7. This figure is inspired by Fig.2 in Qian and Ukkusuri (2021).
Specifically, we take the following dynamics into account:
15
Fig. 7. The schematic diagram for the state transition process in the Queue-Epidemic model.
i). According to the theory of epidemiology, the five statuses of each subgroup in the queue model are further di-
vided into the corresponding statuses of the extended SEIAR model, as shown in Eq. 40-Eq. 42. The superscript
e∈ P ={S,E,I,A,R}is used to index the status. The mobility dynamics of travellers of each status follow the
proposed feedback fluid queuing network model and are represented by yellow dashed lines in Fig. 7. Under
the Assumption 3 of evenly mixed people of different statuses, the flow rates of the population of each status
can be calculated by multiplying the subgroups’ flow rates by the corresponding ratio, such as x1,e
i,u(t)/x1
i,u(t) in
Fig. 7. The ratio of individuals of status eoutside station irepresented by ζe
iin Fig. 7 is fixed based on the
16
epidemic situation of station i.
xd
i,u(t)=X
e∈P
xd,e
i,u(t),∀i∈ N,u∈ U,d∈ D (40)
yd
i,u(t)=X
e∈P
yd,e
i,u(t),∀i∈ N,u∈ U,d∈ D (41)
zk,u(t)=X
e∈P
ze
k,u(t),∀k∈ K,u∈ U (42)
ii). The QEM takes into account the complex passenger flow structure of the metro system, implying that suscep-
tible individuals within a certain subgroup may be at risk of exposure from all carriers within the same facility
(hall, platform, and train). The model uses different functions (gx,d
i,u(t), gy,d
i,u(t), gz
k,u(t)) to account for the new
contagion (newly exposed travellers) in each subgroup at time t. In the hall, the number of susceptible indi-
viduals of a certain subgroup is denoted by xd,S
i,u(t), and the number of virus carriers in the hall includes four
subgroups’ infected people (Pu∈U Pd∈D xd,I
i,u(t)), asymptomatic people (Pu∈U Pd∈D xd,A
i,u(t)), and exposed people
(Pu∈U Pd∈D xd,E
i,u(t)). The new contagion at time tof a certain subgroup in hall ican be expressed by Eq. 43.
Similarly to the hall, the new contagion of the subgroup in the platform ican be calculated by Eq. 44. The
passenger flow structure in the train is relatively simple and follows the traditional SEIAR model as Eq. 45.
gx,d
i,u(t)=β1xd,S
i,u(t)X
u∈U X
d∈D
xd,I
i,u(t)+X
u∈U X
d∈D
kxd,A
i,u(t).xi(t)+β2xd,S
i,u(t)X
u∈U X
d∈D
xd,E
i,u(t).xi(t),∀u∈ U,d∈ D (43)
gy,d
i,u(t)=β1yd,S
i,u(t)X
u∈U X
d∈D
yd,I
i,u(t)+X
u∈U X
d∈D
kyd,A
i,u(t).yi(t)+β2yd,S
i,u(t)X
u∈U X
d∈D
yd,E
i,u(t).yi(t),∀u∈ U,d∈ D (44)
gz
k,u(t)=β1zS
k,u(t)zI
k,u(t)+kzA
k,u(t).zk,u(t)+β2zS
k,u(t)zE
k,u(t).zk,u(t),∀u∈ U,k∈ K (45)
iii). The remaining infectious disease dynamics within each subgroup are entirely consistent with the SEIAR model
represented by the blue line in Fig. 7.
The system dynamics of the QEM is a combination of infectious disease dynamics and mobility dynamics, as
shown in Fig. 7. This integration allows us to account for the interplay between the spread of the disease and the
movement of people. The infectious disease dynamics within each subgroup of system dynamics can be formulated
mathematically as follows.
For boarding subgroups in the hall, we have
x1,S
i,u(t)′=f inx,1
i,u(t)·ξS
i−gx,1
i,u(t)+θx1,R
i,u(t)−f outx,1
i,u(t)·x1,S
i,u(t)/x1
i,u(t)
x1,E
i,u(t)′=f inx,1
i,u(t)·ξE
i+gx,1
i,u(t)−ωx1,E
i,u(t)−f outx,1
i,u(t)·x1,E
i,u(t)/x1
i,u(t)
x1,I
i,u(t)′=f inx,1
i,u(t)·ξI
i+(1 −η)ωx1,E
i,u(t)−γx1,I
i,u(t)−f outx,1
i,u(t)·x1,I
i,u(t)/x1
i,u(t),∀i∈ N,u∈ U
x1,A
i,u(t)′=f inx,1
i,u(t)·ξA
i+ηωx1,E
i,u(t)−γx1,A
i,u(t)−f outx,1
i,u(t)·x1,A
i,u(t)/x1
i,u(t)
x1,R
i,u(t)′=f inx,1
i,u(t)·ξR
i+γx1,I
i,u(t)+x1,A
i,u(t)−θx1,R
i,u(t)−f outx,1
i,u(t)·x1,R
i,u(t)/x1
i,u(t) (46)
For boarding subgroups on the platform, we have
y1,S
i,u(t)′=f iny,1
i,u(t)·x1,S
i,u(t)/x1
i,u(t)−gy,1
i,u(t)+θy1,R
i,u(t)−f outy,1
i,u(t)·y1,S
i,u(t)/y1
i,u(t)
y1,E
i,u(t)′=f iny,1
i,u(t)·x1,E
i,u(t)/x1
i,u(t)+gy,1
i,u(t)−ωy1,E
i,u(t)−f outy,1
i,u(t)·y1,E
i,u(t)/y1
i,u(t)
y1,I
i,u(t)′=f iny,1
i,u(t)·x1,I
i,u(t)/x1
i,u(t)+(1 −η)ωy1,E
i,u(t)−γy1,I
i,u(t)−f outy,1
i,u(t)·y1,I
i,u(t)/y1
i,u(t),∀i∈ N,u∈ U
y1,A
i,u(t)′=f iny,1
i,u(t)·x1,A
i,u(t)/x1
i,u(t)+ηωy1,E
i,u(t)−γy1,A
i,u(t)−f outy,1
i,u(t)·y1,A
i,u(t)/y1
i,u(t)
y1,R
i,u(t)′=f iny,1
i,u(t)·x1,R
i,u(t)/x1
i,u(t)+γy1,I
i,u(t)+y1,A
i,u(t)−θy1,R
i,u(t)−f outy,1
i,u(t)·y1,R
i,u(t)/y1
i,u(t) (47)
17
For subgroups on the train, we have
zS
k,u(t)′=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t)·y1,S
i,u(t)/y1
i,u(t)−gz
k,u(t)+θzR
k,u(t)−f outz
k,u(t)·zS
k,u(t)/zk,u(t)
zE
k,u(t)′=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t)·y1,E
i,u(t)/y1
i,u(t)+gz
k,u(t)−ωzE
k,u(t)−f outz
k,u(t)·zE
k,u(t)/zk,u(t)
zI
k,u(t)′=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t)·y1,I
i,u(t)/y1
i,u(t)+(1 −η)ωzE
k,u(t)−γzI
k,u(t)−f outz
k,u(t)·zI
k,u(t)/zk,u(t),∀i∈ N,u∈ U
zA
k,u(t)′=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t)·y1,A
i,u(t)/y1
i,u(t)+ηωzE
k,u(t)−γzA
k,u(t)−f outz
k,u(t)·zA
k,u(t)/zk,u(t)
zR
k,u(t)′=X
i∈N
PTi,k,u(t)·f outy,1
i,u(t)·y1,R
i,u(t)/y1
i,u(t)+γzI
k,u(t)+zA
k,u(t)−θzR
k,u(t)−f outz
k,u(t)·zR
k,u(t)/zk,u(t) (48)
For leaving subgroups on the platform, we have
y2,S
i,u(t)′=X
k∈K
PTi,k,u(t)·f outz
k,u(t)·zS
k,u(t)/zk,u(t)−gy,2
i,u(t)+θy2,R
i,u(t)−f outy,2
i,u(t)·y2,S
i,u(t)/y2
i,u(t)
y2,E
i,u(t)′=X
k∈K
PTi,k,u(t)·f outz
k,u(t)·zE
k,u(t)/zk,u(t)+gy,2
i,u(t)−ωy2,E
i,u(t)−f outy,2
i,u(t)·y2,E
i,u(t)/y2
i,u(t)
y2,I
i,u(t)′=X
k∈K
PTi,k,u(t)·f outz
k,u(t)·zI
k,u(t)/zk,u(t)+(1 −η)ωy2,E
i,u(t)−γy2,I
i,u(t)−f outy,2
i,u(t)·y2,I
i,u(t)/y2
i,u(t),∀i∈ N,u∈ U
y2,A
i,u(t)′=X
k∈K
PTi,k,u(t)·f outz
k,u(t)·zA
k,u(t)/zk,u(t)+ηωy2,E
i,u(t)−γy2,A
i,u(t)−f outy,2
i,u(t)·y2,A
i,u(t)/y2
i,u(t)
y2,R
i,u(t)′=X
k∈K
PTi,k,u(t)·f outz
k,u(t)·zR
k,u(t)/zk,u(t)+γy2,I
i,u(t)+y2,A
i,u(t)−θy2,R
i,u(t)−f outy,2
i,u(t)·y2,R
i,u(t)/y2
i,u(t) (49)
For leaving subgroups in the hall, we have
x2,S
i,u(t)′=f inx,2
i,u(t)·y2,S
i,u(t)/y2
i,u(t)−gx,2
i,u(t)+θx2,R
i,u(t)−f outx,2
i,u(t)·x2,S
i,u(t)/x2
i,u(t)
x2,E
i,u(t)′=f inx,2
i,u(t)·y2,E
i,u(t)/y2
i,u(t)+gx,2
i,u(t)−ωx2,E
i,u(t)−f outx,2
i,u(t)·x2,E
i,u(t)/x2
i,u(t)
x2,I
i,u(t)′=f inx,2
i,u(t)·y2,I
i,u(t)/y2
i,u(t)+(1 −η)ωx2,E
i,u(t)−γx2,I
i,u(t)−f outx,2
i,u(t)·x2,I
i,u(t)/x2
i,u(t),∀i∈ N,u∈ U
x2,A
i,u(t)′=f inx,2
i,u(t)·y2,A
i,u(t)/y2
i,u(t)+ηωx2,E
i,u(t)−γx2,A
i,u(t)−f outx,2
i,u(t)·x2,A
i,u(t)/x2
i,u(t)
x2,R
i,u(t)′=f inx,2
i,u(t)·y2,R
i,u(t)/y2
i,u(t)+γx2,I
i,u(t)+x2,A
i,u(t)−θx2,R
i,u(t)−f outx,2
i,u(t)·x2,R
i,u(t)/x2
i,u(t) (50)
The system dynamics of the QEM, as described by Eq. 15-Eq. 50, can be separated into two components: (1)
mobility dynamics and (2) infectious disease dynamics. The mobility dynamics are captured by the feedback queuing
network model expressed by Eq. 15-Eq. 39, which models the passenger flow behaviour in the metro system. The
infectious disease dynamics, expressed by Eq. 40-Eq. 50, reflect the spread of the infectious disease by taking into
account the movement of passengers and the interactions between different subgroups.
4. Model analysis
In this section, we present our findings on the solution properties of the proposed QEM. First, we prove the
existence and uniqueness of the solution for the QEM in Proposition 1. Then, we compare the solutions of the
proposed QEM and the benchmark model MEM in Proposition 2 to analyze the difference between them.
Proposition 1. There exists a unique solution for the QEM defined by Eq. 15-Eq. 50.
Proof. The QEM, defined by the system of equations in Eq. 15-Eq. 50, describes the dynamic behaviour of the metro
system. By discretizing the timeline Twith the time step ∆and using tto index the timestamp (the beginning of each
time step), the system state of the metro system facilities can be recursively derived. As the time step ∆approaches 0,
the system state at any given time can be obtained theoretically, resulting in a unique solution for the QEM.
18
Proposition 1 inspires us to develop a recursive algorithm, which will be presented in Section 5, to efficiently solve
the proposed QEM.
To measure the risk associated with the metro system, we introduce the metric of the Total (accumulated) Number
of Newly Exposed Travellers (TNNET).
Definition 1. (TNNET). TNNET refers to the total (accumulated) number of newly exposed travellers during the
timeline. A traveller is considered ”newly exposed” if he/she is in a susceptible status before arriving the metro
system and becomes exposed status after coming into contact with a virus carrier within the metro system. If we
ignore the contagious process outside the station, the TNNET can be calculated by Eq. 51 as follows:
T N NET =Z
TX
i∈N X
u∈U X
d∈D
gx,d
i,u(t)dt +Z
TX
i∈N X
u∈U X
d∈D
gy,d
i,u(t)dt +Z
TX
k∈K X
u∈U
gz
k,u(t)dt.(51)
The three terms in the equation for TNNET, given by Eq. 51, correspond to the number of passengers who become
newly exposed in the halls, platforms, and trains, respectively.
Proposition 2. MEM will underestimate TNNET compared to QEM under the same travel demand. That is:
T N NET ME M ≤T NN ET QEM .(52)
where T N N ET MEM and T N N ETQ EM represent the TNNET of MEM and QEM, respectively.
Proof. The new contagion in each subgroup at timestamp tcan be calculated by Eq. 43-Eq. 45. As an example,
consider hall i. The overall new contagion of hall iat the timestamp tcan be expressed as Eq. 53. Therefore, the total
number of newly exposed travellers in the hall iduring the whole timeline can be calculated by Eq. 54.
X
u∈U X
d∈D
gx,d
i,u(t)=β1X
u∈U X
d∈D
xd,S
i,u(t)·X
u∈U X
d∈D
xd,I
i,u(t)+X
u∈U X
d∈D
kxd,A
i,u(t).xi(t)+β2X
u∈U X
d∈D
xd,S
i,u(t)·X
u∈U X
d∈D
xd,E
i,u(t).xi(t)
(53)
Z
TX
u∈U X
d∈D
gx,d
i,u(t)dt =Z
T
β1X
u∈U X
d∈D
xd,S
i,u(t)·X
u∈U X
d∈D
xd,I
i,u(t)+X
u∈U X
d∈D
kxd,A
i,u(t).xi(t)dt (54)
+Z
T
β2X
u∈U X
d∈D
xd,S
i,u(t)·X
u∈U X
d∈D
xd,E
i,u(t).xi(t)dt,∀i∈ N
The proportion of susceptible individuals Pu∈U Pd∈D xd,S
i,u(t)/xi(t) can be assumed to remain constant and denoted
as ϕover the entire timeline, as the commuting time is too short compared to the switch time of the passengers’
epidemiological status (e.g. the latent period 1/ω). At the same time, as a numerically dominant population, the
influence of small changes of this proportion on the left side of Eq. 54 can be ignored. Thus, Eq. 54 can be simplified
to Eq. 55.
Z
TX
u∈U X
d∈D
gx,d
i,u(t)dt =β1ϕZ
TX
u∈U X
d∈D
xd,I
i,u(t)dt +β1kϕZ
TX
u∈U X
d∈D
xd,A
i,u(t)dt +β2ϕZ
TX
u∈U X
d∈D
xd,E
i,u(t)dt,∀i∈ N (55)
The total number of newly exposed travellers of the hall idepends on the number of virus carriers in the hall i.
According to Assumption 3, passengers of different epidemiological statuses are evenly mixed. Hence, the number of
people in a specific status will increase with the overall number of people.
In the following proof, to distinguish QEM and MEM, we will add a superscript ”∗” for the variables of MEM. In
MEM, the walking velocity of passengers in hall iis constant and equal to free velocity vx
0, as given by Eq. 56. Thus,
by substituting Eq. 56 into Eq. 27, we can derive the service rate of hall ifor MEM, as given by Eq. 57.
vx
i(xi(t))∗≡vx
0,∀i∈ N (56)
µx
i(xi(t))∗=xi(t)∗·vx
0/Lx
i,∀i∈ N (57)
19
According to Eq. 25 and Eq. 30, the output flow rates of hall iare subject to the non-empty probability of the output
facility (1 −PE), and the unblocking probability of the input facility (1 −PB). For the case of MEM, the blocking and
idle probability, namely PE(X)∗and PB(X)∗, are always equal to 0. By substituting Eq. 57, PE(X)∗, and PB(X)∗into
Eq. 25, the output flow rate of boarding passengers (d=1) for each OD in hall iis derived and expressed as Eq. 58.
For comparison, by substituting Eq. 27 into Eq. 25, we represent the hall’s output flow rate of boarding passengers for
each OD as Eq. 59 for QEM.
f outx,1
i,j,u(t)∗=x1
i,j,u(t)∗·vx
0/Lx
i,∀i,j∈ N,u∈ U (58)
f outx,1
i,j,u(t)=x1
i,j,u(t)·vx
i(xi(t))/Lx
i1−PEx
i(xi(t))1−PBy
i(yi(t)),∀i,j∈ N,u∈ U (59)
In a similar manner, the output flow rate of leaving passengers (d=2) for hall ican be expressed as Eq. 60 for MEM
and Eq. 61 for QEM.
f outx,2
i,u(t)∗=x2
i,u(t)∗·vx
0/Lx
i,∀i,j∈ N,u∈ U (60)
f outx,2
i,u(t)=x2
i,u(t)·vx
i(xi(t))/Lx
i1−PEx
i(xi(t)),∀i,j∈ N,u∈ U (61)
It is evident that when the number of passengers is identical, the output flow rates of halls for MEM are higher than
those for QEM, as demonstrated by the inequalities Eq. 62 and Eq. 63.
f outx,1
i,j,u(t)∗≥f outx,1
i,j,u(t) (62)
f outx,2
i,u(t)∗≥f outx,2
i,u(t) (63)
when
x1
i,j,u(t)∗=x1
i,j,u(t),x2
i,u(t)∗=x2
i,u(t)
This indicates that passengers move through the metro system faster in MEM compared to QEM. When travel de-
mand is equal, QEM is greater than MEM in terms of the integral of the system state of the hall over the timeline
(R
T
xi(t)dt). Proportionally, in terms of the integrals of the number of virus carriers in the hall over the timeline
(R
TPu∈U Pd∈D xd,I
i,u(t)dt,R
TPu∈U Pd∈D xd,A
i,u(t)dt, and R
TPu∈U Pd∈D xd,E
i,u(t)dt of Eq. 55), QEM is greater than MEM.
This leads QEM be with higher total number of newly exposed people in the hall (R
TPi∈N Pu∈U Pd∈D gx,d
i,u(t)dt). Sim-
ilarly, the total number of newly exposed people in platform and trains are also higher, that is, for the three items on
the right of Eq. 51, QEM is higher than MEM. Therefore, QEM leads to a higher TNNET compared to MEM.
Proposition 2 highlights the fact that the MEM model is less accurate in estimating the time passengers spend in
the metro system due to its assumption of constant walking velocity of passengers and neglect of blocking and idle
probabilities of facilities, as well as congestion propagation. In comparison, QEM provides a more accurate estimate
of system congestion as it accounts for various realistic features of metros such as the stochastic walking time of
passengers, state-dependent speed of passengers, stochastic arrival of passengers at the metro, and the congestion
propagation among facilities such as trains, platforms, and halls. Thus, QEM estimates the TNNET more accurately
than MEM. The numerical experiments in Section 6.1 will further demonstrate the superiority of QEM in estimating
the TNNET compared to MEM. Please note that when the TNNET considers the accumulated number of newly
exposed travellers that queue outside the station, Proposition 2 still holds because MEM also underestimates the
nonlinear stochastic effect of the congestion propagation outside the station. The numerical experiments in Section
6.2 will provide insights into how to balance the risks outside and inside the metro facilities under different anti-
epidemic measures.
5. Algorithm
In order to solve QEM defined by Eq. 15-Eq. 50, we discretize the entire timeline Twith the time step ∆and use
tin the original equation to represent the beginning of the t-th timestep. Because there is a strong parameter transfer
20
Train
NO.
Station
NO.
Station
NO.
Train
NO.
Fig. 8. The illustration of matrix multiplication.
0 1 0
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
×
10
6
0
=
0 × 10 + 1 × 6 + 0 × 0
1 × 10 + 0 × 6 + 0 × 0
0 × 10 + 0 × 6 + 0 × 0
0×10 + 0 × 6 + 0 × 0
0×10 + 0 × 6 + 0 × 0
0 × 10 + 0 × 6 + 0 × 0
=
6
10
0
0
0
0
Fig. 9. A example of matrix multiplication.
relationship among the facilities (halls, platforms, and trains) in the metro system, a recursive algorithm is designed
to calculate the system dynamics. The time complexity of the algorithm has a linear relationship with the product
of three terms (the numbers of timestamps, stations, and trains), namely O(|T | × |N| × |K |). It is independent of the
passenger volumes and the capacity of stations and trains, making QEM a suitable approach for solving large-scale
metro networks. The pseudocode for the recursive algorithm is shown in Algorithm 1.
Our model offer a significant advantage in representing and analyzing complex transportation systems. The model
can be the use of matrix representation for all of equations, which simplifies the analysis process and enables more
efficient computation. The corresponding variables for all stations can be calculated simultaneously. Moreover, our
sparse matrix representation of the variable PTi,k,u(t), which connects trains and platforms, greatly improves the ability
to write recursive algorithms with generality and extensibility.
To illustrate this point, let’s show the calculation of Eq. 38 as an example. At a certain timestamp t, we focus on
the down-direction, where uis fixed at 1. The Eq. 38 can be easily represented by matrix, as shown in Fig. 8, where
the timestamp tis omitted. The row number of the matrix PTi,k,u(t) represents the train service index (red) and the
column number represents the platform index (green). By matrix multiplication, the matrix PTi,k,u(t) is multiplied by
a column vector composed of the station output flow rate to obtain a column vector whose element represents the train
flow input rate.
Consider a system with three stations and six train services. The variables in Fig. 8 can be assumed as values
displayed in Fig. 9. In the first row of the leftmost sparse matrix, only the second element is equal to 1, indicating
that the first train is at the second platform at timestamp t. At this time, the flow rates from the three platforms to
trains are 10, 6, and 0 successively. Since the first train is dwelling at the second platform, the flow rate from the
second platform to the first train is 6. Similarly, only the first element in the second row of the sparse matrix equals
1, indicating that the second train is stopping at the first platform. The flow rate from the first platform to the second
train is 10. The remaining four rows of the matrix are all zero, which means they are not at any platform. In fact,
train services 3-6 have not yet departed from the depot. So the input flow rates of these trains are zero. This matrix
representation not only simplifies computation but also enhances the scalability and applicability of our model and
algorithm to real-world transportation systems.
6. Numerical Experiment
In this section, the performances of the proposed QEM and the existing MEM are compared and analyzed through
numerical experiments. These experiments consist of two sets: a small-scale case with 6 stations and a real-world
21
Algorithm 1 Recursion algorithm for the fluid queueing network
Input: Enter the input parameters of QEM: Metro system network parameters:PTi,k,u(t), Lx
i,Ly
i,Cx
i,Cy
i,Cz
k,u,
td
i,k,u,tw
j,k,u,xd,e
i,u(1), yd,e
i,u(1), ze
k,u(1), xd
i,u(1), yd
i,u(1), zk,u(1), x1
i,j,u(1), y1
i,j,u(1); Commuting data:T OD(t); Disease
parameters:β1,β2,ω,γ,θ,η,k,ζe
i; and Time step:∆.
Output: The system dynamics performance indicators: xd,e
i,u(t), yd,e
i,u(t), ze
k,u(t), xd
i,u(t), yd
i,u(t), zk,u(t), x1
i,j,u(t), y1
i,j,u(t).
1: Initialize the states of halls xd,e
i,u(1), xd
i,u(1), x1
i,j,u(1); platforms yd,e
i,u(1), yd
i,u(1), y1
i,j,u(1); and trains ze
k,u(1), zk,u(1).
2: Definite Function:hxd,e
i,u(t+1), yd,e
i,u(t+1), ze
k,u(t+1), xd
i,u(t+1), yd
i,u(t+1), zk,u(t+1), x1
i,j,u(t+1), y1
i,j,u(t+1)i=
FunMetroEpixd,e
i,u(t), yd,e
i,u(t), ze
k,u(t), xd
i,u(t), yd
i,u(t), zk,u(t), x1
i,j,u(t), y1
i,j,u(t)
3: Compute xd
i,u(t), yd
i,u(t) and zk,u(t) by Eq. 40-Eq. 42; xi(t), yi(t) by Eq. 15 and Eq. 16;
4: Compute λi,j,u(t) by Eq. 21;
5: If t=1then
6: λi,j,u(t)=λi,j,u(t)
7: else
8: Compute λi,j,u(t), by Eq.(22)
9: End If
10: Compute PEx
i(xi(t)), PEy
i(yi(t)) and PEz
i(zk,u(t)) by Eq. B.4;
11: Compute PBx
i(xi(t)), PBy
i(yi(t)) and PBz
i(zk,u(t)) by Eq. B.5;
12: Compute µx
i(xi(t)) and µy
i(yi(t)) by Eq. 27;
13: Compute f inx,1
i,j,u(t) by Eq. 23; f inx,1
i,u(t) by Eq. 24;
14: Compute f outx,1
i,j,u(t) by Eq. 25; f outx,1
i,u(t) by Eq. 26;
15: Compute f iny,1
i,j,u(t) by Eq. 31; f iny,1
i,u(t) by Eq. 32;
16: Compute f outy,1
i,j,u(t) by Eq. 33; f outy,1
i,u(t) by Eq. 34;
17: Compute f inz
k,u(t) by Eq. 38;
18: Compute x1
i,j,u(t)′by Eq. 28;
19: Compute y1
i,j,u(t)′by Eq. 35;
20: Compute f outz
k,u(t) by Eq. 39;
21: Compute zk,u(t)′by Eq. 19;
22: Compute gx,d
i,u(t), gy,d
i,u(t), and gz
k,u(t) by Eq. 43-Eq. 45;
23: Compute ze
k,u(t)′by Eq. 48;
24: Compute f iny,2
i,u(t) by Eq. 36; f outy,2
i,u(t) by Eq. 37;
25: Compute yd
i,u(t)′by Eq. 18;
26: Compute yd,e
i,u(t)′by Eq. 47 and Eq. 49;
27: Compute f inx,2
i,u(t) by Eq. 29; f outx,2
i,u(t) by Eq. 30;
28: Compute xd
i,u(t)′by Eq. 17;
29: Compute xd,e
i,u(t)′by Eq. 46 and Eq. 50;
30: Update x1
i,j,u(t+1) =x1
i,j,u(t)+x1
i,j,u(t)′·∆;y1
i,j,u(t+1) =y1
i,j,u(t)+y1
i,j,u(t)′·∆;zk,u(t+1) =zk,u(t)+zk,u(t)′·∆;
ze
k,u(t+1) =ze
k,u(t)+ze
k,u(t)′·∆;yd
i,u(t+1) =yd
i,u(t)+yd
i,u(t)′·∆;yd,e
i,u(t+1) =yd,e
i,u(t)+yd,e
i,u(t)′·∆;
xd
i,u(t+1) =xd
i,u(t)+xd
i,u(t)′·∆;xd,e
i,u(t+1) =xd,e
i,u(t)+xd,e
i,u(t)′·∆;
31: Return xd,e
i,u(t+1), yd,e
i,u(t+1), ze
k,u(t+1), xd
i,u(t+1), yd
i,u(t+1), zk,u(t+1), x1
i,j,u(t+1), y1
i,j,u(t+1).
32: End Definite
33: For t =1:(T/∆)−1
34: Invoking Function:hxd,e
i,u(t+1), yd,e
i,u(t+1), ze
k,u(t+1), xd
i,u(t+1), yd
i,u(t+1), zk,u(t+1), x1
i,j,u(t+1), y1
i,j,u(t+1)i
=FunMetroEpixd,e
i,u(t), yd,e
i,u(t), ze
k,u(t), xd
i,u(t), yd
i,u(t), zk,u(t), x1
i,j,u(t), y1
i,j,u(t)
35: End For
22
Fig. 10. Alluvial flow diagram for the travel demand between pairs of zones
case with 22 stations in Chengdu Metro, China. The numerical experiments are implemented using MATLAB 2016b
on a Windows 10 personal computer with a 3.3 GHz, 4-Core Intel(R) Xeon(R) CPU and 16 GB of RAM. The results
obtained from the proposed recursive algorithm are used to evaluate the performance of both models. In the small
case, we use an agent-based simulation to demonstrate the accuracy of the proposed QEM in describing the spread of
infectious diseases in metro systems compared to the existing MEM. The differences between the QEM and MEM in
their descriptions of the epidemic dynamics are analyzed and the impact of travel demand on the TNNET is studied.
The results of the real-world case reveal the impact of four prevention measures on the spread of infectious disease
in the metro system and provide valuable guidance for decision-makers in metro management to optimize epidemic
prevention and control strategies.
6.1. A small case
We first present the parameters of the small case in Section 6.1.1. In Section 6.1.2, we compare the contagion
indicators estimated by the proposed QEM and the existing MEM with the Anylogic simulation software. In Section
6.1.3, We specifically observe the differences between QEM and MEM from the system and station levels. Finally,
in Section 6.1.4, we investigate the impact of travel demand on the gap between the TNNETs estimated by QEM and
MEM.
6.1.1. Parameter setting for small case
Let us consider a bi-directional metro line with six stations: A, B, C, D, E, and F, where the down-direction (u=1)
is defined as ”A to F.” The travel demand for the 30 peak-hour OD pairs along the metro line is recorded in Eq. 64
and is represented by an alluvial flow diagram in Fig. 10. This diagram depicts the relative proportion of travellers
(width of the ribbon) who board at one station and leave at another. The observation shows that stations B and D have
relatively high boarding passenger pressure.
OD =
0 500 600 1100 800 500
1500 0 1000 1100 1200 600
1000 500 0 500 700 600
1600 1000 500 0 1000 400
500 600 700 800 0 400
400 300 600 400 500 0
(64)
The time-varying arrival rate follows the normal distribution with mean µ=70 and standard deviation σ=40.
The metro system operates a total of 26 train services, with 13 train services in each direction, each with a capacity
of 500 passengers. The headway between adjacent trains is 210 seconds, consisting of dwell time and running time.
Dwell time refers to the amount of time that a train spends at a station, with a duration of 30 seconds in this case.
23
Running time is the time taken by the train to travel between two stations or the time spent in motion, which is 180
seconds in this case. The dimensions of the halls are length 40 m, width 40 m, average walking distance 22.4 m, and
space capacity 8000 ped, whereas the platforms have length 150 m, width 6 m, average walking distance 15.3 m, and
space capacity 4500 ped. The total simulation time is 3600 seconds, with a time step of ∆ = 10 seconds. At the start
of the simulation, the metro system is empty, namely the initial states of all facilities are zero.
In the infectious disease model, the parameters 1/ω, 1/γ, 1/θ,η, and kare set to 5.2 days, 14 days, 60 days,
2% and 2, respectively, based on the characteristics of COVID-19. It is assumed that an exposed person is half as
contagious as an infected person, with β2=β1/2. The average number of susceptible people that can be successfully
exposed by an infected individual per unit of time (β1) is dependent on the density of the current queue system. The
value of β1is set to 20 per/day when the density of the queue system is close to 100% and 10 per/day when the density
is close to 0%. The value of β1increases linearly with the density. For the proportions of all statuses of passengers
outside all stations, we assume that they are the same and set as ζS
i:ζE
i:ζI
i:ζA
i:ζR
i=0.95 : 0 : 0.04 : 0.01 : 0.
This means that 95% of passengers are susceptible, 0% are exposed, 4% are infected, 1% are asymptomatic and 0%
are recovered.
6.1.2. Comparison of QEM, MEM and agent-based simulation results
As obtaining real data on the changes in the status of individuals after a commute is difficult, we validate the
accuracy of the QEM and MEM through an agent-based simulation. The simulation software, Anylogic, is used and
both pedestrians and trains are modelled as agents. The 3D simulation diagram of a station in the small-scale case
is shown in Fig. 11. The assumption is made that the pedestrian agents carrying the virus (with the red aperture in
Fig. 11) can effectively expose susceptible agents within a 2.5 m distance, and the other parameter settings are kept
consistent with the small case scenario. The visualization of the agent-based simulation can be found in Supplemental
Fig. 11. 3D simulation of a station.
Material 1. For infectious disease dynamics, we divide the population into five statuses. However, we find that the
travel time is relatively short compared to the recovery process of infected individuals and the onset process of exposed
individuals, so we only consider the number of newly exposed travellers (E) in our analysis. We collect the simulation
data on the total number of newly exposed travellers over time, and we get the TNNET mentioned earlier at the last
moment of the simulated time. Then the results calculated by QEM and MEM are contrasted and shown in Fig. 12.
Overall, our study highlights the superiority of the proposed QEM in capturing the dynamics of infectious diseases
within metro systems. Through observation of the total number of exposed travellers curves, we find that QEM
exhibited an exponential upward trend, whereas MEM exhibited an S-shaped growth pattern. Furthermore, the trend
24
0 50 100 150 200 250 300 350
Timestamp
0
100
200
300
400
500
600
Total number of the newly exposed travellers (per)
Metro system
TNNETQEM
TNNETMEM
TNNETQEM
TNNETMEM
TNNETQEM
TNNETMEM
QEM
MEM
Simulation
Fig. 12. Comparison of QEM, MEM, and simulation results.
of the results of the Anylogic simulation aligns with that of QEM, providing additional support for its effectiveness.
We evaluate the precision of both QEM and MEM using the coefficient of determination (R-Squared). Our results
reveal that QEM had a significantly better R Squared (0.945), indicating a much stronger interpretive effect, while
MEM had a very poor R-Squared (0.374), indicating a large deviation. This finding demonstrates the potential for
QEM to be a valuable tool in predicting and controlling the spread of infectious diseases.
6.1.3. Detailed differences between QEM and MEM results
Fig. 13 illustrates the difference between QEM and MEM at the overall system level. Specifically, the two solid
0 50 100 150 200 250 300 350
Timestamp
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Number of travellers of all status (per)
104
0
50
100
150
200
250
300
350
400
450
500
Number of newly exposed travellers (per)
Metro System
QEM
MEM
QEM
MEM
Fig. 13. The change of the numbers of the overall travellers and the newly exposed travellers inside the metro system.
lines (left axis) depict the number of all passengers inside the system over time. The MEM assumes a constant walking
velocity and disregards randomness and state-dependence. This results in faster system service for passengers and a
25
lower number of individuals inside the system. Contrarily, the QEM takes into account the system’s stochastic nature,
the spillback among facilities (trains, platforms, and halls), and the state-dependence of service rates in facilities. This
leads to longer times spent in the system, particularly during periods of congestion. Two dotted lines (right axis) in
Fig. 13 explain the number of newly exposed individuals that are inside the system at each timestamp. The travellers
counted in Fig. 13 only include the individuals currently inside the metro system, and individuals leaving the metro
system are not considered in the subsequent timestamp. Therefore, the results presented in this figure differ from those
in Section 6.1.2. For MEM, the peak of the number of newly exposed travellers appears 100 timestamps later than the
peak of travellers of whole travellers, and the number of newly exposed individuals decreases as passengers leave the
system. However, since QEM fully considers the close contact of passengers while queuing in the station, the number
of newly exposed people in the system even increases when passengers leave. This suggests that the number of newly
exposed travellers generated during queuing in the system is greater than the number of newly exposed travellers who
leave the system.
Fig. 14 illustrates the difference between QEM and MEM at the station level. It further emphasizes the effect of
queues on disease transmission. An interesting observation from Fig. 14 is that at stations B and D, the number of
0 100 200 300
Timestamp
0
500
1000
1500
2000
Number of travellers of all status (per)
0
2
4
6
8
10
Number of newly exposed travellers (per)
Station A
QEM
MEM
QEM
MEM
0 100 200 300
Timestamp
0
1000
2000
3000
4000
5000
Number of travellers of all status (per)
0
50
100
150
Number of newly exposed travellers (per)
Station B
QEM
MEM
QEM
MEM
0 100 200 300
Timestamp
0
500
1000
1500
2000
2500
3000
Number of travellers of all status (per)
0
10
20
30
40
50
60
70
Number of newly exposed travellers (per)
Station C
QEM
MEM
QEM
MEM
0 100 200 300
Timestamp
0
1000
2000
3000
4000
5000
Number of travellers of all status (per)
0
20
40
60
80
100
120
140
Number of newly exposed travellers (per)
Station D
QEM
MEM
QEM
MEM
0 100 200 300
Timestamp
0
500
1000
1500
2000
2500
Number of travellers of all status (per)
0
5
10
15
20
25
Number of newly exposed travellers (per)
Station E
QEM
MEM
QEM
MEM
0 100 200 300
Timestamp
0
100
200
300
400
500
600
700
Number of travellers of all status (per)
0
0.5
1
1.5
2
2.5
Number of newly exposed travellers (per)
Station F
QEM
MEM
QEM
MEM
Fig. 14. The change of the number of travellers of all status and exposed travellers inside the station
newly exposed travellers estimated by QEM is much larger than the one estimated by MEM. It is intuitive because
stations B and D are high-demand and have relatively great traffic pressure, and thereby congestion is easy to happen.
Compared with QEM, MEM ignores the nonlinear stochastic effect of congestion and overestimates the service ef-
ficiency. Consequently, MEM takes an optimistic view of the transmission intensity of infectious diseases. Another
unexpected finding is that station C (a low-demand station between two high-demand stations) becomes high-risk
according to the proposed QEM, while MEM does not think so. This observation from QEM is counterintuitive since
it is expected that low-demand stations (e.g. stations A, E and F) are low-risk. After further investigation, it is found
that after trains in both directions pass through stations B and D, the available capacity is less, resulting in fewer
passengers getting on trains at station C between them. That is, it leads to congestion propagation at station C. It
highlights the significance of coordinating the numbers of passengers that board trains at these stations.
6.1.4. Influence of travel demands on TNNET
The relationship between the TNNET and travel demand is further analyzed in Fig. 15. Concretely speaking, we
vary the traffic demand from 50% to 150% of the base travel demand.
The results of our study show that as the traffic demand increases, the TNNET of the system also increases.
Furthermore, the difference between QEM and MEM becomes increasingly pronounced with higher traffic demand.
This suggests that MEM underestimates the spread of infectious diseases in the metro system, especially in crowded
26
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Travel demand
0
200
400
600
800
1000
1200
TNNET (per)
QEM
MEM
Fig. 15. The change of TNNET under different travel demands.
conditions. Our findings highlight the relevance of considering the nonlinear stochastic effect of the congestion
propagation in the transmission mechanism of infectious diseases in the metro system. As such, QEM is more effective
in accurately capturing the transmission mechanism, compared to MEM.
6.2. A real-world case
We employ data from Chengdu Metro line 1 to conduct a real-world case study. In Section 6.2.1, we specify
the parameter settings. In Scetion 6.2.2, we evaluate the computational efficiency of QEM approach. Section 6.2.3
utilizes QEM to simulate the spread of the epidemic in the halls and platforms, as measured by the change in the
number of newly exposed individuals. In Sections 6.2.4, 6.2.5, 6.2.6 and 6.2.7, we compare the effects of various
control measures (allowed entering proportion control, train dwell time adjusting, social distancing, and mandatory
mask-wearing) on TNNETs (including the accumulated number of newly exposed travellers that queue outside the
station) under both QEM and MEM. Our findings offer valuable insights.
6.2.1. Parameter setting for real-world case
The real-world case study is based on data from Chengdu Metro line 1, which has 22 stations. The data on the
time-varying arrival rate is obtained from the AFC (Automatic Fare Collection) system. The metro line operates 30
service trains with a capacity of 1500 passengers each, in both directions. The running time and dwell time are set
at 160 s and 20 s, respectively. The total simulation time is 9000 s and the time step ∆is set to 1 s. The remaining
parameters are set based on the small-scale case study.
6.2.2. Computational efficiency of the QEM
The computational efficiency of the proposed QEM is visualized in Fig. 16. We compare the run times of QEM
and MEM in three travel demand scenarios. The results show that although the calculation time of QEM is slightly
longer than that of MEM, both are within 1 s and are not affected by the number of passengers. This highlights the
computational efficiency of the proposed QEM, making it suitable for solving large-scale metro networks.
6.2.3. The epidemic situation in halls and platforms
To evaluate the spread of the epidemic at each station, we track the number of newly exposed travellers over time
inside each hall and platform. The results are depicted in Fig. 17, where the station of Tianfu Square (TFS), located
in the central business district, is particularly noticeable. Tianfu Square station has the most significant number of
27
Fig. 16. The time of calculation of MEM and QEM
Fig. 17. The number of exposed travellers in halls and platforms of Chengdu Metro line 1
exposed passengers and is the only station with congestion, bringing the number of newly exposed people in the hall
to more than 500 (as Fig. 17 a)), and platform to more than 350 (as Fig. 17 b)) at the end of the timestamp.
For mobility dynamics, due to the limited capacity of the station (hall and platform) and the overloaded travel
demand, we find that passengers are forced to gather outside the station, increasing the risk of an epidemic. This
observation prompts us to consider the transmission that occurs outside the stations and leads to a more comprehensive
understanding of the spread of infectious diseases in realistic metro systems.
28
6.2.4. Effects of allowed entering proportion control
We further analyze the effect of a controllable factor, the proportion of passengers allowed to enter the station
(allowed entering proportion) on the epidemic in the metro system. The experiments are performed using both QEM
and MEM. In prior research on passenger flow control (Chen, 2022), the allowed entering proportion is considered
as a decision variable. In this paper, for the sake of simplicity, the same level of control is applied to all stations.
A low allowed entering proportion leads to passengers gathering outside the station, while a high allowed entering
proportion, indicating few restrictions on entry, can result in congestion inside the system, both of which may lead
to an outbreak. One possible trade-offbetween the contagion of the inside and outside system is to minimize the
overall transmission by controlling the allowed entering proportion. To verify this idea, we vary the allowed entering
proportion under three levels of travel demands and observe the change in the TNNET (including the accumulated
number of newly exposed travellers that queue outside the station). Suppose that the infectivity outside is half that
inside the system, and then the result of QEM is shown in Fig. 18.
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Allowed Entering Proportion
1800
2000
2200
2400
2600
2800
3000
3200
TNNET (per)
n = 0.95
n = 1
n = 1.05
Fig. 18. TNNET curve of QEM under three travel demand through
variation of allowed entering proportion.
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Allowed Entering Proportion
1400
1600
1800
2000
2200
2400
2600
TNNET (per)
n = 0.95
n = 1
n = 1.05
Fig. 19. TNNET curve of MEM under three travel demand through
variation of allowed entering proportion.
Observations show that, under medium travel demand (n=1), increasing the allowed entering proportion (up to
0.76) leads to an increase in inside system transmission of infectious diseases. But it also provides greater relief to
outside transmission, resulting in a decline in the TNNET curve. However, if the allowed entering proportion exceeds
0.76, a further increase can lead to congestion within the system. This significantly increases the contagion inside the
system and causes a surge in the TNNET curve. In short, a trade-offexists between inside and outside contagion at all
travel demand levels (low, medium, and high), and the optimal allowed entering proportion varies, with values of 0.80,
0.76, and 0.72, respectively. As travel demand increases, the optimal value decreases. Moreover, the optimization
effects are significant, with reductions in the TNNET by 29.09%, 27.82%, and 26.94% under low, medium, and high
travel demand levels, respectively.
However, the results obtained with the MEM (as depicted in Fig. 19) show a continuous decline in the TNNET
curve as the allowed entering proportion increases. In MEM, the nonlinear stochastic effect of the congestion prop-
agation within the metro system is ignored and thus the contagion inside system is greatly underestimated. Despite
controlling the allowed entering proportion has a little bit of positive effect on the restraint of epidemic spread in
halls and platforms, MEM thinks that it brings more risk of the epidemic spreading for passengers outside the station
because increasing their commuting time. Therefore, under three travel demand levels, MEM believes that allowed en-
tering proportion control is counterproductive to epidemic prevention, that is, the optimal allowed entering proportion
are 1.
We also consider an extreme scenario (n=1.3) where the travel demand is 1.3 times the benchmark travel demand
of the medium level. The result is shown in Fig. 20. We can see that there also exists a trade-offfor MEM under an
extreme scenario. But its optimal allowed entering proportion is much larger than the QEM’s. This finding suggests
that the allowed entering proportion control in MEM has a positive effect only when the travel demand is extremely
high. The reason for this is that MEM ignores the nonlinear stochastic effect of the congestion propagation compared
to QEM.
29
0.4 0.5 0.6 0.7 0.8 0.9 1
Allowed Entering Proportion
2400
2600
2800
3000
3200
3400
3600
3800
4000
4200
4400
TNNET (per)
MEM
QEM
Fig. 20. TNNET curve under extreme scenario (n=1.3) through variation of allowed entering proportion.
6.2.5. Effects of train dwell time adjusting
Some studies have mitigated the nonlinear stochastic effect of congestion propagation by adjusting the train dwell
time (Cornet et al., 2019). In previous small cases, congestion has been discovered to exacerbate the spread of the
epidemic, so we also want to control COVID-19 by adjusting the train dwell time. For simplicity, this paper assumes
that all train dwell times keep identical at all stations. However, train capacity is limited, and longer dwell time will
not reduce the congestion on the platform but increase the commuting time of passengers. Conversely, too short
dwell time causes passengers who should be boarding trains to stay on the platform. Both of them increase the risk
of infection. To analyze the control effects, we vary the train dwell time under three levels of travel demands and
measured the TNNET. The results of the QEM are shown in Fig. 21.
20 25 30 35 40 45 50
The dwell time of the trains (s)
2200
2400
2600
2800
3000
3200
3400
TNNET (per)
n=0.95
n=1
n=1.05
Fig. 21. TNNET curve of QEM under three travel demands through
adjusting the train dwell time.
20 25 30 35 40 45 50
The trains dwell time (s)
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
TNNET (per)
n=0.95
n=1
n=1.05
Fig. 22. TNNET curve of MEM under three travel demands through
adjusting the train dwell time.
The results show that under medium travel demand (n=1), the TNNET curve falls very slowly as the train dwell
time increases from 20 seconds to 38 seconds. But when it is over 38 seconds, the TNNET spikes. The TNNET curves
exhibit a similar trend under low, medium, and high travel demands, with some fluctuations. The optimal dwell times
are 42 seconds, 38 seconds, and 30 seconds, respectively. Fluctuations in the curves may be attributed to the uneven
distribution of passenger flow in time and space. Optimization effects are observed to be 10.18%, 2.12%, and 2.43%,
respectively. In summary, there exists an optimal dwell time with a moderate value that minimizes the TNNET. As
travel demand increases, the optimal dwell time decreases. However, the results of MEM, shown in Fig. 22, indicate
30
that the TNNET curves continue to rise with an increase in dwell time. In MEM, the walking speed of passengers
is constant, and thus the boarding efficiency of passengers is higher than that in QEM, particularly during periods of
congestion. A shorter dwell time does not result in a large number of stranded passengers, and extending dwell time
would only increase the commuting time without any added benefits.
The simultaneous implementation of both control measures - the allowed entering proportion control and train
dwell time adjusting - does not result in any conflict and can lead to improved outcomes. For medium travel demand,
the impact of simultaneously altering both allowed entering proportion and train dwell time with QEM is represented
in Fig. 23.
2000
1
2200
2400
50
0.9
2600
TNNET (per)
45
2800
0.8 40
3000
Allowed entering proportion
35
3200
0.7 30
The trains dwell time (s)
25
0.6 20
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
Fig. 23. TNNET surface diagram of QEM under medium travel demand through the simultaneous implementation of both control measures.
The result indicates that the implementation of the combined control strategy can lead to improved performance
compared to single control measures. Our findings suggest that the optimal allowed entering proportion increases
from 0.76 to 0.83 and the optimal dwell time decreases from 38 s to 29 s under medium travel demand. By comparing
TNNET, we find that the epidemic prevention effect of combined measures is 30.90% with medium travel demand,
improved by 3.08% compared with only controlling the allowed entering proportion, and by 28.78% compared with
only adjusting the train dwell time.
6.2.6. Effects of social distancing
Several recent studies have highlighted the relevance of social distancing in ensuring public safety (Ku et al., 2021;
H¨
orcher et al., 2022). This paper builds on this work by investigating the impact of social distancing on the spread
of infectious diseases using both the QEM and MEM, and examines the effects under different travel demands. In
this paper, the capacity of halls and platforms is considered to be the number of passengers that can be supported
during congestion. Each square meter can accommodate up to five passengers. Since our model is a population-based
aggregated model, assuming that we limit the number of people that can be accommodated per square meter (i.e.,
allowed density), passengers can be considered to automatically maintain social distance. In addition, assuming that
31
the space occupied by each passenger is a regular hexagon, we can convert the allowed density to the allowed social
distance. By gradually increasing the allowed density in halls and platforms from 0.25 to 5 per/m2(namely, gradually
decreasing the social distance from 2.15 m to 0.48 m), we observe changes in TNNET using the QEM approach. The
results are shown in the Fig. 24.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Allowed Density (per/m 2)
1800
2000
2200
2400
2600
2800
3000
3200
3400
TNNET (per)
low
medium
high
Fig. 24. TNNET curve of QEM under three travel demand through
variation of social distance.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Allowed Density (per/m 2)
1300
1400
1500
1600
1700
1800
1900
TNNET (per)
low
medium
high
Fig. 25. TNNET curve of MEM under three travel demand through
variation of social distance.
In consideration of the potential of outside station infection, both measures of social distancing and allowed enter-
ing proportion control achieve similar results with a trade-offbetween inside and outside contagion. Observations of
TNNET with low travel demand show a decrease in TNNET as allowed density decreases until it reaches 1.75 per/m2.
This result suggests that social distancing is effective, despite a potential sacrifice of traffic efficiency. However, when
allowed density drops below 1.75 per/m2, station capacity becomes too small and outside station contagion surges.
Thus, setting the allowed social distance at 0.81 m (allowed density at 1.75 per/m2) is cost-effective for low travel
demand. When the travel demand increases from the low level to the medium or high level, the allowed social distance
also increases to 0.96 m (density of 1.25 per/m2). The social distancing effects are calculated as 17.03%, 20.38%, and
20.31% for low, medium, and high travel demands, respectively. The results of the same experiments under the MEM
are shown in the Fig. 25. It can be seen that because MEM ignores the nonlinear stochastic effect of the congestion
propagation, TNNETs are basically not affected by social distance and in straight lines.
6.2.7. Effects of mandatory mask-wearing
According to Ku et al. (Ku et al., 2021), mandatory mask-wearing can reduce the rate of infection by 93.5% in
public transportation, which means that it is likely to be a dominant preventive measure. Therefore, we use the QEM
and MEM to assess the effect of mandatory mask-wearing on low, medium and high travel demand. We hypothesize
that mandatory mask-wearing could reduce the transmissibility (β1, β2) of the virus by 93.5% (Ku et al., 2021). The
results are shown as Fig. 26
Overall, the effects of mandatory mask-wearing are similar in both the QEM and MEM, both of which achieve
excellent results. Specifically, under low, medium, and high travel demands, mandatory mask-wearing in the MEM
leads to decreases in TNNET of 93.96%, 93.97%, and 93.99%, respectively, while in the QEM method, the decreases
are 94.41%, 94.44%, and 94.46%, respectively. This indicates that QEM is more sensitive to virus transmissibility
parameters, β1and β2.
We explore the effectiveness of a combination of allowed entering proportion control, train dwell time adjusting,
and mandatory mask-wearing measures, but find only marginal improvements over mandatory mask-wearing alone.
This is likely due to the fact that the allowed entering proportion control and train dwell time adjusting measures
are uniformly implemented for all stations, regardless of congestion levels. Tailored and time segment-customized
optimization for each station may hold potential for further improvement.
7. Conclusion
This paper presents a novel approach, QEM, to analyze the transmission of COVID-19 in metro systems from a
meso-level perspective. The QEM integrates an extended multiple-subgroup SEIAR model into a time-varying and
32
low medium high
Travel demand
0
500
1000
1500
2000
2500
3000
3500
TNNET (per)
QEM without mask
QEM with mask
MEM without mask
MEM with mask
Fig. 26. Comparison of QEM and MEM for evaluating the effects of mandatory mask-wearing.
state-dependent feedback queuing network model. This approach explicitly models the nonlinear stochastic effect
of congestion propagation. That is, it considers various realistic features of metro systems, such as the dynamic
stochastic arrival and walking time of passengers, the state-dependent speed of passengers, the congestion propagation
among facilities (trains, platforms, and halls), and the interaction and cross infection among multiple subgroups of
passengers. Thus, the proposed QEM can provide an accurate understanding of the spread of the epidemic, changes
in passenger subgroups, and congestion propagation in metro systems. We validate the QEM versus the micro-level
agent-based simulation. Compared with the simulation, the fitting degree (R-squared) of QEM is 0.945, while the
R-squared of the benchmark model MEM which ignores the nonlinear stochastic effect of congestion propagation
is only 0.374. We also theoretically analyze the differences between the proposed QEM and the benchmark model
MEM. It is proved that MEM would always underestimate the spread of the epidemic in the subway system compared
with QEM. A recursive algorithm is developed for the QEM. Its time complexity is independent of the passenger
volumes and capacities of stations and trains, and thus is suitable for the analysis and decision-making of large-scale
metro networks. We compare the CPU times of QEM and MEM for a metro line with 22 stations. Although the
CPU time of QEM is slightly longer than that of MEM, both are within 1 s and are not affected by the travel demand.
The practicality of the QEM is demonstrated through numerical experiments on a small-scale case (6 stations) and
a real-world case (22 stations). The insights into the potential impact of four prevention measures (including the
allowed entering proportion control, train dwell time adjusting, social distancing and mandatory mask-wearing) on
the epidemic spread are evaluated. Some interesting results are revealed:
(1) The difference between the epidemic spreads estimated by QEM and MEM will become larger with the increase
in travel demand and congestion. That is, without considering the nonlinear stochastic effect of congestion
propagation, the risk of epidemic spreads will be increasingly underestimated as the travel demand increases.
(2) From the proposed QEM, low-demand and low-risk stations that are located between high-demand stations may
become high-risk because of the congestion propagation, while the MEM does not think so. Our model displays
the importance of coordinating the numbers of passengers that board trains at these stations.
(3) There exists a moderate level of allowed entering proportion, train dwell time, and social distance to minimize
the risk of epidemic spread. Interestingly, it reveals the trade-offof risks inside and outside metro facilities.
When the allowed entering proportion, train dwell time, or social distance are large or small, the overall risk
cannot be minimized.
33
(4) According to the QEM, the optimal social distance increases but the optimal allowed entering proportion and
train dwell time decrease as the travel demand grows, while the MEM is insensitive to these measures. Our
model reveals that the optimal allowed entering proportion, train dwell time, and social distance should be
tailored by the travel demand.
(5) Mandatory mask-wearing indeed is the most effective prevention measure. The train dwell time adjusting is
unremarkable. The combination of prevention measures may further reduce the risk. Unexpectedly, the allowed
entering proportion control is more effective than social distancing. It reveals that the former provides a better
balance between the risks inside and outside metro systems.
We identify a few directions in studying metro systems with epidemics. In this paper, we only consider a metro
line that does not require a transfer, and the metro network with intersections of multiple lines is future work. Many
studies have shown that increasing ventilation and disinfection levels can reduce the transmission rate of COVID-19,
but at a high cost (Sun and Zhai, 2020). Finding a balance between economic costs and epidemic prevention is worth
studying. As demonstrated in the numerical section, the combination of prevention measures can be a useful tool
to reduce the risk of metro systems. In this paper, control parameters in the prevention measure are identical for all
stations or trains. Using an optimization method to obtain time-dependent control plans for each station and each train
would be a challenging but meaningful research question.
8. ACKNOWLEDGMENTS
The work is supported by the National Natural Science Foundation of China (No. 62203367, 71901183), the Ap-
plied Basic Research Project in the Department of Science and Technology of Sichuan Province (No. 2021YJ0066),
the Fundamental Research Funds for the Central Universities (No.2682022CX026) and the Chengdu Science and
Technology Project (No. 2021-RK00-00057-ZF). The authors are responsible for any remaining errors.
9. Author Contributions
The authors confirm their contribution to the paper as follows: study conception and design: Aoping Wu, Lu Hu;
data collection: Juanxiu Zhu; analysis and interpretation of results: Aoping Wu, Lu Hu; draft manuscript preparation:
Aoping Wu, Lu Hu, Pan Shang. All authors reviewed the results and approved the final version of the manuscript.
Appendix A. Notations
Table A1: Notations for the SEIAR model
Parameters
β1The average number of susceptible people exposed by an infected individual per unit of time.
β2The average number of susceptible people exposed by an exposed individual per unit of time.
1/ω The average latent period length of exposed individuals.
1/γ The average length of the recovery period of the infected and asymptomatic individuals.
1/θ The average length of immunity period of the recovered individuals.
ηThe probability of exposed individuals turning to asymptomatic after the latent period.
k The contagion intensity of asymptomatic individuals relative to infected individuals.
Variables
N Total number of population.
S The number of susceptible population.
E The number of exposed (latent) population.
I The number of infected population with symptoms.
A The number of asymptomatic population.
R The number of recovered population.
34
Table A2: Notations for the feedback queuing network model
Sets, indexes and parameters
NThe set of the stations.
KThe set of the train services.
UThe set of the operating direction of the trains.
DThe set of the purposes of passengers.
TThe timeline.
i,j The index of the stations (halls and platforms). i,j∈ N
k The index of the train services. k ∈ K
u The index of the operating direction of the trains. u ∈ U
d The index of the purposes of passengers. d ∈ D
t The moment of the timeline. t ∈ T
Cx
i,Cy
i,Cz
k,uThe capacity of hall i, platform i, train k in direction u.
Lx
i,Ly
iThe average walking distance of passengers in hall i, on platform i.
td
i,k,uThe departure time of train k in direction u from platform i.
tw
j,k,uThe dwell time of train k in direction u at platform i.
odi,j(t)The number of the passengers arriving station i with destination station j
at time t.
Variables
xi(t), yi(t)The total number of passengers of hall i and platform i at time t.
zk,u(t)The number of passengers of train k in direction u at time t.
xd
i,u(t), yd
i,u(t)The number of passengers of subgroup of hall i and platform i
with purpose d for (or from) trains in direction u at time t.
x1
i,j,u(t), y1
i,j,u(t)The number of passengers of boarding subgroups of hall i and platform i
with destination j at time t.
f inx,d
i,u(t),f outx,d
i,u(t)The input and output flow rate of xd
i,u(t)at time t.
f inx,1
i,j,u(t),f outx,1
i,j,u(t)The input and output flow rate of x1
i,j,u(t)at time t.
f iny,d
i,u(t),f outy,d
i,u(t)The input and output flow rate of yd
i,u(t)at time t.
f iny,1
i,j,u(t),f outy,1
i,j,u(t)The input and output flow rate of y1
i,j,u(t)at time t.
f inz
k,u(t),f outz
k,u(t)The input and output flow rate of zk,u(t)at time t.
PTi,k,u(t)Binary variable indicating whether the service train k in direction
u dwells at platform i at time t.
λi,j,u(t)The arrival rate of OD pair (i,j)at time t.
λi,j,u(t)The current arrival rate of OD pair (i,j)in feedback queuing at time t.
vx
i(xi(t)),vy
i(yi(t)) The walking velocity of passengers in hall i and on platform i at time t.
µx
i(xi(t)), µy
i(yi(t)) The service rate of hall i and platform i at time t.
PBx
i(xi(t)),PBy
i(yi(t)),PBz
k,u(zk,u(t)) The state-dependent blocking probability of hall i, platform i,
and train k at time t.
PEx
i(x),PEy
i(y),PEz
k,u(z)The state-dependent idle probability of hall i, platform i,
and train k at time t.
Appendix B. Parameter calibration
Appendix B.1. The passengers’ mean walking velocity
From Smith (1991); Xu et al. (2014); Hu et al. (2019), the passengers’ mean walking velocity in a system is
usually inversely related to the density and is described by the two most commonly used models. The first is the
linear model as Eq. B.1, where Crepresents the capacity of the system and xdenotes the system state. The second is
the exponential function, which is given by Eq. B.2. This function requires three representative points (0,v0), (a,va),
(b,vb) to calibrate its parameters γvand βv. In this paper, we assume a maximum density of 5 passengers per square
35
Table A3: Notations for QEM
Set, indexes and parameters
PThe set of epidemiological statuses of SEIAR model.
e The index of statuses. e ∈ P ={S,E,I,A,R}
ζe
iThe ratio of individuals of status e outside station i.
Variables
xd,e
i,u(t)The number of passengers of status e within xd
i,u(t)at time t.
yd,e
i,u(t)The number of passengers of status e within yd
i,u(t)at time t.
ze
k,u(t)The number of passengers of status e within zk,u(t)at time t.
gx,d
i,u(t)The new contagion between the susceptible individuals of the subgroup xd
i,u(t)
with all virus carriers in the hall i at time t.
gy,d
i,u(t)The new contagion between the susceptible individuals of the subgroup yd
i,u(t)
with all virus carriers in the platform i at time t.
gz
i,u(t)The new contagion between the susceptible individuals with all virus carriers
in the train k in the direction u at time t.
meter and set (0, 1.6 m/s), (0.4C, 0.85 m/s), and (0.8C, 0.47 m/s) as the representative points.
v(x)=v0·1−x
C(B.1)
v(x)=v0·exp−x
βvγv(B.2)
where
γv=ln[ln(va/v0)/ln(vb/v0)]
ln[a/b], βv=a
[ln(v0/va)]1/γv
=b
[ln(v0/vb)]1/γv
Therefore, the mean service rate can be calculated by Eq. B.3 accordingly. In this equation, Lrepresents the
average walking distance of passengers.
µ(x)=x·v(x)/L(B.3)
Fig. B.27 shows the relationship between the mean velocity or service rate and the density. In Fig. B.27 a), it can be
seen that the mean velocity decreases as the density increases in both linear and exponential models. However, the
mean velocity in the linear model reaches zero at maximum density, whereas in the exponential model, the velocity
remains at a low level. Fig. B.27 b) shows that the mean service rate first increases and then decreases with the
increase in density. In terms of the maximum service rate, the linear model gives a higher value than the exponential
model, but the jam service rate (when density equals 5) is the opposite.
According to Hu et al. (2019); Smith (1991); Xu et al. (2014), it is more practical to adopt the exponential model.
By utilizing the exponential model, we obtain vx
i(xi(t)) and vy
i(yi(t)) to describe the passengers’ mean walking velocity
in the hall and platform, respectively.
Appendix B.2. The idle and blocking probabilities
To obtain the state-dependent idle and blocking probabilities PE (x) and PB(x) of the Mt/G(x)/C/Cqueuing model,
we approximate them using the exponential functions as shown in Eq. B.4 and Eq. B.5, where the γE,γB,βE, and βB
are the parameters to be determined. Eq. B.4 and Eq. B.5 are similar to the pointwise mapping functions proposed by
(Hu et al., 2019).
PE(x)=exp−x
βEγE(B.4)
PB(x)=exp−C−x
βBγB(B.5)
36
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Density (per/m 2)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Mean velocity (m/s)
a) Mean velocity vs. density
Exponential model
Linear model
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Density (per/m 2)
0
20
40
60
80
100
120
Mean service rate (per/s)
b) Mean service rate vs. density
Exponential model
Linear model
Fig. B.27. The relationship between the mean velocity or service rate and the density.
Through taking the log of both sides of these two equations twice, it is found that only two groups of representative
points are needed to determine the parameters. For the idle probability PE(x), we use (a1,pea1) and (a2,pea2) to
calibrate γEand βE(see Eq. B.6). For the blocking probability PB(x), (a1,pba1) and (a2,pba2) are used to calibrate
γBand βBas shown in Eq. B.7.
γE=ln[ln(pea1)/ln( pea2)]
ln(a1/a2), βE=a1
[ln(1/pea1)]1/γE
=a2
[ln(1/pea2)]1/γE(B.6)
γB=ln[ln(pba1)/ln( pba2]
ln[(C−a1)/(C−a2)], βB=C−a1
[ln(1/pba1)]1/γB
=C−a2
[ln(1/pba2)]1/γB(B.7)
The four representative points can be obtained by Eq. B.8-Eq. B.11. These analytical expressions are derived from
the corresponding steady-state queuing model M/G(n)/C/Cunder the exponential velocity model.
aj=1+
C
X
n=1
n
Y
i=1
ρi j−1C
X
n=1n
n
Y
i=1
ρi j,∀j=1,2 (B.8)
peaj=1+
C
X
n=1
n
Y
i=1
ρi j−1,∀j=1,2 (B.9)
pbaj=1+
C
X
n=1
n
Y
i=1
ρi j−1C
Y
i=1
ρi j,∀j=1,2 (B.10)
with the state-dependent traffic intensity
ρi j =λj
i(v0/L)·exp[−((i−1)/βv)γv],∀i=1,2, ..., C,j=1,2 (B.11)
where parameters βvand γvare the same as those of Eq. B.2. λjis the demand that corresponds to the representative
points. It values bjtimes (e.g. b1=0.5 and b2=1) of the maximum traffic capacity of the facility as shown in Eq. B.12.
λj=bj·max
i∈[1,C]{i(v0/L)·exp[−((i−1)/βv)γv},∀j=1,2 (B.12)
37
The relationship between the idle or blocking probability and the relative density is visualized in Fig. B.28. The
idle probability PE(x) decreases exponentially with the relative density x/C, while the blocking probability PB(x)
increases exponentially with the relative density x/C.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative Density x/C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Idle probability PE(x)
b) Idle probability vs. relative density
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative Density x/C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Blocking probability PB(x)
a) Blocking probability vs. relative density
Fig. B.28. The relationship between the idle or blocking probability and the relative density.
In this manner, we obtain the state-dependent idle probabilities PEx
i(xi(t)), PEy
i(yi(t)), PEz
k,u(zk,u(t)) for the halls,
platforms, and trains, as well as the state-dependent blocking probabilities PBx
i(xi(t)), PBy
i(yi(t)), PBz
k,u(zk,u(t)).
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