Virus spreading in public transport networks: the alarming consequences of the business as usual scenario

Panchamy Krishnakumari and Oded Cats, Dittlab | SmartPTLab, TU Delft

Highlights

- An 80% capacity reduction is required to adhere to 1.5m (~5ft) social-distancing 

- Smart card data can be leveraged for contact tracing 

- Public transport contact networks are highly connected with an average of 1200 interactions per passenger due to network topology and demand patterns

- With the same demand as pre-corona, merely 3 initially infected passengers will lead to 55% of the passengers population getting infected within 20 days 

Public transport worldwide is heavily affected by the coronavirus pandemic. In a period where sharing and crowding are becoming a public health concern, the connectivity and efficiency offered by mass transport become a potential peril. How quickly and adversely does a virus spread in a public transport system? How many will potentially get infected if pre-pandemic ridership levels are expected while aiming to maintain social distancing? We explore this by studying the evolution of contract networks and an epidemiological model using a sample of smart card data from the Washington DC metro network. 

On an average day, 600 000 passengers travel in the DC metro network. Each metro train consists of 6 to 8 railcars, with an average maximum capacity of 1700 people. During the morning period (5AM-12PM) a metro train carries an average of 133 travellers at any given time. What are the implications of transitioning into a 1.5 meter (~5 feet) society? Specifically, what is the capacity drop that adhering to social-distancing implies?

Assuming that passengers are spaced across platforms and metro trains seeking to ensure a minimum of 1.5 meter distance from any fellow traveller, the metro can carry only 312 people at a given time based on the number of rail cars, rail car width, and length. This implies a capacity reduction of more than 80%. The capacity increases to 703 for 1m (~3.3 ft) distance and drops to 176 for 2m (~6.5 ft) distance. For a typical weekday morning, the ridership that violates this minimum distance of 1m (>703 people) is visualized in the video below. 

Click the image to watch the video showing the ridership of each track segment at a given time during the morning period of 5 March 2018.

It is important to realize that crowding in public transport has two negative effects for virus spreading that exacerbate each other: increasing likelihood to be exposed to infected co-passengers as well as increasing proximity to fellow travellers and the resulting transmission probability. Smart card data allows us to (i) reconstruct the contact network reflecting the passengers one potentially gets in contact with during the course of his/her public transport journey, and; (ii) assess and assign probabilities to get infected from connections in the contact graph based on crowding estimates. 

Contact networks

Contact networks can be used to represent the extent of potential person-to-person interaction. In this study, we use contact graphs to examine virus spreading in public transport networks. The contact network depends on the travel demand patterns, the transit network design (lines, interchange stations), resource allocation (frequencies, rolling stock) and its resulting utilization (passenger flows). 

The contact network for the Washington DC metro network reflects for each passenger how many other passengers’ does he/she potentially experience in proximity on a given working day. We define a person as being connected to another person if he/she has been at the same station/train at the same time. Based on this definition, there are more than 337 million person-to-person interactions in just the morning period in the DC metro network when the total ridership of the network is ~ 270 000 passengers. There is thus an average of ~1200 interactions per passenger. The histogram of the number of interactions per passenger has a long tail, implying that some of the passengers are super-connectors that have a particularly high potential to spread the virus as they come in contact with many co-riders.

For illustration, we show below the contact network of a randomly selected passenger who could have come in contact with approximately 1000 other passengers, i.e. s/he was on-board the same train or at waiting at the same station at the same time as 1000 other people during his/her 15 minutes travel in the metro between 8:07 am and 8:21 am. We can see that this passenger could have been in contact with people who were traveling from all these other locations even though his/her trip was limited to a few stations (highlighted in red). This demonstrates the potential of using smart card data for contact tracing in public transport systems.

Assumptions

We construct the contact networks for the entire passenger population from the smart card data. We then adopt some known parameters to describe how the virus spread to analyse how the epidemic may spread in public transport systems. Note that there is great uncertainty about some of these aspects (e.g. whether survivors develop antibodies to resist renewed infection, the impact of proximity duration on the probability to get infected) and we can revisit these assumptions based on findings from medical research.

The key parameters used are an incubation period of 5 days and a quarantine period of 21 days. After the quarantined days, passengers are assumed to become immune and they re-enter the metro system and thus contribute again to crowding. We assume that social-distancing is not enforced yet passengers comply with it to the extent possible by spacing themselves evenly across platforms and trains without making changes to their travel plans.

In addition, we need to make a couple of assumptions related to (i) what is the probability that one is in spending (waiting or travelling) time in proximity to an infected passenger given the number of potential contacts implied by the contact network, and; (ii) what is the probability that one gets infected if one is in proximity to an infected passenger. We hereby assume that: 

• the probability of a person getting infected given that one stands in proximity to someone who is infected (P(A/B)) - which depends on the (platform and on-board) crowding conditions and adherence to social distancing - is assumed to be 0 if the distance is greater than 1.5m (~5 ft), be 1 if less than 1m, and follow an exponential decay between 1m and 1.5m

Each distance relates to certain crowding conditions based on the train and platform area. In order to adhere to 1m social distancing, only 703 passengers should be allowed at a given time. Similarly, only 312 and 176 passengers should be allowed for 1.5m and 2m respectively.

• the probability that a person is standing next to someone who is infected (P(B)), which is related to the number of people infected is defined by the ratio between the number of infected passengers (waiting on the platform or travelling on-board) and the total number of passengers in a given track segment or station. An example of this probability for different passenger load levels is shown below for the case that 3 of the passengers waiting on the waiting on the platform or travelling on-board at the same time are infected.

Epidemic model

Employing these assumptions, the probability that each person within the infected person's contact network is infected along his/her journey on any given day, can be calculated as:

The combined probability 1 - P(A/B)P(B) is the probability that a passenger is infected when traversing a track segment or station m located along that passenger's trajectory. The video below shows how the infection spreads throughout the network when merely 3 passengers carry the virus to start with. 

Click the image to watch the video simulating how the virus spreading evolves geographically throughout the metro system over 50 days.

We simulate the epidemiological consequences of the cases where 3 and 3000 passengers are initially infected. 55% of the population is infected within 20 days when 3 people are infected whereas 60% of the population is infected within already 10 days when 3000 people are infected initially in the network. As could be expected, the more passengers are initially infected, the more rapidly the virus spreads and the larger the share of the passengers population that the virus reaches. However, what is truly remarkable is how this does not scale. The impact of an initial rate of 3000 passengers (about 1 in 100 passengers) as opposed to an initial rate which is 3 orders of magnitude smaller, i.e. merely 3 passengers (about 1 in 10,000 passengers), is only a 5% difference in infection rate. This stems from the long incubation period and the connectivity of the contact graph resulting from network topology and demand patterns.

The state matrix shows the different states of the 270 000 passengers. Each passenger is on any given day in one of the following states: susceptible (not infected), infected (and travelling), quarantined (infected and not travelling), and immune (and travelling again). It can be seen how the virus gradually spreads, in the case where 3000 people are initially infected (right), after 5 days already most people are infected. This chilling finding indicates how terribly things could have gone if severe stay-at-home measures such as those imposed in most countries were not instantly implemented or how they may still go wrong in the event of an irresponsible exit strategy.

All the results reported in this study are averaged over a large number of simulation instances due to the randomness of designating initially infected passengers. The convergence criterion for the end of the simulation is the absence of new infections. We simulated the epidemic model for scenarios with a different number of people are initially infected (3, 30, 300, 3000, 30000), and for each scenario, we ran 100 instances. The convergence day until there are no new infections from these simulations is presented below. It shows that for cases of the fewer initially infected case it takes more time until no new infections occur. However, the number of people getting infected in the process increases as the share of initially infected increases. Thus, more people are infected in a shorter period of time leading to stress in finite medical resources. This relates directly to efforts to flatten the curve. 

Data limitations

The smart card dataset does not contain card identifiers. We, therefore, choose to focus on the morning period only and consider 270 000 trajectories to correspond to 270 000 individual passengers. This is a conservative assumption since multiple trips by the same individual increase the respective exposure. 

Further research

Understanding where the new capacity limits are binding and where there is still residual capacity can support public transport service providers in determining how to allocate their resources: which lines to run and their service frequencies. Certain demand connections may prevent downstream passengers from travelling while an alternative solution such as a new designated (bus) line or an on-demand service might be offered. Hence, a blanket capacity reduction without considering the demand distribution across the network is not recommended. There is, therefore, a need to understand the interaction between people in these networks and use that information to proactively limit ridership at different locations.

The insights gained in this study can aid in preparing exit strategies and transitioning to a 1.5 meter (5ft) society. The model can be applied to assess intervention scenarios such as limiting the ridership for certain segments or the impact of closing certain stations. Furthermore, we can investigate how self-quarantine based on contact tracing information (as opposed to the incubation period) can impact the spread of the virus.

Acknowledgments

We thank Washington Metropolitan Area Transit Authority and in particular Jordan Holt for their valuable cooperation and for providing the data that made this study possible. We also thank the Transport Institute of TU Delft for supporting this research.