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.
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) [
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 [
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
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
Automobile transportation parameters.
City | Province | Time | Distance | Toll | Gas fee |
---|---|---|---|---|---|
Xiaogan | Hubei | 90 | 76.5 | 30 | 43 |
Huanggang | Hubei | 77 | 75.3 | 30 | 42 |
Jingzhou | Hubei | 180 | 220 | 90 | 123 |
Xianning | Hubei | 90 | 92.6 | 30 | 52 |
E’zhou | Hubei | 90 | 75 | 20 | 42 |
Xiangyang | Hubei | 227 | 313 | 150 | 175 |
Huangshi | Hubei | 100 | 100 | 40 | 56 |
Jingmen | Hubei | 180 | 240 | 110 | 134 |
Suizhou | Hubei | 120 | 170 | 80 | 95 |
Xiantao | Hubei | 110 | 102 | 40 | 57 |
Yichang | Hubei | 240 | 322 | 140 | 180 |
Tianmen | Hubei | 120 | 142 | 50 | 80 |
Enshi | Hubei | 420 | 519 | 250 | 291 |
Shiyan | Hubei | 300 | 443 | 210 | 248 |
Qianjiang | Hubei | 131 | 155 | 70 | 87 |
Shijiazhuang | Hebei | 660 | 898 | 440 | 503 |
Taiyuan | Shanxi | 720 | 944 | 450 | 529 |
Shenyang | Liaoning | 1320 | 1812 | 890 | 1015 |
Changchun | Jilin | 1560 | 2088 | 1030 | 1169 |
Harbin | Heilongjiang | 1440 | 2354 | 1160 | 1318 |
Nanjing | Jiangsu | 408 | 550 | 260 | 308 |
Hangzhou | Zhejiang | 660 | 827 | 350 | 463 |
Hefei | Anhui | 300 | 388 | 180 | 217 |
Fuzhou | Fujian | 672 | 919 | 450 | 515 |
Nanchang | Jiangxi | 330 | 355 | 170 | 199 |
Ji’nan | Shandong | 600 | 864 | 420 | 484 |
Zhengzhou | Henan | 360 | 514 | 250 | 288 |
Changsha | Hunan | 289 | 345 | 150 | 193 |
Guangzhou | Guangdong | 672 | 955 | 480 | 535 |
Haikou | Hainan | 1152 | 1566 | 750 | 877 |
Chengdu | Sichuan | 840 | 1130 | 550 | 633 |
Guiyang | Guizhou | 720 | 1011 | 510 | 566 |
Kunming | Yunnan | 1170 | 1558 | 760 | 872 |
Xi’an | Shanxi | 510 | 740 | 360 | 414 |
Lanzhou | Gansu | 930 | 1360 | 670 | 762 |
Xi’ning | Qinghai | 1188 | 1594 | 790 | 893 |
Taibei | Taiwan | — | — | — | 5600 |
Beijing | — | 780 | 1174 | 580 | 657 |
Tianjin | — | 720 | 1144 | 560 | 641 |
Shanghai | — | 606 | 825 | 390 | 462 |
Chongqing | — | 690 | 897 | 440 | 502 |
Hohhot | Inner Mongolia | 960 | 1380 | 650 | 773 |
Nanning | Guangxi | 810 | 1209 | 590 | 677 |
Lhasa | Tibet | 3060 | 3482 | 1110 | 1950 |
Yinchuan | Ningxia | 966 | 1448 | 710 | 811 |
Urumqi | Xinjiang | 2160 | 3267 | 1600 | 1830 |
Hong Kong | — | 840 | 1107 | 510 | 620 |
Macao | — | 960 | 1210 | 600 | 678 |
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.
Transportation parameters of high-speed railway, common railway, coach, and flight.
City | Province | High-speed railway | Common railway | Coach | Flight | ||||
---|---|---|---|---|---|---|---|---|---|
Time | Ticket fee | Time | Ticket fee | Time | Ticket fee | Time | Ticket fee | ||
Xiaogan | Hubei | 30 | 58 | 60 | 14.5 | 60 | 32 | — | — |
Huanggang | Hubei | 90 | 22 | 37 | 20 | — | — | — | — |
Jingzhou | Hubei | 90 | 76 | 89 | 32.5 | 300 | 60 | — | — |
Xianning | Hubei | 24 | 40 | 60 | 12.5 | 60 | 28 | — | — |
E’zhou | Hubei | 24 | 20 | 80 | 12.5 | 60 | 28 | — | — |
Xiangyang | Hubei | 90 | 130 | 190 | 50.5 | 240 | 88 | — | — |
Huangshi | Hubei | 37 | 30 | 100 | 16.5 | 120 | 42 | — | — |
Jingmen | Hubei | — | — | 190 | 40.5 | 180 | 101 | — | — |
Suizhou | Hubei | 55 | 70 | 128 | 26.5 | 180 | 66 | — | — |
Xiantao | Hubei | 60 | 50 | — | — | 90 | 35 | — | — |
Yichang | Hubei | 150 | 121 | 310 | 53.5 | 270 | 105 | — | — |
Tianmen | Hubei | 60 | 45 | 90 | 19.5 | 180 | 55 | — | — |
Enshi | Hubei | 270 | 187 | 420 | 78 | 480 | 130 | — | — |
Shiyan | Hubei | 144 | 217 | 330 | 72 | 390 | 135 | — | — |
Qianjiang | Hubei | 90 | 65 | — | — | 150 | 57 | — | — |
Shijiazhuang | Hebei | 240 | 415 | 540 | 124 | — | — | 450 | 1450 |
Taiyuan | Shanxi | 390 | 486 | 1332 | 173.5 | 750 | 320 | 105 | 727 |
Shenyang | Liaoning | 660 | 800 | 1320 | 217 | 1440 | 480 | 180 | 1820 |
Changchun | Jilin | 680 | 904 | 1527 | 243 | — | — | 180 | 1800 |
Harbin | Heilongjiang | 753 | 1012 | 1396 | 268.5 | — | — | 180 | 1800 |
Nanjing | Jiangsu | 180 | 200 | — | — | 480 | 200 | — | — |
Hangzhou | Zhejiang | 300 | 300 | 720 | 120 | 700 | 285 | 80 | 830 |
Hefei | Anhui | 120 | 134 | — | — | 360 | 150 | — | — |
Fuzhou | Fujian | 371 | 267 | 720 | 120 | 700 | 280 | 90 | 870 |
Nanchang | Jiangxi | 150 | 120 | 344 | 53.5 | 390 | 120 | — | — |
Ji’nan | Shandong | 360 | 525 | 720 | 130 | 780 | 280 | 95 | 1000 |
Zhengzhou | Henan | 140 | 244 | 300 | 75 | 480 | 140 | — | — |
Changsha | Hunan | 92 | 165 | 240 | 53.5 | 300 | 120 | — | — |
Guangzhou | Guangdong | 260 | 464 | 750 | 138.5 | 960 | 340 | 110 | 1800 |
Haikou | Hainan | — | — | 1440 | 250 | 1290 | 430 | 150 | 1700 |
Chengdu | Sichuan | 560 | 375 | 990 | 185 | 960 | 330 | 120 | 1350 |
Guiyang | Guizhou | 300 | 481 | 930 | 163.5 | 960 | 320 | 115 | 1000 |
Kunming | Yunnan | 420 | 665 | 1373 | 217 | 1500 | 480 | 135 | 1660 |
Xi’an | Shanxi | 270 | 455 | 900 | 135.5 | 560 | 240 | 85 | 1200 |
Lanzhou | Gansu | 400 | 654 | 1200 | 190 | 1200 | 430 | 135 | 1330 |
Xi’ning | Qinghai | — | — | — | — | — | — | 130 | 1300 |
Taibei | Taiwan | — | — | — | — | — | — | 155 | 1400 |
Beijing | — | 270 | 520 | 720 | 152.5 | 900 | 320 | 120 | 2200 |
Tianjin | — | 300 | 525 | 840 | 156.5 | 960 | 300 | 115 | 1150 |
Shanghai | — | 300 | 336 | 900 | 140 | 720 | 250 | 95 | 1880 |
Chongqing | — | 390 | 279 | 540 | 140 | 780 | 280 | 95 | 1650 |
Hohhot | Inner Mongolia | — | — | 1828 | 229 | — | — | 130 | 1050 |
Nanning | Guangxi | 450 | 478 | 840 | 170 | 1050 | 320 | 120 | 1180 |
Lhasa | Tibet | — | — | — | — | — | — | 230 | 1070 |
Yinchuan | Ningxia | — | — | 1560 | 198 | — | — | 135 | 1300 |
Urumqi | Xinjiang | — | — | 2310 | 345 | — | — | 260 | 2000 |
Hong Kong | — | 280 | 679 | — | — | — | — | 130 | 1574 |
Macao | — | — | — | — | — | — | — | 100 | 1380 |
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 Auto mode: where High-speed railway mode: where Common railway mode: where Coach mode: where Flight mode: where
To cater for the consideration of both mode choice and destination choice, we propose a multimodal network user equilibrium model as follows:
We denote the original data of historical demand distribution ratio, population, and travel distance of a destination region
Parameters for nested logit model-based traffic assignment.
1.2 | 1.1 | 1.25 | 1.3 | 1.0 |
0.6 | 2 | 0.05 | 0.01 | 0.0 |
0.003 | 10 | 5 | 0.3 | 0.5 |
The objective function (
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:
To solve the nested logit model-based problem, one can first compute the generalized cost
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
Wuhan and other 35 regions outside the province of Hubei.
Wuhan and other 16 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
The historical demand distribution data, the population of cities and provinces, and the road distance from Wuhan to other destination regions.
Destination | Historical demand distribution ratio | Population in millions | Road distance |
---|---|---|---|
Xiaogan | 0.118615 | 4.8780 | 76.5 |
Huanggang | 0.110331 | 6.2910 | 75.3 |
Jingzhou | 0.055153 | 5.7442 | 220 |
Xianning | 0.043188 | 2.4626 | 92.6 |
E’zhou | 0.031015 | 1.0487 | 75 |
Xiangyang | 0.036824 | 5.6140 | 313 |
Huangshi | 0.033994 | 2.4293 | 100 |
Jingmen | 0.029304 | 2.8737 | 240 |
Suizhou | 0.028677 | 2.1622 | 170 |
Xiantao | 0.029431 | 1.1660 | 102 |
Yichang | 0.024785 | 4.1150 | 322 |
Tianmen | 0.020131 | 1.4189 | 142 |
Enshi | 0.016231 | 3.2903 | 519 |
Shiyan | 0.017129 | 3.3830 | 443 |
Qianjiang | 0.011413 | 0.9463 | 155 |
Hebei | 0.01537 | 11.0312 | 898 |
Shanxi | 0.00676 | 4.4619 | 944 |
Liaoning | 0.00442 | 8.3160 | 1812 |
Jilin | 0.00177 | 7.6770 | 2088 |
Heilongjiang | 0.00246 | 10.8580 | 2354 |
Jiangsu | 0.01780 | 25.3086 | 550 |
Zhejiang | 0.01188 | 10.3600 | 827 |
Anhui | 0.02734 | 24.5670 | 388 |
Fujian | 0.00958 | 7.8000 | 919 |
Jiangxi | 0.02451 | 5.5455 | 355 |
Shandong | 0.01502 | 7.4604 | 864 |
Henan | 0.07515 | 40.5440 | 514 |
Hunan | 0.04200 | 32.6188 | 345 |
Guangdong | 0.02316 | 45.9177 | 955 |
Hainan | 0.00337 | 2.3023 | 1566 |
Sichuan | 0.02397 | 16.3300 | 1130 |
Guizhou | 0.00746 | 4.8819 | 1011 |
Yunnan | 0.00627 | 6.8500 | 1558 |
Shanxi | 0.01332 | 10.0037 | 740 |
Gansu | 0.00433 | 3.7536 | 1360 |
Qinghai | 0.00097 | 2.3871 | 1594 |
Beijing | 0.01147 | 21.536 | 1174 |
Tianjin | 0.00222 | 15.6183 | 1144 |
Shanghai | 0.00792 | 24.2814 | 825 |
Chongqing | 0.01638 | 31.2432 | 897 |
Inner Mongolia | 0.00195 | 3.1260 | 1380 |
Guangxi | 0.00840 | 7.5687 | 1209 |
Ningxia | 0.00074 | 2.2931 | 3267 |
Xinjiang | 0.00182 | 3.5058 | 1107 |
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.
Based on the model proposed in Section
Demand distribution estimation and the error of estimation for cities inside the province of Hubei.
Destination | Real condition | Est. results | Error | Error ratio (%) |
---|---|---|---|---|
Xiaogan | 690000.00 | 665702.75 | −24297.25 | −3.52 |
Huanggang | 652000.00 | 612343.63 | −39656.37 | −6.08 |
Jingzhou | 327000.00 | 320223.65 | −6776.35 | −2.07 |
Xianning | 250500.00 | 274428.51 | 23928.51 | 9.55 |
E’zhou | 198500.00 | 235207.15 | 36707.15 | 18.49 |
Xiangyang | 196500.00 | 193181.29 | −3318.71 | −1.69 |
Huangshi | 188500.00 | 236658.71 | 48158.71 | 25.55 |
Jingmen | 165000.00 | 156216.92 | −8783.08 | −5.32 |
Suizhou | 160500.00 | 199004.52 | 38504.52 | 23.99 |
Xiantao | 148500.00 | 202067.37 | 53567.37 | 36.07 |
Yichang | 140500.00 | 131028.35 | −9471.65 | −6.74 |
Tianmen | 104000.00 | 183119.51 | 79119.51 | 76.08 |
Shiyan | 93000.00 | 89343.10 | −3656.90 | −3.93 |
Enshi | 90500.00 | 64965.07 | −25534.93 | −28.22 |
Qianjiang | 57000.00 | 140509.64 | 83509.64 | 146.51 |
Demand distribution estimation and the error of estimation for other province-level destination regions outside the province of Hubei.
Destination | Real condition | Est. results | Error | Error ratio (%) |
---|---|---|---|---|
Henan | 284000.00 | 250214.37 | −33785.63 | −11.90 |
Hunan | 174000.00 | 187131.66 | 13131.66 | 7.55 |
Anhui | 113500.00 | 144540.75 | 31040.75 | 27.35 |
Jiangxi | 106000.00 | 128569.12 | 22569.12 | 21.29 |
Guangdong | 97000.00 | 44218.78 | −52781.22 | −54.41 |
Jiangsu | 73000.00 | 88634.88 | 15634.88 | 21.42 |
Chongqing | 63500.00 | 42893.78 | −20606.22 | −32.45 |
Sichuan | 62000.00 | 20641.30 | −41358.70 | −66.71 |
Shandong | 55000.00 | 27805.30 | −27194.70 | −49.44 |
Zhejiang | 53500.00 | 45031.19 | −8468.81 | −15.83 |
Hebei | 46500.00 | 40775.05 | −5724.95 | −12.31 |
Fujian | 45500.00 | 36048.21 | −9451.79 | −20.77 |
Beijing | 43000.00 | 26050.15 | −16949.85 | −39.42 |
Guangxi | 39500.00 | 14846.45 | −24653.55 | −62.41 |
Shanxi | 36000.00 | 37590.59 | 1590.59 | 4.42 |
Shanghai | 33000.00 | 36512.91 | 3512.91 | 10.65 |
Shanxi | 29500.00 | 22472.40 | −7027.60 | −23.82 |
Guizhou | 27500.00 | 21864.32 | −5635.68 | −20.49 |
Yunnan | 26500.00 | 8502.62 | −17997.38 | −67.91 |
Hainan | 19000.00 | 2387.83 | −16612.17 | −87.43 |
Gansu | 17500.00 | 8962.46 | −8537.54 | −48.79 |
Liaoning | 16500.00 | 2702.27 | −13797.73 | −83.62 |
Heilongjiang | 14000.00 | 1237.75 | −12762.25 | −91.16 |
Xinjiang | 10000.00 | 68.07 | −9931.93 | −99.32 |
Inner Mongolia | 9000.00 | 6731.88 | −2268.12 | −25.20 |
Jilin | 8500.00 | 1706.55 | −6793.45 | −79.92 |
Tianjin | 7500.00 | 18275.56 | 10775.56 | 143.67 |
Ningxia | 4000.00 | 3969.71 | −30.29 | −0.76 |
Qinghai | 3000.00 | 3140.77 | 140.77 | 4.69 |
Tibet | 1000.00 | 4156.12 | 3156.12 | 315.61 |
Others | 19500.00 | 18317.04 | −1182.96 | −6.07 |
Table
Results of the aggregated demand distribution ratio and the aggregated error ratio of the estimated demand distribution for all destinations, destinations inside Hubei, and destinations outside Hubei.
Destination range | Aggregated traffic distribution ratio | Aggregated error ratio of estimation (%) | |
---|---|---|---|
Real condition (%) | Estimation (%) | ||
All | — | — | 18.60 |
Des. inside Hubei | 69.24 | 74.08 | 6.99 |
Des. outside Hubei | 30.76 | 25.92 | 15.73 |
Traffic flow distribution results for the real condition.
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 (
Estimation of the number of incidence cases and the error of estimation for cities inside the province of Hubei.
Destination | Real condition | Est. results | Error | Error ratio (%) |
---|---|---|---|---|
Xiaogan | 3518.00 | 4166.77 | 648.77 | 18.44 |
Huanggang | 2907.00 | 3832.78 | 925.78 | 31.85 |
Jingzhou | 1580.00 | 2004.34 | 424.34 | 26.86 |
Xianning | 836.00 | 1717.70 | 881.70 | 105.47 |
E’zhou | 1394.00 | 1472.21 | 78.21 | 5.61 |
Xiangyang | 1175.00 | 1209.16 | 34.16 | 2.91 |
Huangshi | 1015.00 | 1481.29 | 466.29 | 45.94 |
Jingmen | 928.00 | 977.79 | 49.79 | 5.37 |
Suizhou | 1307.00 | 1245.61 | −61.39 | −4.70 |
Xiantao | 575.00 | 1264.78 | 689.78 | 119.96 |
Yichang | 931.00 | 820.13 | −110.87 | −11.91 |
Tianmen | 496.00 | 1146.18 | 650.18 | 131.09 |
Shiyan | 672.00 | 559.22 | −112.78 | −16.78 |
Enshi | 252.00 | 406.63 | 154.63 | 61.36 |
Qianjiang | 198.00 | 879.48 | 681.48 | 344.18 |
Estimation of the number of incidence cases and the error of estimation for other province-level destination regions outside the province of Hubei.
Destination | Real condition | Est. results | Error | Error ratio (%) |
---|---|---|---|---|
Henan | 1274.00 | 1566.14 | 292.14 | 22.93 |
Hunan | 1018.00 | 1171.29 | 153.29 | 15.06 |
Anhui | 990.00 | 904.71 | −85.29 | −8.62 |
Jiangxi | 936.00 | 804.74 | −131.26 | −14.02 |
Guangdong | 1415.00 | 276.77 | −1138.23 | −80.44 |
Jiangsu | 633.00 | 554.78 | −78.22 | −12.36 |
Chongqing | 577.00 | 268.48 | −308.52 | −53.47 |
Sichuan | 543.00 | 129.20 | −413.80 | −76.21 |
Shandong | 768.00 | 174.04 | −593.96 | −77.34 |
Zhejiang | 1238.00 | 281.86 | −956.14 | −77.23 |
Hebei | 319.00 | 255.22 | −63.78 | −19.99 |
Fujian | 313.00 | 225.63 | −87.37 | −27.91 |
Beijing | 537.00 | 163.05 | −373.95 | −69.64 |
Guangxi | 254.00 | 92.93 | −161.07 | −63.41 |
Shanxi | 248.00 | 235.29 | −12.71 | −5.13 |
Shanghai | 404.00 | 228.54 | −175.46 | −43.43 |
Shanxi | 133.00 | 140.66 | 7.66 | 5.76 |
Guizhou | 146.00 | 136.85 | −9.15 | −6.26 |
Yunnan | 176.00 | 53.22 | −122.78 | −69.76 |
Hainan | 168.00 | 14.95 | −153.05 | −91.10 |
Gansu | 136.00 | 56.10 | −79.90 | −58.75 |
Liaoning | 127.00 | 16.91 | −110.09 | −86.68 |
Heilongjiang | 484.00 | 7.75 | −476.25 | −98.40 |
Xinjiang | 76.00 | 0.43 | −75.57 | −99.44 |
Inner Mongolia | 75.00 | 42.14 | −32.86 | −43.82 |
Jilin | 93.00 | 10.68 | −82.32 | −88.51 |
Tianjin | 141.00 | 114.39 | −26.61 | −18.87 |
Ningxia | 75.00 | 24.85 | −50.15 | −66.87 |
Qinghai | 18.00 | 19.66 | 1.66 | 9.21 |
Tibet | 1.00 | 26.01 | 25.01 | 2501.40 |
Hong Kong | 233.00 | 114.65 | −118.35 | −50.79 |
Macao | 155.00 | 70.40 | −84.60 | −54.58 |
Taiwan | 11.00 | 31.83 | 20.83 | 189.38 |
Figure
Comparison of the estimated number and the real number of incidence cases.
Mode flow distribution results of cities inside the province of Hubei.
City | Road | High-speed rail | Rail | Coach | Flight |
---|---|---|---|---|---|
Xiaogan | 170285.59 | 178569.34 | 223626.44 | 93221.37 | — |
Huanggang | 161473.63 | 25124.46 | 425745.54 | — | — |
Jingzhou | 5018.37 | 45290.82 | 269914.25 | 0.22 | — |
Xianning | 33196.64 | 151630.82 | 61341.02 | 28260.04 | — |
E’zhou | 25378.08 | 188460.03 | 8447.63 | 12921.41 | — |
Xiangyang | 3884.20 | 174498.63 | 14686.33 | 112.13 | — |
Huangshi | 28096.44 | 200465.62 | 7468.12 | 628.54 | — |
Jingmen | 129761.11 | — | 24305.32 | 2150.49 | — |
Suizhou | 30586.40 | 155330.54 | 13007.88 | 79.70 | — |
Xiantao | 87947.54 | 90625.94 | — | 23493.89 | — |
Yichang | 29596.79 | 101257.41 | 94.67 | 79.47 | — |
Tianmen | 26117.57 | 109137.47 | 47827.86 | 36.61 | — |
Shiyan | 9449.63 | 79120.77 | 771.80 | 0.90 | — |
Enshi | 5196.79 | 59325.61 | 441.77 | 0.90 | — |
Qianjiang | 76189.15 | 62691.08 | — | 1629.41 | — |
In Table
Results of the aggregated number of incidence case distribution ratio and the aggregated error ratio of the number of estimated incidence case distribution for all destinations, destinations inside Hubei, and destinations outside Hubei.
Destination range | Aggregated number of incidence case distribution ratio | Aggregated error ratio of the estimation (%) | |
---|---|---|---|
Real condition (%) | Estimation (%) | ||
All | — | — | 39.52 |
Within Hubei | 56.40 | 74.08 | 30.36 |
Outside Hubei | 43.60 | 25.92 | 39.94 |
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
Mode flow distribution results of other province-level destination regions outside the province of Hubei.
Destination | Road | High-speed rail | Rail | Coach | Flight |
---|---|---|---|---|---|
Henan | 44850.03 | 118548.74 | 81885.68 | 4929.93 | — |
Hunan | 39450.61 | 92948.86 | 43774.41 | 10957.78 | — |
Anhui | 32300.71 | 107671.97 | — | 4568.06 | — |
Jiangxi | 28926.26 | 82248.90 | 13420.05 | 3973.91 | — |
Guangdong | 5998.14 | 36177.92 | 2020.71 | 21.68 | 0.33 |
Jiangsu | 16969.83 | 70065.96 | — | 1599.09 | — |
Chongqing | 3815.81 | 26847.00 | 12063.14 | 166.99 | 0.84 |
Sichuan | 4396.64 | 15655.80 | 343.30 | 115.42 | 130.14 |
Shandong | 16523.83 | 6598.24 | 3179.75 | 345.35 | 1158.13 |
Zhejiang | 5446.38 | 36087.67 | 1047.02 | 255.62 | 2194.49 |
Hebei | 3402.03 | 26799.17 | 10573.71 | — | 0.13 |
Fujian | 3006.54 | 29630.07 | 1349.57 | 346.38 | 1715.65 |
Beijing | 1164.38 | 21849.67 | 2971.85 | 64.20 | 0.05 |
Guangxi | 1994.34 | 10993.61 | 1415.26 | 25.41 | 417.83 |
Shanxi | 21077.62 | 13574.76 | 131.76 | 2740.70 | 65.76 |
Shanghai | 7129.73 | 28941.40 | 113.64 | 328.00 | 0.13 |
Shanxi | 4241.40 | 5456.97 | 1.04 | 258.44 | 12514.56 |
Guizhou | 2971.35 | 17850.25 | 164.81 | 24.04 | 853.86 |
Yunnan | 122.45 | 8364.34 | 5.24 | 0.08 | 10.50 |
Hainan | 2237.15 | — | 26.92 | 26.92 | 96.83 |
Gansu | 978.49 | 7754.39 | 36.45 | 3.31 | 189.81 |
Liaoning | 183.87 | 2345.41 | 149.94 | 2.56 | 20.50 |
Heilongjiang | 33.87 | 751.06 | 267.06 | — | 185.77 |
Xinjiang | 2.69 | — | 1.79 | — | 63.58 |
Inner Mongolia | 1385.89 | — | 0.02 | — | 5345.97 |
Jilin | 22.74 | 1600.39 | 23.20 | — | 60.22 |
Tianjin | 2057.90 | 15236.40 | 689.83 | 38.92 | 252.51 |
Ningxia | 2615.52 | — | 2.36 | — | 1351.83 |
Qinghai | 545.38 | — | — | — | 2595.38 |
Tibet | — | — | — | — | 4156.12 |
Hong Kong | 2933.43 | 8307.63 | — | — | 6.39 |
Macao | 4666.53 | — | — | — | 419.13 |
Taiwan | — | — | — | — | 1983.93 |
Aggregated mode flow distribution ratio of destination regions: (a) inside Hubei and (b) outside Hubei.
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.
Table
Results of the aggregated traffic flow ratio of cities inside Hubei under cases applying lockdown on different travel modes.
Original case (%) | Locked down travel mode | ||||
---|---|---|---|---|---|
Automobile (%) | High-speed railway (%) | Common railway (%) | Coach (%) | Flight (%) | |
74.08 | 75.06 | 78.63 | 74.41 | 74.16 | 74.34 |
Results of the aggregated mode flow increment in percentage under cases applying lockdown on different travel modes for destinations inside Hubei.
Aggregated mode flow increment (%) | Locked down travel mode | |||
---|---|---|---|---|
Automobile (%) | High-speed railway (%) | Common railway (%) | Coach (%) | |
Automobile | — | 121.59 | 63.65 | 5.49 |
High-speed railway | 25.54 | — | 32.78 | 4.31 |
Common railway | 35.29 | 62.22 | — | 4.02 |
Coach | 42.98 | 102.42 | 36.47 | — |
Results of the aggregated mode flow increment in percentage under cases applying lockdown on different travel modes for destinations outside Hubei.
Aggregated mode flow increment (%) | Locked down travel mode | ||||
---|---|---|---|---|---|
Automobile (%) | High-speed railway (%) | Common railway (%) | Coach (%) | Flight (%) | |
Automobile | — | 127.41 | 14.82 | 2.42 | 4.13 |
High-speed railway | 19.98 | — | 14.34 | 1.94 | 1.37 |
Common railway | 19.77 | 98.77 | — | 2.32 | 0.50 |
Coach | 28.28 | 114.78 | 17.71 | — | 1.13 |
Flight | 29.38 | 63.29 | 4.03 | 0.77 | — |
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 [
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
Derivatives of the aggregated mode flows of auto as well as the aggregated demands of destination regions with high incidence rates with respect to different input parameters.
Aggr. mode flow | 0.1 ∗ | |||||
---|---|---|---|---|---|---|
−14934.37 | 16010.48 | 11616.38 | 2172.01 | 69.88 | 26895.38 | |
21.96 | −232.22 | 168.19 | 21.8 | 3.76 | 1954.73 |
As we discussed in Section
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.
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 (
The authors declare that they have no conflicts of interest.
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.