Journal of Advanced Transportation

Journal of Advanced Transportation / 2020 / Article
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Innovative Methods for Data Informed Multimodal Transport Flow and System Analysis 2020

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Research Article | Open Access

Volume 2020 |Article ID 8898923 | https://doi.org/10.1155/2020/8898923

Yu Zheng, "Estimation of Disease Transmission in Multimodal Transportation Networks", Journal of Advanced Transportation, vol. 2020, Article ID 8898923, 16 pages, 2020. https://doi.org/10.1155/2020/8898923

Estimation of Disease Transmission in Multimodal Transportation Networks

Academic Editor: Weiwei Qi
Received27 Mar 2020
Revised21 May 2020
Accepted16 Jun 2020
Published01 Aug 2020

Abstract

Mathematical models are important methods in estimating epidemiological patterns of diseases and predicting the consequences of the spread of diseases. Investigation of risk factors of transportation modes and control of transportation exposures will help prevent disease transmission in the transportation system and protect people’s health. In this paper, a multimodal traffic distribution model is established to estimate the spreading of virus. The analysis is based on the empirical evidence learned from the real transportation network which connects Wuhan with other cities. We consider five mainstream travel modes, namely, auto mode, high-speed railway mode, common railway mode, coach mode, and flight mode. Logit model of economics is used to predict the distribution of trips and the corresponding diseases. The effectiveness of the model is verified with big data of the distribution of COVID-19 virus. We also conduct model-based tests to analyze the role of lockdown on different travel modes. Furthermore, sensitivity analysis is implemented, the results of which assist in policy-making for containing infection transmission through traffic.

1. Introduction

Despite tremendous efforts to reduce and control infectious diseases, infections continue to be a global threat to worldwide public health. Understanding the virus propagation is quite essential for the implementation of antivirus methods. While research studies about the antivirus policy have been extensively investigated, the viewpoint from the perspective of the propagation along transportation modals is relatively ignored. Consideration of risk factors of transportation modes and control of transportation exposures will help prevent disease transmission in the transportation system and protect people’s health. When an infectious disease case occurs at a location, investigators need to understand the mechanisms of disease propagation in the transportation network.

On December 31, 2019, the outbreak of novel coronavirus was first reported in China. The global outbreak of COVID-19 was mainly caused by transmission through different transportation modes. To prevent the spreading of virus, all the transportation system from Wuhan to the outside was closed in the morning of January 23, 2020. On January 30, 2020, the WHO (World Health Organization) declared a global emergency. On March 11, WHO declared the COVID-19 outbreak to be a global pandemic. For weeks after the first reports of a mysterious new virus of COVID-19, millions of people poured out of the central Chinese city, cramming onto buses, trains, and planes as the first wave of China’s great Lunar New Year migration broke across the nation, and some of them are virus carriers. The travel patterns broadly track with the early spread of the virus. The majority of confirmed cases and deaths have occurred in China, within Hubei province, followed by high numbers of cases in central China, with pockets of infections in Chongqing, Shanghai, and Beijing as well. The initial spread of travelers to provinces in central China is with large pools of migrant workers. There might be a “high correlation” between the early spread of coronavirus cases and the distribution of travel destinations. The atmosphere in the transportation vessels is closed, and it is easy for the virus to spread. And the transmission speed is different in different traffic modals, due to the different air fluency in the traffic vessels.

Mathematical models have become important tools in epidemiology in understanding epidemiological patterns of diseases and predicting the consequences of the introduction of public health interventions to control the spread of diseases. There are two lines of studies in epidemics spreading. The first line is the spreading model of differential equation, and the second line is the complex network theory. In the literature, there are three spreading models widely used in modeling virus transmission, namely, SIR model, SIS model, and SI model (acronyms such as M, S, E, I, and R are often used for the epidemiological classes. The class M represents individuals with passive immunity. The class S represents susceptible individuals who can become infected. The class E represents the exposed individuals in the latent period, who are infected but not yet infectious. The class I represents the individuals of infective, who are infectious in the sense that they are capable of transmitting the infection. The class R represents recovered individuals with permanent infection-acquired immunity. The choice of which epidemiological class to include in a model depends on the characteristics of the particular disease being modeled and the purpose of the model) [14]. To solve the models, three kinds of algorithms have been developed based on percolation theory [5, 6], mean field theory [7, 8], and Markov chain theory [9, 10].

Researchers also developed models to investigate the propagation of different types of viruses including some nonbiological viruses, such as the computer virus, the flash disk virus, the Bluetooth phone virus, and the email virus. Otero-Muras et al. presented a systematic approach to the biochemical network dynamic analysis and control based on both thermodynamic and control theoretic tools [11]. Based on biological control strategy in pest management, Pang and Chen constructed a pest-epidemic model with impulsive control, i.e., periodically spraying microbial pesticide and releasing infected pests at different fixed moments [12]. Jin and Wang developed a new dynamic propagation model of FD-SEIR, namely, flash disk virus susceptible-exposed-infectious-recovered, which is embodied by introducing the FD state and new propagation rate [13]. Huang et al. developed an epidemic model of Bluetooth phone virus [14]. Li et al. formulated a novel deterministic SEIS model for the transmission of email viruses in growing communication networks [15]. Jackson and Chen-Charpentier presented two plant virus propagation models, one with no delays and the other with two delays [16]. Jia and Lv established a stochastic rumor propagation model. Sufficient conditions for extinction and persistence in the mean of the rumor have been examined [17]. Zhang et al. established a spreading model based on contact strength and SI model, and a weighted network with community structure based on a network model proposed by Barrat et al. [18].

In the following, a multimodal traffic distribution model is established to estimate the spreading of virus. The analysis is based on the empirical evidence learned from the real transportation network which connects Wuhan with other cities. Five travel modes are considered, namely, auto mode, high-speed railway mode, common railway mode, coach mode, and flight mode. Logit model of economics is used to predict the distribution of trips and the corresponding diseases. The effectiveness of the model is verified with big data of the distribution of COVID-19 virus. The main contributions of the paper are in four aspects. First, we propose a multimodal traffic distribution model using data of the real transportation system. Second, we study the relation between the state of disease transmission and the traffic flows distribution based on the numerical results of the proposed model and the big data of the distribution of COVID-19 virus. Third, we use the model to predict the role of lockdown on different transport means and analyze its impact on the disease transmission. Fourth, we present a sensitivity analysis for the proposed model and derive various transportation improvement policies to control large-scale transportation exposure.

The remainder of this paper is organized as follows. Section 2 establishes a multimodal traffic distribution model to estimate the spreading of virus. The proposed model is validated in Section 3, using a real traffic distribution from Wuhan to other regions in China during the outbreak of COVID-19. Conclusions are made in Section 4.

2. Multimodal User Equilibrium Model

2.1. Multimode Travel Cost Functions

The multimode travel cost functions are based on the empirical evidence learned from the real transportation network connecting Wuhan to other cities. We consider five mainstream travel modes, namely, auto mode, high-speed railway mode, common railway mode, coach mode, and flight mode (Tables 1 and 2). Let represent the travel cost in the travel mode () to a destination region indexed by a region name (, D = {Xiaogan, Huanggang, Jingzhou, Xianning, E’zhou, Xiangyang, Huangshi, Jingmen, Suizhou, Xiantao, Yichang, Tianmen, Shiyan, Enshi, Qianjiang, Henan, Hunan, Anhui, Jiangxi, Guangdong, Jiangsu, Chongqing, Sichuan, Shandong, Zhejiang, Hebei, Fujian, Beijing, Guangxi, Shanxi, Shanghai, Shanxi, Guizhou, Yunnan, Hainan, Gansu, Liaoning, Heilongjiang, Xinjiang, Inner Mongolia, Jilin, Tianjin, Ningxia, Qinghai, Tibet, Hong Kong, Macao, Taiwan}). The set of destination regions D includes 15 cities within the province of Hubei and 33 province-level regions in China. We further define the set of 15 cities within the province of Hubei as ( = {Xiaogan, Huanggang, Jingzhou, Xianning, E’zhou, Xiangyang, Huangshi, Jingmen, Suizhou, Xiantao, Yichang, Tianmen, Shiyan, Enshi, Qianjiang}) and the set of other 33 province-level regions as ( = {Henan, Hunan, Anhui, Jiangxi, Guangdong, Jiangsu, Chongqing, Sichuan, Shandong, Zhejiang, Hebei, Fujian, Beijing, Guangxi, Shanxi, Shanghai, Shanxi, Guizhou, Yunnan, Hainan, Gansu, Liaoning, Heilongjiang, Xinjiang, Inner Mongolia, Jilin, Tianjin, Ningxia, Qinghai, Tibet, Hong Kong, Macao, Taiwan}), and D = .


CityProvinceTimeDistanceTollGas fee

XiaoganHubei9076.53043
HuanggangHubei7775.33042
JingzhouHubei18022090123
XianningHubei9092.63052
E’zhouHubei90752042
XiangyangHubei227313150175
HuangshiHubei1001004056
JingmenHubei180240110134
SuizhouHubei1201708095
XiantaoHubei1101024057
YichangHubei240322140180
TianmenHubei1201425080
EnshiHubei420519250291
ShiyanHubei300443210248
QianjiangHubei1311557087
ShijiazhuangHebei660898440503
TaiyuanShanxi720944450529
ShenyangLiaoning132018128901015
ChangchunJilin1560208810301169
HarbinHeilongjiang1440235411601318
NanjingJiangsu408550260308
HangzhouZhejiang660827350463
HefeiAnhui300388180217
FuzhouFujian672919450515
NanchangJiangxi330355170199
Ji’nanShandong600864420484
ZhengzhouHenan360514250288
ChangshaHunan289345150193
GuangzhouGuangdong672955480535
HaikouHainan11521566750877
ChengduSichuan8401130550633
GuiyangGuizhou7201011510566
KunmingYunnan11701558760872
Xi’anShanxi510740360414
LanzhouGansu9301360670762
Xi’ningQinghai11881594790893
TaibeiTaiwan5600
Beijing7801174580657
Tianjin7201144560641
Shanghai606825390462
Chongqing690897440502
HohhotInner Mongolia9601380650773
NanningGuangxi8101209590677
LhasaTibet3060348211101950
YinchuanNingxia9661448710811
UrumqiXinjiang2160326716001830
Hong Kong8401107510620
Macao9601210600678

Note that, in this part, the unit of measurement is kilometers for the distance, minutes for the time, and CNY for all kinds of tolls and fees.

CityProvinceHigh-speed railwayCommon railwayCoachFlight
TimeTicket feeTimeTicket feeTimeTicket feeTimeTicket fee

XiaoganHubei30586014.56032
HuanggangHubei90223720
JingzhouHubei90768932.530060
XianningHubei24406012.56028
E’zhouHubei24208012.56028
XiangyangHubei9013019050.524088
HuangshiHubei373010016.512042
JingmenHubei19040.5180101
SuizhouHubei557012826.518066
XiantaoHubei60509035
YichangHubei15012131053.5270105
TianmenHubei60459019.518055
EnshiHubei27018742078480130
ShiyanHubei14421733072390135
QianjiangHubei906515057
ShijiazhuangHebei2404155401244501450
TaiyuanShanxi3904861332173.5750320105727
ShenyangLiaoning660800132021714404801801820
ChangchunJilin68090415272431801800
HarbinHeilongjiang75310121396268.51801800
NanjingJiangsu180200480200
HangzhouZhejiang30030072012070028580830
HefeiAnhui120134360150
FuzhouFujian37126772012070028090870
NanchangJiangxi15012034453.5390120
Ji’nanShandong360525720130780280951000
ZhengzhouHenan14024430075480140
ChangshaHunan9216524053.5300120
GuangzhouGuangdong260464750138.59603401101800
HaikouHainan144025012904301501700
ChengduSichuan5603759901859603301201350
GuiyangGuizhou300481930163.59603201151000
KunmingYunnan420665137321715004801351660
Xi’anShanxi270455900135.5560240851200
LanzhouGansu400654120019012004301351330
Xi’ningQinghai1301300
TaibeiTaiwan1551400
Beijing270520720152.59003201202200
Tianjin300525840156.59603001151150
Shanghai300336900140720250951880
Chongqing390279540140780280951650
HohhotInner Mongolia18282291301050
NanningGuangxi45047884017010503201201180
LhasaTibet2301070
YinchuanNingxia15601981351300
UrumqiXinjiang23103452602000
Hong Kong2806791301574
Macao1001380

Note that, in this part, the unit of measurement is kilometers for the distance, minutes for the time, and CNY for all kinds of tolls and fees.

The function of travel cost for each mode from Wuhan to a destination region indexed by is described as follows:(1)Auto mode:where represents the cost of gasoline consumed per kilometer and denotes the auto travel time. The cost function consists of three terms. The first one stands for the monetary cost of travel time captured by the product of the value of in-vehicle travel time and the travel time ; the second one is the cost of gasoline consumed by this trip; the third one is the highway tolls charged along the highway captured by the product of the highway toll charged per kilometer and the total highway length. The average vehicle occupancy is the average number of occupants in a vehicle. We set the n-piece of auto utility function taking account of the actual traffic situation. The Transport Bureau of Wuhan announced that the average vehicle occupancy of a privately owned automobile in Wuhan is 1.8 persons/vehicle. Besides, it was reported by the Spring Festival Transport Office of the province of Hubei that advantages such as the trip cost shared by several relatives and friends, larger space for luggage, and no need to transfer are attracting more and more individuals traveling back home by a private car during the Spring Festival travel season. It is therefore reasonable to assume that the number of occupants in a vehicle within the Spring Festival travel season should be no less than the average vehicle occupancy, i.e., 1.8 persons/vehicle. Taking account of this, the value of is set as 2.(2)High-speed railway mode:where is the high-speed railway travel time and is the value of time spent in the high-speed railway. The travel cost consists of 2 terms: the first term is the cost of the travel time and the second term represents the high-speed railway ticket price.(3)Common railway mode:where is the common railway travel time and is the value of time spent in the common railway. The travel cost consists of 2 terms: the first term is the cost of the travel time and the second term represents the common railway ticket price.(4)Coach mode:where is the coach travel time and is the value of time spent in a coach. The travel cost consists of 2 terms: the first term is the cost of the travel time and the second term represents the coach ticket price.(5)Flight mode:where is the travel time by taking a plane and is the value of time spent in a flight. The travel cost consists of 2 terms: the first term is the cost of the travel time and the second term is the flight ticket price.

2.2. Multimodal User Equilibrium Model

To cater for the consideration of both mode choice and destination choice, we propose a multimodal network user equilibrium model as follows:subject towhere is the impedance parameter associated with the travel mode choice to the destination region d, is the impedance parameter associated with destination choice, is the exogenous attractiveness of the travel mode m to the destination d, and is the exogenous attractiveness of the destination region d. indicates travel demand in the travel mode m from Wuhan to the destination region d. indicates travel demand from Wuhan to the destination region d. As to , we develop the following weighted destination attractiveness measure to quantify the attractiveness of each destination to Wuhan:where is the normalized historical demand distribution ratio of the destination region d; is the normalized population of the destination region d; is the normalized distance between the destination and Wuhan; is a weighting parameter for the parameter ; is a weighting parameter for the parameter ; is a weighting parameter for the parameter . The weighting parameters , , and reflect the extent of effects of historical demand distribution, population, and distance on the attractiveness of a destination region . Learned from real-life experiences, the historical demand distribution ratio of a destination region () is correlated oppositely to its travel distance to Wuhan (). Furthermore, referring to equation (2l), for a destination region , its generalized travel cost to Wuhan () varies incrementally with the travel time which is determined by the travel distance (). To mitigate the interrelation between and , the term is multiplied by the term .

We denote the original data of historical demand distribution ratio, population, and travel distance of a destination region as , , and . As , , and are incommensurable, namely, all are measured in different units, they cannot be directly added and need to be normalized before the use of the weighted-sum method. To do so, we define , , and as follows:where is a function to obtain the minimum item in a list, for example, is to get the minimum item in the list . is a function to obtain the maximum item in a list, for example, is to get the maximum item in the list . According to equation (2i), it is easy to get that , from which we can infer that the value of the weighted distance parameter ranges from 0 to 1 for any (). Besides, it can be learned from experiences that passengers are more sensitive to travel cost in a short trip than in a long trip. This phenomenon has been studied in the area of stochastic traffic flow distribution [19]. In this work, passengers within the province of Hubei are more sensitive to the travel cost than those traveling out of the province of Hubei because of shorter travel distance. That is to say, the magnitude of travel cost takes a greater effect on the attractiveness of a destination region inside Hubei than that outside Hubei, meaning the value of should be greater for than for . Along with the already known condition that , it consequently requires the travel distance-related weighting parameter taking a smaller value for than . Furthermore, as the provincial capital of Hubei, Wuhan attracts a good many of migrant workers and students working or studying there each year for its abundant employment opportunities and diverse educational resources. The migrant population constitutes the majority of travel demands in the Spring Festival travel season in Wuhan. Without unexpected disruption, the historical traffic distribution of Wuhan in recent Spring Festival travel seasons will provide high-quality evidence for predicting the traffic distribution of this year. To reflect the significant impacts of the historical traffic distribution on the assessment of a destination’s attractiveness, we suggest that the historical traffic distribution-related weighting parameter takes a larger value than the population-related weighting parameter . The detailed value setting for various parameters defined in this part can be found in Table 3.



1.21.11.251.31.0
()
0.620.050.010.0
0.0031050.30.5

The objective function (2a) is a two-level nested logit choice model to deal with the interrelated decisions in a multimodal network. The first level focuses on destination choice and the second level on mode choice. Equation (2b) ensures that the amount of flow assigned to different destination regions from Wuhan sums to the total travel demand which, in this work, amounts to 5,000,000. Equation (2c) represents the mode flow conservation constraint. Equations (2d) and (2e) are the nonnegativity conditions for destination demands and mode flows, respectively.

By deriving the first-order optimality conditions of the proposed program, we have the following nested logit model for destination choice and mode choice, respectively:where is users’ perception of the generalized cost of traveling from the origin city Wuhan to the destination city d, which is computed as a “log-sum” of travel cost of each mode, i.e.,

To solve the nested logit model-based problem, one can first compute the generalized cost according to equation (2l) and then carry out multiproportional traffic assignment (2j)-(2k) to obtain the combined destination distribution and modal split, i.e., and .

3. Case Study

The outbreak of COVID-19, which started in December last year, took Wuhan as the center and soon spread to all regions of China (including Hong Kong, Macao, and Taiwan). In the early morning of January 22, the province of Hubei launched level II emergency response to public health emergencies, and then cities in Hubei successively stopped public transportation. As of 11 : 00 on January 24, public transportation in 12 cities in Hubei had been shut down, including Wuhan, E’zhou, Xiantao, Zhijiang, Qianjiang, Huanggang, Chibi, Jingmen, Xianning, Huangshi, Dangyang, and Enshi, among which Wuhan, as the transport hub of more than 10 million people, temporarily closed its airports, rail stations, and all main roads out of town, as well as suspended public buses and subways. The government announced that citizens should not leave Wuhan without special reasons, and the lift of the lockdown will be announced separately. On January 26, the Information Office of the People’s Government of Hubei held a press conference, pointing out that from the beginning of the Spring Festival to the closure of Wuhan, more than 5 million people left Wuhan, and more than 9 million remained in the city.

In this section, we will use the transportation model proposed in Section 2 to analyze the traffic flow distribution for the 5 million people outbound from Wuhan and then estimate the epidemic situation based on the demand distribution results. We are mainly concerned about the distribution of people within the province of Hubei as well as outside the province of Hubei. Figure 1 shows the map of the province of Hubei and 35 other regions of China, and Figure 2 shows the map of Wuhan and 16 other cities in the province of Hubei.

To facilitate the computation of the travel utility to a destination province outside Hubei, instead of calculating the travel utility to each city in the destination province, we only calculate the travel utility to the provincial capital city. For the calculation of the normalized historical demand distribution ratio parameter in equation (2f), we collect the data of migration from Wuhan to other destination regions of the year 2017 on the Tencent social network’s Spring Festival geographic positioning data platform. The data show that except several provinces including Henan, Hunan, Anhui, Jiangsu, and Guangdong, for other provinces, the majority of the traffic out of Wuhan flowed into their provincial capitals. As a result, we replace the population of a province by the population of its provincial capital for the computation of the parameter in equation (2f). As to other five provinces, i.e., Henan, Hunan, Anhui, Jiangsu, and Guangdong, we use the sum of population of cities which occupied the most amount of immigration from Wuhan in 2017 instead of the population of the province. Besides, to obtain the travel distance parameter in equation (2f), we use the road length from Wuhan to other destination regions to measure the travel distance. The data of historical demand distribution, the population of cities and provinces, and the road distance from Wuhan to other destination regions can be found in Table 4. parameters for the computation of the nested logit model-based traffic assignment are listed in Table 3. Learned from the real traveling experiences, parameters are set as follows: . Furthermore, in this study, we consider different travel cost sensitivities of passengers with different scale of travel path sizes. The related research results [19] reveal that passengers in short trip are more sensitive to travel distance or travel cost than those in long trip, which causes the value of for is 5 times that for .


DestinationHistorical demand distribution ratioPopulation in millionsRoad distance

Xiaogan0.1186154.878076.5
Huanggang0.1103316.291075.3
Jingzhou0.0551535.7442220
Xianning0.0431882.462692.6
E’zhou0.0310151.048775
Xiangyang0.0368245.6140313
Huangshi0.0339942.4293100
Jingmen0.0293042.8737240
Suizhou0.0286772.1622170
Xiantao0.0294311.1660102
Yichang0.0247854.1150322
Tianmen0.0201311.4189142
Enshi0.0162313.2903519
Shiyan0.0171293.3830443
Qianjiang0.0114130.9463155
Hebei0.0153711.0312898
Shanxi0.006764.4619944
Liaoning0.004428.31601812
Jilin0.001777.67702088
Heilongjiang0.0024610.85802354
Jiangsu0.0178025.3086550
Zhejiang0.0118810.3600827
Anhui0.0273424.5670388
Fujian0.009587.8000919
Jiangxi0.024515.5455355
Shandong0.015027.4604864
Henan0.0751540.5440514
Hunan0.0420032.6188345
Guangdong0.0231645.9177955
Hainan0.003372.30231566
Sichuan0.0239716.33001130
Guizhou0.007464.88191011
Yunnan0.006276.85001558
Shanxi0.0133210.0037740
Gansu0.004333.75361360
Qinghai0.000972.38711594
Beijing0.0114721.5361174
Tianjin0.0022215.61831144
Shanghai0.0079224.2814825
Chongqing0.0163831.2432897
Inner Mongolia0.001953.12601380
Guangxi0.008407.56871209
Ningxia0.000742.29313267
Xinjiang0.001823.50581107

Note that, in this part, the unit of measurement is kilometers for the distance, minutes for the time, and CNY for all kinds of tolls and fees.
3.1. Demand Assignment

Based on the model proposed in Section 2, we calculate the traffic flows from Wuhan to other 48 destination regions which include 15 cities within Hubei, and 33 destination regions outside Hubei. Note that we exclude several regions which include the Shennongjia Forest District in Hubei, the Diaoyu Islands, and the South China Sea Islands from the calculation of destination demand distribution for that the traffic flows of these regions are very small. Besides, the demand distribution in real condition is collected from the Baidu Migration Big Data Platform. The error ratio of estimation is defined as the ratio of the estimation error to the result in real condition. Data of the estimated demand distribution, the demand distribution in real condition, and the error of estimation are listed in Tables 5 and 6 for cities within the province of Hubei and destination regions outside Hubei, respectively.


DestinationReal conditionEst. resultsErrorError ratio (%)

Xiaogan690000.00665702.75−24297.25−3.52
Huanggang652000.00612343.63−39656.37−6.08
Jingzhou327000.00320223.65−6776.35−2.07
Xianning250500.00274428.5123928.519.55
E’zhou198500.00235207.1536707.1518.49
Xiangyang196500.00193181.29−3318.71−1.69
Huangshi188500.00236658.7148158.7125.55
Jingmen165000.00156216.92−8783.08−5.32
Suizhou160500.00199004.5238504.5223.99
Xiantao148500.00202067.3753567.3736.07
Yichang140500.00131028.35−9471.65−6.74
Tianmen104000.00183119.5179119.5176.08
Shiyan93000.0089343.10−3656.90−3.93
Enshi90500.0064965.07−25534.93−28.22
Qianjiang57000.00140509.6483509.64146.51


DestinationReal conditionEst. resultsErrorError ratio (%)

Henan284000.00250214.37−33785.63−11.90
Hunan174000.00187131.6613131.667.55
Anhui113500.00144540.7531040.7527.35
Jiangxi106000.00128569.1222569.1221.29
Guangdong97000.0044218.78−52781.22−54.41
Jiangsu73000.0088634.8815634.8821.42
Chongqing63500.0042893.78−20606.22−32.45
Sichuan62000.0020641.30−41358.70−66.71
Shandong55000.0027805.30−27194.70−49.44
Zhejiang53500.0045031.19−8468.81−15.83
Hebei46500.0040775.05−5724.95−12.31
Fujian45500.0036048.21−9451.79−20.77
Beijing43000.0026050.15−16949.85−39.42
Guangxi39500.0014846.45−24653.55−62.41
Shanxi36000.0037590.591590.594.42
Shanghai33000.0036512.913512.9110.65
Shanxi29500.0022472.40−7027.60−23.82
Guizhou27500.0021864.32−5635.68−20.49
Yunnan26500.008502.62−17997.38−67.91
Hainan19000.002387.83−16612.17−87.43
Gansu17500.008962.46−8537.54−48.79
Liaoning16500.002702.27−13797.73−83.62
Heilongjiang14000.001237.75−12762.25−91.16
Xinjiang10000.0068.07−9931.93−99.32
Inner Mongolia9000.006731.88−2268.12−25.20
Jilin8500.001706.55−6793.45−79.92
Tianjin7500.0018275.5610775.56143.67
Ningxia4000.003969.71−30.29−0.76
Qinghai3000.003140.77140.774.69
Tibet1000.004156.123156.12315.61
Others19500.0018317.04−1182.96−6.07

Table 7 provides the results of the aggregated demand distribution ratio, which shows that, in both the real and estimation conditions, the traffic flows within the province of Hubei account for most part (about 70%) of the total demands. It is also shown in Table 7 that the aggregated error ratio of demand estimation for cities inside Hubei (6.99%) is smaller than that for destinations outside Hubei (15.73%), meaning it performs better in demand estimation within the province of Hubei, and the aggregated error ratio of demand estimation for all destination regions is 18.60%. The demand distribution results which take a decreasing order are shown in Figure 3. The destination name marked with an asterisk denotes a city within the province of Hubei. Note that in Figure 3, we put the demand distribution results of Hong Kong, Macao, and Taiwan into one item named “others” for brevity. According to results in Figure 3, in both the real and estimation conditions, Xiaogan, Huanggang, Jingzhou, Xianning, and E’zhou are the top five cities within the province of Hubei with the largest distribution of people, and Henan, Hunan, Anhui, and Jiangxi are the top four destination regions outside Hubei with the largest distribution of people. All results indicate that in a tolerable error range, our model delivers a desirable performance on the estimation of traffic flow distribution.


Destination rangeAggregated traffic distribution ratioAggregated error ratio of estimation (%)
Real condition (%)Estimation (%)

All18.60
Des. inside Hubei69.2474.086.99
Des. outside Hubei30.7625.9215.73

3.2. Numerical Estimation of Incidence Cases

According to statistics released by the Chinese health authority, after March 18, all the increased confirmed cases in China are imported from overseas. Therefore, we use statistics of the day, March 18, to obtain the number of confirmed cases resulted from the travelers from Wuhan. The average incidence rate () of people leaving Wuhan is defined as follows:where is the number of confirmed cases nationwide apart from Wuhan. With , we have . We then further estimate the number of incidence cases in different destination regions which is equal to , . Results of the real number of incidence cases, the estimated number of incidence cases, the error of estimation for the number of incidence cases as well as the error ratio estimation for the number of incidence cases are listed in Tables 8 and 9 for destinations within and outside Hubei, respectively.


DestinationReal conditionEst. resultsErrorError ratio (%)

Xiaogan3518.004166.77648.7718.44
Huanggang2907.003832.78925.7831.85
Jingzhou1580.002004.34424.3426.86
Xianning836.001717.70881.70105.47
E’zhou1394.001472.2178.215.61
Xiangyang1175.001209.1634.162.91
Huangshi1015.001481.29466.2945.94
Jingmen928.00977.7949.795.37
Suizhou1307.001245.61−61.39−4.70
Xiantao575.001264.78689.78119.96
Yichang931.00820.13−110.87−11.91
Tianmen496.001146.18650.18131.09
Shiyan672.00559.22−112.78−16.78
Enshi252.00406.63154.6361.36
Qianjiang198.00879.48681.48344.18


DestinationReal conditionEst. resultsErrorError ratio (%)

Henan1274.001566.14292.1422.93
Hunan1018.001171.29153.2915.06
Anhui990.00904.71−85.29−8.62
Jiangxi936.00804.74−131.26−14.02
Guangdong1415.00276.77−1138.23−80.44
Jiangsu633.00554.78−78.22−12.36
Chongqing577.00268.48−308.52−53.47
Sichuan543.00129.20−413.80−76.21
Shandong768.00174.04−593.96−77.34
Zhejiang1238.00281.86−956.14−77.23
Hebei319.00255.22−63.78−19.99
Fujian313.00225.63−87.37−27.91
Beijing537.00163.05−373.95−69.64
Guangxi254.0092.93−161.07−63.41
Shanxi248.00235.29−12.71−5.13
Shanghai404.00228.54−175.46−43.43
Shanxi133.00140.667.665.76
Guizhou146.00136.85−9.15−6.26
Yunnan176.0053.22−122.78−69.76
Hainan168.0014.95−153.05−91.10
Gansu136.0056.10−79.90−58.75
Liaoning127.0016.91−110.09−86.68
Heilongjiang484.007.75−476.25−98.40
Xinjiang76.000.43−75.57−99.44
Inner Mongolia75.0042.14−32.86−43.82
Jilin93.0010.68−82.32−88.51
Tianjin141.00114.39−26.61−18.87
Ningxia75.0024.85−50.15−66.87
Qinghai18.0019.661.669.21
Tibet1.0026.0125.012501.40
Hong Kong233.00114.65−118.35−50.79
Macao155.0070.40−84.60−54.58
Taiwan11.0031.8320.83189.38

Figure 4 illustratively presents results in Tables 9 and 10. It can be observed from Figure 4 that our estimation overestimates the number of incidence cases in most cities within Hubei, as well as two provinces, i.e., Henan and Hunan. According to results in Section 3.1, these cities/provinces are the destination regions with the largest traffic flow distributions. The fact of the lower incidence rate of these destination regions with the most immigration from Wuhan than the average incidence rate implies that measures adopted by the Chinese government played an effective role in preventing a more serious situation from developing. Measures in the prevention and control of the epidemic involve lockdown on public transportations in 12 cities in Hubei, a two-week mandatory self-quarantine for people immigrated from Wuhan, residential community-based management, constructing temporary treatment centers, the centralized schedule of medical staff and supplies to the scarce areas, and so on. At the same time, results in Figure 4 also reveal that the model-based results underestimate the incidence rate of several destination regions including Guangdong, Chongqing, Sichuan, Shandong, Zhejiang, Beijing, Shanghai, Heilongjiang, and Hong Kong. This is because high frequent commercial activities that involve face-to-face or close contact with other people lead to the higher incidence rates of the economically developed provinces, such as Guangdong, Zhejiang, Beijing, Shanghai, and Hong Kong. It is also interesting to see that the province of Heilongjiang, far away from Wuhan, not as commercially active as the provinces mentioned above, is featured by its high incidence rate. The high incidence rate of Heilongjiang may be attributed to the mass contact transmission of virus in gathering activities. According to news reported in Heilongjiang, to the date of February 7, there had been 48 family aggregating activities which were the source of 194 cases of cluster infection.


CityRoadHigh-speed railRailCoachFlight

Xiaogan170285.59178569.34223626.4493221.37
Huanggang161473.6325124.46425745.54
Jingzhou5018.3745290.82269914.250.22
Xianning33196.64151630.8261341.0228260.04
E’zhou25378.08188460.038447.6312921.41
Xiangyang3884.20174498.6314686.33112.13
Huangshi28096.44200465.627468.12628.54
Jingmen129761.1124305.322150.49
Suizhou30586.40155330.5413007.8879.70
Xiantao87947.5490625.9423493.89
Yichang29596.79101257.4194.6779.47
Tianmen26117.57109137.4747827.8636.61
Shiyan9449.6379120.77771.800.90
Enshi5196.7959325.61441.770.90
Qianjiang76189.1562691.081629.41

In Table 11, we compute the aggregated error ratio of estimation for the number of incidence cases which is equal to 39.52%, almost two times of the aggregated error ratio of estimation for the traffic flow distribution (18.60%), indicating that the spread of the epidemic is not linear with respect to the model-based traffic flow distribution. For cities within Hubei, the aggregated number of incidence case distribution ratio in real condition (56.40%) is much lower than the estimation (74.08%) while for provinces outside Hubei, the aggregated number of incidence case distribution ratio in real condition (43.60%) is much higher than the estimation (25.92%). This is because the spread of disease within Hubei is well controlled by means of transport restriction, medical assistance, and other effective methods while the high economic activity frequency as well as the high occurrence of mass gatherings in some provinces outside Hubei will potentially increase the incidence rate outside Hubei.


Destination rangeAggregated number of incidence case distribution ratioAggregated error ratio of the estimation (%)
Real condition (%)Estimation (%)

All39.52
Within Hubei56.4074.0830.36
Outside Hubei43.6025.9239.94

3.3. Mode Flow Distribution

Public transport as the main mode of transportation in big cities carries the highest risk of transmission of infection for a number of reasons. The high density of passengers confined in relatively small spaces was the primary cause. Besides, the in-vehicle air conditioning system featured by the low ventilation rates makes it easy for virus to spread. And the indirect infection from the contaminated public facilities in transport vessels is also one of the major danger sources. Furthermore, for passengers taking a long trip, multiple public transportation transfers are often involved, the fact of which potentially increases the incidence rate. In contrast, self-driving or taking a ride in a privately owned vehicle has several advantages over public transport in containing the transmission of infection. First, passengers are separated by vehicles. The spatial isolation reduces the risk of cross infection. Second, in the self-driving travel mode, passengers drive to destinations directly without any transfer most of the time. Third, people who are friends or familiar with each other often travel together in a privately owned vehicle. It is easy for them to learn the health condition of each other which helps to raise their awareness of health security and as a result mitigates the risk of infection. Comparisons of different transportation means’ impacts on the virus spreading reveal that it is important to enhance the epidemic prevention from the perspective of public transport control.

In this section, we first calculate the mode flow distribution based on the proposed model. The mode flow distributions of each destination region are listed in Tables 10 and 12. The results of the aggregated mode flow ratio for destinations inside Hubei as well as outside Hubei are shown in Figure 5. It can be seen from Figure 5 that, for destinations both inside and outside Hubei, public transports are the mainstream transportation means accounting for about 80% of the total demands. Besides, the most popular travel mode of public transport is the high-speed railway for trips both inside and outside Hubei, which indicates that enhanced measurements, such as disinfection and disease detection, should be adopted by the high-speed railway transportation system. Furthermore, the proportion of aggregated mode flow ratio of the common railway inside Hubei (29.63%) is much higher than that outside Hubei (13.55%), indicating that, for trips from Wuhan to cities inside Hubei, extra efforts should also be paid on the epidemic control in the common railway transportation system.


DestinationRoadHigh-speed railRailCoachFlight

Henan44850.03118548.7481885.684929.93
Hunan39450.6192948.8643774.4110957.78
Anhui32300.71107671.974568.06
Jiangxi28926.2682248.9013420.053973.91
Guangdong5998.1436177.922020.7121.680.33
Jiangsu16969.8370065.961599.09
Chongqing3815.8126847.0012063.14166.990.84
Sichuan4396.6415655.80343.30115.42130.14
Shandong16523.836598.243179.75345.351158.13
Zhejiang5446.3836087.671047.02255.622194.49
Hebei3402.0326799.1710573.710.13
Fujian3006.5429630.071349.57346.381715.65
Beijing1164.3821849.672971.8564.200.05
Guangxi1994.3410993.611415.2625.41417.83
Shanxi21077.6213574.76131.762740.7065.76
Shanghai7129.7328941.40113.64328.000.13
Shanxi4241.405456.971.04258.4412514.56
Guizhou2971.3517850.25164.8124.04853.86
Yunnan122.458364.345.240.0810.50
Hainan2237.1526.9226.9296.83
Gansu978.497754.3936.453.31189.81
Liaoning183.872345.41149.942.5620.50
Heilongjiang33.87751.06267.06185.77
Xinjiang2.691.7963.58
Inner Mongolia1385.890.025345.97
Jilin22.741600.3923.2060.22
Tianjin2057.9015236.40689.8338.92252.51
Ningxia2615.522.361351.83
Qinghai545.382595.38
Tibet4156.12
Hong Kong2933.438307.636.39
Macao4666.53419.13
Taiwan1983.93

To contain the COVID-19 outbreak, many countries have implemented flight restrictions to China. At the same time, China itself has imposed a lockdown of the transportation system of Wuhan as well as the entire Hubei province. In this context, it is reasonable to investigate how the mode flow distribution changes with different outbound transport restrictions in Wuhan. We will use the proposed nested logit model to analyze the role of lockdown on each transport means in the following content.

3.3.1. Lockdown Test

Table 13 shows the results of the aggregated demands ratio inside Hubei under cases applying lockdown on different travel modes and it reveals that a lockdown on any travel mode will lead to an increase of the aggregated demands ratio inside Hubei, among which shutting down the high-speed railway will cause the maximum rise of the aggregated demands ratio inside Hubei from 73.87% to 78.15%. This indicates that a lockdown on any travel mode will not make a big difference to the change of the aggregated demand distribution between destinations inside and outside Hubei. We then check the effects of transport restriction on the change of mode flow distribution, and related results are listed in Tables 14 and 15 for destinations inside Hubei and outside Hubei, respectively. From Tables 14 and 15, it can be seen that, for destinations both inside and outside Hubei, lockdown on a certain transportation means leads to the growth of traffic flows of other travel modes, and particularly lockdown on the high-speed railway has the most prominent impact on the traffic flow increment of other travel modes, indicating that in the case of a lockdown on the high-speed railway, enforcement on the control of transportation exposures should be conducted for all the other public transport systems. It also reveals that a lockdown on a certain travel mode may cause different extent of aggregated mode flow increment of other travel modes. For example, the common railway restriction has the most significant impact on the increase of the aggregated mode flows of automobile (63.65%) for destinations inside Hubei. And the automobile restriction leads to higher aggregated mode flow growth of coach (42.98%) for destinations inside Hubei than any other aggregated mode flow increment. This indicates that it is important to measure the magnitude of correlation between lockdown on a certain travel mode and the traffic flow increase of other travel modes, and the public transport mode which has a high correlation with the lockdown needs intensified management to contain virus spreading through transportation.


Original case (%)Locked down travel mode
Automobile (%)High-speed railway (%)Common railway (%)Coach (%)Flight (%)

74.0875.0678.6374.4174.1674.34


Aggregated mode flow increment (%)Locked down travel mode
Automobile (%)High-speed railway (%)Common railway (%)Coach (%)

Automobile121.5963.655.49
High-speed railway25.5432.784.31
Common railway35.2962.224.02
Coach42.98102.4236.47


Aggregated mode flow increment (%)Locked down travel mode
Automobile (%)High-speed railway (%)Common railway (%)Coach (%)Flight (%)

Automobile127.4114.822.424.13
High-speed railway19.9814.341.941.37
Common railway19.7798.772.320.50
Coach28.28114.7817.711.13
Flight29.3863.294.030.77

3.4. Sensitivity Analysis

In this work, the logit-based probability expression for both destination and mode choice ensures that the solution to the lower-level programming is unique. Hence, the standard sensitivity analysis method for nonlinear programming problem can be used directly to derive the sensitivity information. The detailed derivation can be referred to Yang and Chen [20] and Yang et al. [21]. In this section, we conduct sensitivity analyses to explore how changes in input parameters including road tolls, high-speed railway ticket fees, common railway ticket fees, coach ticket fees, flight ticket fees, and the average vehicle occupancy affect certain traffic flows we are interested in. To ease the work of analysis, we investigate the change of traffic flows with respect to the same amount of perturbations of a particular parameter for all destinations rather than for each destination, respectively. For example, the term of derivative indicates the change of the traffic flow with respect to an increase of 1 Chinese Yuan (CNY) in the road tolls for all destinations, different from the derivative term () which represents the change of the traffic flow with respect to an increase of 1 CNY in the road toll for the specific destination indexed by .

As aforementioned, compared with other public transport modes, traveling in privately owned vehicles contributes to less transportation exposures. As a result, measures taken to encourage traffic flows shifting from the public transport modes to the auto mode will mitigate transmission risks. We check the derivatives of the aggregated mode flows of auto which is defined as with respect to perturbations of input parameters in Table 16. According to the results in Table 16, an increase of has the most direct positive impact on , which is followed by an increase of . This indicates that the increase of the average vehicle occupancy and the increase of high-speed railway ticket fee for all destinations are the most effective way to boost the aggregated mode flow of auto:


Aggr. mode flow0.1 ∗ 

−14934.3716010.4811616.382172.0169.8826895.38
21.96−232.22168.1921.83.761954.73

As we discussed in Section 3.2, the actual incidence rates in economically developed destinations as well as destinations with high occurrence of big gathering activities are much higher than the estimated incidence rates. It is a natural thought to prevent the virus spreading from seriously developing by curbing the demands of these destinations. The aggregated demands of the economically developed destinations as well as destinations with high occurrence of big gathering activities is defined as in equation (3b), which is the sum of demands of multiple destinations involving Zhejiang, Jiangsu, Guangdong, Beijing, Shanghai, Hong Kong, and Heilongjiang. We check the sensitivity of the aggregated demands of these destinations () with respect to different parameters in Table 16. One useful application of the derivatives of is to identify effective measures to induce negative growth of . The negative value of indicates that an increase in a particular parameter leads to a decline in . In this study, decreasing will cause the most decline in , and the second largest decline is generated by increasing . These results indicate that the decrease of average vehicle occupancy as well as the increment of the high-speed railway pricing will be good candidates for the control of traffic flows from Wuhan to those destinations with high incidence rate risks. Remember that reducing will also lead to a decrease of , which is against the original aim of promoting the aggregated auto flows. This kind of contradictive effect of the adjustment of a certain parameter should be taken account of and considerate measures should be conducted. With the aim of increasing and decreasing at the same time, we find a compromise in the conflict mentioned above, which raises the prices of high-speed railway tickets by a large amount while making a small improvement on the average vehicle occupancy or keeping the average vehicle occupancy without any change.

4. Conclusion

In this paper, a nested logit-based multimodal traffic flow distribution model and a solution algorithm are proposed. The model is designed taking account of experiences learned from historical data as well as making use of information collected from the real transportation system. The proposed model is verified by the application to a real-life problem of the demand distribution from Wuhan to other nationwide regions during the outbreak of COVID-19. The estimation results in the case show that the model proposed in this work delivers a desirable performance on demand distribution estimation. The results of the estimation of the number of incidence cases reveal that the spread of the epidemic is not linear with respect to the estimated traffic flow distribution results. And further analysis on this result inspires us that the spread of the crisis is not purely dependent on the transportation situation, but also affected on the one hand by the control methods conducted by the public power and on the other hand by the frequency of local economic activities as well as the occurrence number of crowd-collected activities. The analysis of the role of lockdown on different travel modes reflects that lockdown on the high-speed railway has the most prominent impact on the traffic flow increment of other travel modes, and a lockdown on a certain travel mode causes different extent of aggregated mode flow increment of other travel modes. It is important to measure the magnitude of correlation between lockdown on a certain travel mode and the traffic flow increase of other travel modes. The public transport mode which has a high correlation with the lockdown policy needs intensified management to prevent virus from spreading through transportation. Furthermore, sensitivity analysis is implemented in this study, and based on the results of which, we work out a compromise solution for stimulating the traffic flow of automobile and reducing the demands of target regions with high incidence rates at the same time.

Data Availability

The data used to support this study are available at Tencent social network’s 2017 Spring Festival geographic positioning data platform, Baidu Migration Big Data Platform (https://qianxi.baidu.com/2020/), and https://news.qq.com/zt2020/page/feiyan.htm#/global.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work has been substantially supported by the National Natural Science Foundation of China through several projects (no. 71890970/71890973 and 71531011) and a project sponsored by the Program of Shanghai Academic Research Leader.

References

  1. R. M. Anderson, R. M. May, and B. Anderson, “The mathematics of infection. (Book reviews: infectious diseases of humans. dynamics and control),” Science, vol. 254, no. 2, pp. 591-592, 1991. View at: Google Scholar
  2. N. Bailey, “The mathematical theory of infectious diseases and its applications,” Immunology, vol. 34, no. 5, pp. 955-956, 1978. View at: Google Scholar
  3. H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at: Publisher Site | Google Scholar
  4. M. E. J. Newman, “The structure and function of complex networks,” SIAM Review, vol. 45, no. 2, pp. 167–256, 2003. View at: Publisher Site | Google Scholar
  5. C. Li, W. Hu, and T. Huang, “Stability and bifurcation analysis of a modified epidemic model for computer viruses,” Mathematical Problems in Engineering, vol. 2014, Article ID 784684, 14 pages, 2014. View at: Publisher Site | Google Scholar
  6. C. Fraser, C. A. Donnelly, S. Cauchemez et al., “Pandemic potential of a strain of influenza A (H1N1): early findings,” Science, vol. 324, no. 5934, pp. 1557–1561, 2009. View at: Publisher Site | Google Scholar
  7. M. E. J. Newman, S. Forrest, and J. Balthrop, “Email networks and the spread of computer viruses,” Physical Review E, vol. 66, no. 3, Article ID 035101, 2002. View at: Publisher Site | Google Scholar
  8. R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics in finite size scale-free networks,” Physical Review E, vol. 65, no. 3, Article ID 035108, 2002. View at: Publisher Site | Google Scholar
  9. Y. Wang, D. Chakrabarti, C. Wang, and C. Faloutsos, “Epidemic spreading in real networks: an eigenvalue viewpoint,” in Proceedings of the 22nd International Symposium on Reliable Distributed Systems (SRDS’03), pp. 25–34, IEEE, Florence, Italy, October 2003. View at: Publisher Site | Google Scholar
  10. D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos, “Epidemic thresholds in real networks,” ACM Transactions on Information and System Security, vol. 10, no. 4, pp. 1311–1326, 2008. View at: Publisher Site | Google Scholar
  11. I. Otero-Muras, G. Szederkényi, K. M. Hangos, and A. A. Alonso, “Dynamic analysis and control of biochemical reaction networks,” Mathematics and Computers in Simulation, vol. 79, no. 4, pp. 999–1009, 2008. View at: Publisher Site | Google Scholar
  12. G. Pang and L. Chen, “Dynamic analysis of a pest-epidemic model with impulsive control,” Mathematics and Computers in Simulation, vol. 79, no. 1, pp. 72–84, 2008. View at: Publisher Site | Google Scholar
  13. C. Jin and X.-Y. Wang, “Analysis and control stratagems of flash disk virus dynamic propagation model,” Security and Communication Networks, vol. 5, no. 2, pp. 226–235, 2012. View at: Publisher Site | Google Scholar
  14. X. Y. Huang, C. Jin, and S. L. Jin, “Dynamic propagation model of blue-tooth virus on smart phones,” Advanced Materials Research, vol. 121-122, pp. 620–626, 2010. View at: Publisher Site | Google Scholar
  15. Y. H. Li, J. X. Pan, and Z. Jin, “Dynamic modeling and analysis of the email virus propagation,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 472072, 22 pages, 2012. View at: Publisher Site | Google Scholar
  16. M. Jackson and B. M. Chen-Charpentier, “A model of biological control of plant virus propagation with delays,” Journal of Computational and Applied Mathematics, vol. 330, pp. 855–865, 2017. View at: Publisher Site | Google Scholar
  17. F. Jia and G. Lv, “Dynamic analysis of a stochastic rumor propagation model,” Physica A: Statistical Mechanics and Its Applications, vol. 490, pp. 613–623, 2018. View at: Publisher Site | Google Scholar
  18. J. Zhang, J. Jiang, Y. Chen, K. Yang, H. L. You, and Y. W. Chen, “Epidemic spreading characteristics and immunity measures based on complex network with contact strength and community structure,” Mathematical Problems in Engineering, vol. 2015, Article ID 316092, 10 pages, 2015. View at: Publisher Site | Google Scholar
  19. S. Kitthamkesorn and A. Chen, “A path-size weibit stochastic user equilibrium model,” Transportation Research Part B: Methodological, vol. 57, no. 11, pp. 378–397, 2013. View at: Publisher Site | Google Scholar
  20. C. Yang and A. Chen, “Sensitivity analysis of the combined travel demand model with applications,” European Journal of Operational Research, vol. 198, no. 3, pp. 909–921, 2009. View at: Publisher Site | Google Scholar
  21. C. Yang, A. Chen, X. Xu, and S. C. Wong, “Sensitivity-based uncertainty analysis of a combined travel demand model,” Transportation Research Part B: Methodological, vol. 57, pp. 225–244, 2013. View at: Publisher Site | Google Scholar

Copyright © 2020 Yu Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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