Outbreaks of several serious diseases, such as SARS, influenza, measles, and tuberculosis, have occurred during air travel [1] [2] [3] [4] [5] . Concern over such spread of infection led to a sharp decline in air travel during the current COVID-19 outbreak, with passenger traffic estimated to be 44% -59% below normal in 2020 [6] . Airlines have responded to this with several procedural changes. Two key changes include lowering occupancy by keeping the middle seat empty and changes to boarding procedures [7] [8] .
For example, Delta Airlines has introduced a boarding procedure where it boards the plane starting from the last row, although business class still boards first [7] . United Airlines uses a backto-front by rows boarding process with business boarding last typically [8] . The trend toward backto-front boarding by rows is motivated by the desire to reduce the likelihood of passengers passing others when they take their seats [7] . Although such a boarding process is known to be slower than alternative processes [9] , a reduction in social proximity would lead to a reduced risk of exposure to viruses, and thus reduced risk of an infection outbreak inflight.
However, our past results suggest that such a process may not reduce social proximity [10] . In prior work on the risk of Ebola and SARS spread during boarding airplanes, we observed that passengers' clustering while waiting for others to stow their luggage had a significant impact on increasing infection risk. Such clustering tended to increase with more zones that were in contiguous sections of the airplane. Back-to-front boarding by rows is equivalent to one zone per row and can thus be expected to yield increased clustering and exposure to the virus. There is a need to understand better the impact of the new boarding processes on exposure to the virus.
In this work, we simulate several boarding processes and evaluate their impact on social proximity during boarding. We also study the dominant mechanisms through which they generate social proximity. Our results show that while a back-to-front boarding does indeed reduce exposure of seated passengers to those who are walking past them toward their seats, it increases proximity between pairs of seated passengers and pairs of passengers in the aisle. The net impact is to increase exposure by around 100% compared with random boarding. It also increases exposure by approximately 50% compared with a typical boarding process prior to the COVID-19 outbreak. While keeping middle seats empty yields a substantial reduction in exposure, the boarding processes show the same relative trends.
Our results suggest that airlines would have a lower infection risk from their pre-COVID-19 boarding process. They can reduce the risk even further by using a random boarding process.
Pedestrian dynamics models simulate the walking movement of humans, which can be used to determine contact patterns. A variety of models have been developed [11] [12] . Social dynamics models [13] [14] are the most popular and are motivated by the idea of molecular dynamics. They treat each pedestrian as analogous to an atom. The force fi acting on the i th pedestrian is defined as follows. 
where vi(t) is the actual velocity at time t of a pedestrian with mass mi whose desired velocity is v0i(t). The momentum generated by a pedestrian's desire to move toward a goal (governed by the difference between desired and actual velocity and a time constant t) results in self-propulsion toward that goal. This is countered by a pairwise repulsion force fij(t) due to interactions with other pedestrians and fixed surfaces, such as walls. The numerical solution of the above ODE for each pedestrian generates a trajectory for each pedestrian.
The repulsion term fij is modeled through empirically determined force fields, often inspired by molecular dynamics [10] . Social dynamics models differ in the manner in which fij is defined. Additional behavioral features need to be explicitly coded. For example, passengers spend a certain amount of time stowing their luggage, and a passenger going to the window seat experiences a delay if the aisle seat is occupied.
We had earlier developed the Self-Propelled Entity Dynamics (SPED) model to simulate boarding and deplaning with a specific form of fij based on the Lennard-Jones potential from molecular dynamics [10] . We recently proposed the Constrained Linear Movement (CALM) model designed for narrow passages and evaluated its correctness in boarding and deplaning [15] . The CALM model defines fij by setting fik to 0 for all k except the nearest neighbor in the direction of motion. If di is the distance to the nearest neighbor l in the direction of motion, then fil is given by Equation (2). The solution of Equations (1) and (2) yields each passenger's position as a function of time. Certain behavioral features of passenger movement, such as the time to stow luggage, are accounted for directly in the simulation code, with their parameters presented in Section 4.1. The passenger trajectories are then used to determine the extent of social proximity between passengers.
where a, b, and c are constants determined as a = 2.11, b = 0.366, and c = 0.966 in [15] . The mass term cancels out from both sides of Equation (1) when we substitute Equation (2) into Equation (1) , and is assigned the value 0.4 s.
Snapshot from a simulation that boards the business class first, followed by the economy class in a back-to-front manner. Green dots represent passengers, and blue dots fixed objects, such as seats and walls.
The CALM model is around 60 times faster than the SPED model. This computational efficiency leads to its effectiveness in the analysis of different boarding procedures for the following reason. Precise prediction is difficult because human movement behavior has inherent uncertainties and variations [16] [17] , compounded by insufficient data during a new epidemic [18] . We address this challenge by parameterizing the sources of uncertainty, such as passenger walking speed, and then evaluate social proximity under various possible scenarios. This parameter space is large, and it is computationally demanding even on a parallel computer. The computationally efficient CALM model enables us to analyze this large parameter space relatively fast. We use a low discrepancy parameter sweep to reduce further the number of simulations needed [19] * .
We perform a parameter sweep, varying the following parameters, to generate 16,000 scenarios of possible passenger movement patterns during each boarding process. The desired speed of a passenger in the absence of anyone nearby, v0i, has a range between 1.1 m/s and 1.3 m/s. Parameters for behavioral features are selected as follows. Passengers are tracked from when they * A low discrepancy sequence fills a space uniformly. A low discrepancy parameter sweep uses such a sequence to choose values of parameters that are uniformly spread through the space. A pseudorandom choice of points, on the other hand, can yield some clustering in the set of parameters chosen. A conventional lattice-based parameter sweep, where values in each dimension are equally distributed, does not cover a space as uniformly either. For example, a lattice-based parameter sweep with two variables might choose 100 points along each direction, leading to 10,000 parameter combinations being simulated. A two-dimensional low discrepancy sequence with 10,000 points would cover such a space much more uniformly. This can result in one to three orders of magnitude fewer points being needed in a low discrepancy parameter sweep [19] . are just outside the plane door. They start entering the plane once the passenger in front of them is a certain distance away, which is specified by a line-distance threshold that ranges from 0.5 m to 1.6 m. Passengers decrease their speed after entering the plane when they turn toward the aisle. This is simulated by multiplying their speed by an intersection-speed coefficient, ranging between 0.2 and 0.8, when they are within a distance intersection-distance threshold of the aisle. The latter parameter varies between 0.4 m and 0.6 m. Luggage stowing time ranges from 8 s to 14 s * . After stowing the luggage, passengers move into their seats at a fraction of the normal speed. This is simulated by multiplying their speed by the towards-seat-speed coefficient, which has a range from 0.2 to 0.6.
Equation (1) is solved numerically using an Euler solver with a time-step size of 0.005 s. We output results every 250 time-steps so that each output represents 1.25 s of real time. For each simulation, we perform the following analysis. At each time step, we measure social proximity through contacts between pairs of passengers, where passengers whose distance to each other is less than a specified threshold are considered to be in contact. Each such pair yields 2.5 personseconds of contact, indicating that two persons have been in social proximity for 1.25 s each.
We simulated several boarding procedures. We present the results of the following four.
1. 1-Zone: Passengers board in random order with no preference given to business class passengers.
Business class passengers board first, and the rest of the plane is divided into six contiguous zones of a few rows each. These zones are boarded starting from the backmost zone and moving forward. Within each zone, boarding is random.
3. Back-to-front: This simulates row-wise boarding, starting with the last row. No preference is provided to business class passengers. This is similar to a recent United Airlines' boarding process.
Business class boards first, and then the economy class boards, in a back to front manner. This is similar to Delta Airlines' new boarding process.
We show below that the random 1-Zone boarding process is the most effective among those that we studied in reducing social proximity. It differs from the random process employed in airlines such as Southwest. In the latter, passengers may sit anywhere they wish to, and their seating preference may not yield a random order. In our simulations, passengers in this process are assigned specific seats, but the boarding order is random. The 6-Zone business-first procedure is meant to approximate a common procedure before COVID-19, where the business class boards first, and then people board in a large number of zones, with random order within a zone. Both the back-to-front procedures emulate new boarding procedures that have been adopted by different airlines.
Some airlines are keeping the middle seat empty while others are not [20] . We simulate both situations. We use a 1.82 m threshold for contacts, corresponding to the 6-feet threshold advocated by public health agencies. Fig. 2 and Fig. 3 show that 1-Zone performs best, 6-Zones next best, and both the new procedures fare worst, yielding roughly double the contacts of the best and 50% * This value is generated for each passenger using a random number generator, rather than being a simulation parameter generated using a low discrepancy sequence. more than the procedure that they replaced. On the other hand, the policy of keeping middle seats empty, which has been introduced by a few airlines, delivers a substantial reduction in social proximity. 
We now examine how contacts are generated. In earlier work, we had noticed an increase in social proximity for a large number of zones by analyzing the trajectories of specific simulation results [10] . Here, we systematically measure proximity by determining the number of contacts arising from each of the following situations.
1. Contacts between a pair of passengers, both of whom are seated. 2. Contacts between a pair of passengers, both of whom are in the aisle. 3. Contacts between passengers, one of whom is seated and the other in the aisle.
The new boarding procedures adopted by airlines try to lower the value of the last category. Our previous work hypothesizes that it would not be beneficial due to an increase in the contacts from the second category. We look at one specific typical simulation from each boarding process and analyze its contact patterns to generate insight into the mechanisms yielding social proximity. Fig. 4 shows that the back-to-front procedures do, indeed, reduce contacts between seated passengers and those moving in the aisle. However, this is not the primary generator of social proximity. An increase in contacts between passengers who are both in the aisle or both seated makes the new procedures substantially worse than the previous one and even more when compared with using a single zone. These results also show that while the hypothesized second factor is sufficient to deliver better social distancing with fewer zones, it is not the primary driver. The most important factor that yields better social distancing with fewer zones arises from the contacts generated between seated passengers. With more zones, passengers remain seated close to each other during the boarding process, while with one zone, they are distributed around the plane. Some airlines have introduced a new policy of reducing occupancy by keeping middle seats empty. While the motivation for this is to yield social distancing during the flight, we wish to see its impact on the boarding process. In particular, are the relative differences between boarding processes preserved with this seating policy? Fig. 4 shows that the relative differences are maintained and that 1-Zone remains around twice as effective as the new boarding procedures. We also notice that the new seating policy produces a substantial reduction in social proximity during the boarding process.
We next examine whether changing the contact threshold changes the relative trends. If one procedure were to yield poor results mostly due to contacts close to 1.818 m away, but it yielded low counts for close-range contacts, then it might be effective in reducing infection risk, because close contacts are at higher risk of infection spread. Furthermore, if passengers wear masks, then social proximity would arise at a smaller threshold. We consider a contact threshold of 0.606 m, corresponding to a 2-feet distance. We see from Fig. 5 that the same relative trends hold, with 1-Zone having even a slightly greater advantage. Much of this decrease in social proximity with a 0.606 m threshold arises from a reduction in contacts between two seated passengers. Note that with middle seats unoccupied, contacts between seated passengers are generated from the business class alone, because the separation between the window and aisle seats in economy class is higher than 0.606m. The relative trends of the different procedures still hold because fewer zones produce substantially lower contacts between two passengers in the aisle. The net impact is that both with and without middle seats occupied, the relative differences between procedures remain roughly the same.
We next discuss some limitations of this work. First, we consider social proximity only during boarding. The infection could spread during the flight too, which we don't study. There are mixed results in the literature regarding the vulnerabilities during flights. Computational Fluid Dynamics simulations suggest that there could be a considerable effect of the cabin airflow [21] . On the other hand, a model based on empirical measurement of virus particles and passenger behavior suggested that the risk is minimal and that other aspects of air travel, including boarding, could be responsible for outbreaks during air travel [22] .
Evidence from SARS outbreaks in airplanes suggests that both views have some merit. Out of 40 flights with SARS infected persons on board, only five resulted in infections to passengers, with a total of 37 infected persons across these flights [4] . A single flight accounted for the majority of the infections. Passengers in rows close to the index passenger were disproportionately affected. However, half the infections occurred amongst passengers farther than two rows away. Given that inflight transmission risk is primarily to persons within two rows [4] , the above observation suggests that boarding could play a significant role. In any case, given that the new boarding procedures are meant to reduce exposure to the virus during boarding, we focus on the merits of these changes alone in this paper.
Another limitation is that we estimate social proximity, rather than predict infection likelihood. The virus dose required for COVID-19 infection is not known, and we believe that quantitatively accurate predictions are not feasible at this point. Consequently, we focus on the relative benefits of different boarding procedures through the estimation of social proximity generated by them.
Our results show that while the original boarding procedures were substantially worse than they could have been, the new boarding procedures further worsen infection risk by increasing social proximity. Furthermore, we identify the mechanisms leading to social proximity. Sciencebased changes to boarding procedures can decrease COVID-19 infection risk substantially by increasing social distance. In future work, we are working on a close-range infection transmission model, which will enable the estimation of actual infection risk.