Model-based assessment of the risks of viral transmission in non-confined crowds

In principle, the instantaneous transmission rate due to droplets emitted at $t_{e}$ by a contagious individual $E$ and inhaled at $t_{r}>t_{e}$ by a ‘receiver’ $R$ reads

where the characteristic time for infection $T_{0}\propto n_{\mathrm{inf}}/c_{v}$ at a given distance in front of the index patient is related to the specifics of the disease (namely, the viral titer $c_{v}$ in the respiratory fluid and the minimal infectious dose $n_{\mathrm{inf}}$), whereas the function $\tilde{\nu}$ accounts for the fluid dynamics of droplet emission and transport. Its parameters $r$, $\theta^{E}$, and $\theta^{R}$ will be clarified in the following. Unfortunately, an accurate derivation of $\tilde{\nu}$ from the fluid dynamics of droplet and aerosol propagation in these scenarios would not only be extremely demanding computationally, but also hinge on very specific information that is neither available [39] nor transferable between situations, e.g., ambient air flows [29, 40], wind speed [23], humidity [27], and speech details [18].

Therefore, we opted for the development of coarser-grained, ad hoc models that notably overlook propagation delays and ambient air flows. (Relaxing the former hypothesis does not alter our main findings, as we show in Appendix D by bringing back into play more realistic transmission dynamics (see Fig. S5); on the other hand, ambient air flows may only be neglected if there is very little or no wind, in which case the exhaled puff is expected to be fairly similar indoors and outdoors in the short range.) With the insight into droplet propagation gained from computational studies as well as experiments on expiration and inhalation [20, 41, 23, 2, 16, 30], the disease transmission rate thus boiled down to a function $\nu_{ER}(t)=\nu\Big{[}r(t),\theta^{E}(t),\theta^{R}(t)\Big{]}$ that decreases with the horizontal distance $r$ between the individuals’ heads and the orientations $\theta^{E}$ and $\theta^{R}$ of the emitter’s and receiver’s heads relative to the direction of the vector that connects them (note that droplet emission and inhalation are not symmetric [20]).

More precisely, assuming that these variables can be decoupled, we write the instantaneous transmission rate as

where $\bar{f}(x)$ is a function such that $\bar{f}(x\approx 0)=1$ and $\bar{f}(x)$ decays rapidly for $x\gg 1$. To be concrete, we tested the following three functions,

Because of the limited accuracy of our positional measurements and the uncertainties about very near field transmission, the peaks of these functions are leveled at $x\to 0$, viz., $\bar{f}_{k}=\min(1,f_{k})$ for $k=1,\,2,\,3$. We then defined a family of plausible parameter sets for the $\nu$ functions, namely, $r_{0}$, $\tilde{T}_{0}$, $\theta_{0}^{E}$, and $\theta_{0}^{R}$ in Eq. 2) as well as $m$ in $f_{2}$. The parameters were bounded using a broad scope of available empirical data, which suggest a characteristic distance $r_{0}\leqslant 1\,\mathrm{m}$, an infection time (at $r_{0}=0.5\,\mathrm{m}$) $T_{0}=\tilde{T}_{0}/\bar{f}(\frac{0.5}{r_{0}})$ between a dozen minutes and an hour, and an exponent $m\approx 2-2.5$ (more details are given in Appendix B), and within these plausible bounds we explored different sets of values, listed in Table 1. The parameter sets thus span the entire spectrum from highly contagious (‘pessimistic’) to only mildly contagious (‘optimistic’).

An example of such a function is depicted in Fig. 1. The spatial decay of the transmission function is such that $\nu(r,\theta^{E},\theta^{R})$ becomes negligible past a few meters, except for the worst-case model describing uncovered sneezes [17]. The typical time for infection in the event of a close ($r=50\,\mathrm{cm})$ face-to-face contact lies between 5 minutes (an extremely pessimistic estimate) and 30 minutes (except for sneezes), consistently with the epidemiological literature on SARS-CoV-2, outbreak reports and exposure case studies [15, 12, 42, 43, 44, 45, 30].

The resulting models are representative of a disease that may be transmitted to multiple individuals within an hour in unfavorable cases [8, 46], but generally requires close and prolonged contacts for transmission: Even in households, reported attack rates lie in the range 5%-30%[47], often around 15% [48, 49, 50]; moreover, casual episodic contacts at work or in the community, even face to face, do not necessarily trigger an outbreak [48, 43]. More quantitatively, the spatial decay described by the fairly optimistic models agrees well with the decay of the concentration of droplets emitted by a coughing subject [41] (see Fig. S3). The reliability of the transmission models is further tested by mimicking a journey aboard a Chinese train (overlooking its confined nature), where transmission risks for passengers who sat close to an infected traveler were recently assessed using trip records [4]. Overall, the simulated results, detailed in Appendix B, are compatible with the empirical data; the models featuring the most optimistic parameters display the best concordance. Finally, let us mention that the order of magnitude of the parameters (especially the most optimistic ones) is consistent with the putative minimal infectious dose of SARS-CoV-2, estimated to be order 100 particles [51], granted that most exhaled droplets contain 0 or 1 viral copy [52] (see Appendix B).

Although these pieces of evidence favor the optimistic end of the parameter spectrum, our study is conducted with the whole gamut of plausible model parameters. This variety better reflects our current uncertainty with regard to the transmission of SARS-CoV-2, but also the established inter-individual and inter-case variability [16, 18], depending on physiology, talking characteristics [53], viral mutations, etc.

We used a privacy-respective setup to film pedestrian flows and crowds in diverse scenarios in a discrete and passive way with a TomTom Bandit camera covered with a thin plastic layer (to deteriorate the quality of the image) and installed in a zenithal position. This privacy-respective setup was approved by the Data Protection Officer of the French National Centre for Scientific Research (CNRS).

After some pre-processing with the FFMpeg software to correct the ’fish-eye’ effect, downsample the video, and select the area of interest (from $8\,\mathrm{m^{2}}$ to $100\,\mathrm{m^{2}}$), the positions and head orientations of all pedestrians were manually tracked at a rate of 2 frames per second (fps) with the help of a dedicated Python software; linear interpolation then increased the rate to 10 fps. The estimated experimental error on the positions is typically below or around 20 cm, while that on the head orientations is lower than $20^{\circ}$ in most videos.

Because of the limited field of view, some interactions with off-camera people were missed, especially at large separations. We compensated for this by appraising the fraction of interactions thus lost as a function of their range, under the assumption of homogeneous density, and reweighting the detected contacts accordingly. We checked that this rescaling method successfully restores the genuine contact distribution, up to contact distances close to the size of the field of view (Fig. S2 in Appendix A).

Besides, we were able to identify and keep track of groups of pedestrians (i.e., co-walkers who appear to be relatives, co-workers, or friends) by visual inspection. Overall, close to 5,000 pedestrian trajectories were thus obtained. Pixel coordinates were converted into real-world coordinates with a geometric formula whose parameters are fit thanks to predefined calibration points at pedestrian height (see Appendix A).

For each scenario the time and space-resolved pedestrian measurements are then coupled to the above transmission models. This directly yields the instantaneous rate, abbreviated as $\nu_{ij}(t)$, at which a supposedly infected index patient $i$, that we will call Iago, transmits the disease to other pedestrians j around him via virus-laden droplets. Under the independent action hypothesis (IAH) [54, 55], each inhaled virus is equally likely to lead to an infection, with no cooperation or antagonism between viruses. It follows that, over the time interval $[t_{0},t_{0}+\tau_{i}]$ over which Iago was filmed, he infected a number $C_{i}^{(\tau_{i})}$ of other people $j$ (leaving aside his fellow group members $G_{i}$, whose possible infection is not specifically related to the scenario, except at the cafés), given by a Wells-Riley-like equation [56], viz.

where $S_{j}^{0}$ is the probability that $j$ is susceptible (i.e., not already infected) at the beginning of the observation interval, and $N_{ij}=\int_{t_{0}}^{t_{0}+\tau_{i}}\nu_{ij}(t)\,dt$ is the cumulative transmission risk [57].

It is worth noticing that the IAH implies that infections are a stochastic process without threshold: Any encounter can potentially result in a new case, and multiple short interactions with various people are as risky as a single long one, and even riskier because, once infected, agent j can no longer be infected, viz., $S_{j}^{0}:\,1\to 0$. This saturation of the risk complicates the evaluation of $C_{i}^{(\tau_{i})}$ even if we assume that all pedestrians except Iago’s group $G_{i}$ of co-walkers are initially susceptible, because our videos only record part of Iago’s wandering and may thus miss earlier off-camera interactions. Nonetheless, rigorous upper and lower bounds on $C_{i}^{(\tau_{i})}$ can be set by noticing that, on the one hand, $S_{j}^{0}\leqslant 1$ and that, on the other hand, $\sum_{j\notin G_{i}}(1-S_{j}^{0})$ cannot be larger than the number of people actually infected by Iago, which is related to $C_{i}^{(\tau_{i})}$ (see Appendix C). Finally, for comparison purposes, $C_{i}^{(\tau_{i})}$ is recast into an hourly rate of new infections $C_{i}\equiv C_{i}^{(\Delta T)}=\frac{\Delta T}{\tau_{i}}C_{i}^{(\tau_{i})}$ with $\Delta T=1\,\mathrm{h}$, assuming that the recorded videos are representative. As this rate is very sensitive to the chosen characteristic time for infection $T_{0}$, which exhibits great variability, we will mostly present results rescaled by $T_{0}$, which is equivalent to setting $\Delta T=T_{0}$ and showing the number of new cases over a time interval $\Delta T$.

Static scenarios – namely, the cafés and waiting lines – are handled slightly differently, because then Iago’s neighbors do not change significantly as time passes, in which case we set $N_{ij}=\frac{\Delta T}{\tau_{i}}\int_{t_{0}}^{t_{0}+\tau_{i}}\nu_{ij}(t)\,dt$ and $S_{j}^{0}=1$ in Eq. 3. In a nutshell, the proposed framework enables us to quantitatively translate patchy observations, with undetected contacts, into an estimated global risk of viral spread.

Inserting the collected field data into this framework, we obtain upper and lower bounds on the mean rate $C=\left\langle C_{i}\right\rangle_{i}$ of new infections per hour for each scenario and each transmission model. Figure 2 presents a sample of results for four of these models (also see Figs. S6-S8 for the results obtained with other models). These results confirm the efficiency of the proposed bound-setting method, as the bounds are found to confine $C$ to a narrow interval. Most importantly, the ranking of the different scenarios turns out to be robust, that is to say, largely preserved across models. This is our first major result.

Pursuing the analysis of Fig. 2, we observe that street cafés present the highest risks in terms of the mean number of new infections per hour, even though their tables were more spaced when the videos were shot than before the pandemic. These infections at cafés are easily rationalized by the close, face-to-face interactions between people sharing a table, let alone the increased emission of droplets associated with lively discussions and eating, which is overlooked here. This result is in line with case reports of high risks of viral transmission while dining and drinking (indoors or outdoors, unspecifically) [3, 46, 57, 6]. Next in line among the observed scenarios comes the outdoor market alley. Despite its high average density $\rho\simeq 0.5\,\mathrm{ped/m^{2}}$, this scenario never matches the level of risk at cafés, bar with the very pessimistic parameters corresponding to high contagiousness. Further down the list, crowd density explains the considerably higher risks at train and metro stations ($\rho\simeq 0.25\,\mathrm{ped/m^{2}}$) than on fairly busy streets in Lyon ($\rho\simeq 0.05-0.1\,\mathrm{ped/m^{2}}$) and, to an even larger extent, the riverbank walkway that we filmed. Somewhat intriguingly, the estimated infection rate may be as large, or even larger, at the observed testing site in Lyon than it is on these streets, although the overall density there is low and attendants were strictly asked to stay 2 meters apart from each other; yet, their relative proximity was prolonged over considerable time and, besides, they tended to turn and pace around a bit while waiting. One should however bear in mind that our models estimate the risks of viral transmission if no face mask is worn, whereas everybody was wearing a mask at the testing site that we filmed.

Besides the robustness of this qualitative ranking of scenarios, largely maintained across parameter sets, on a more quantitative note we observe that the infection rates are always (except with the worst-case models) at least 10 times scantier in the investigated streets than at cafés, even without taking into account that talking and eating augment droplet emissions. In addition, the pessimistic estimates are generally at most a factor 10 larger than the (possibly more relevant) most optimistic ones. Thus, it is reasonable to conclude from those estimated values, that contagious Iago will infect a number of order 1 person roughly speaking if he sits at a café for one hour, whereas he would probably cause significantly less than $\sim 0.1$ new infections if he spent this time walking on a fairly busy street.

Nonetheless, these average rates brush aside the variety of pedestrian contacts in the different scenarios, which is better reflected in the box plots of Fig. 3. The figure shows that, while the scenarios involving a moving crowd cause fewer infections than cafés on average, their rates of infections $C_{i}$ are more dispersed and, unlike cafés, they feature many values that significantly deviate from the bulk, both at $C_{i}\simeq 0$ and at high-$C_{i}$, the latter being pedestrians that fortuitously turn into super-spreaders because of their pattern of on-street contacts. As we shall see below, the blame does not necessarily rest on the pedestrian, but rather on the ebbs and flows of crowding in each observed situation.

Prior to that, let us remark that accounting for the directionality of droplet propagation and describing the orientations of pedestrians’ heads had a marked effect on our results. Indeed, not only does an isotropic transmission model overestimate risks in crowds by a factor of at least 10 in comparison to its directional counterpart, but it also alters the ranking of scenarios: It predicts considerably more infections at the outdoor market than at the cafés (Fig. S8). Otherwise, such an inversion (along with high risk estimates) is only found for our worst-case transmission models, in particular the model that we introduced to mimic the effect of a contagious patient sneezing every few minutes without covering his or her sneezes. On the other hand, allowing infections within groups, as we did for the cafés, does not dramatically change the picture, even though it substantially heightens the risks associated with sparse situations, for instance, the riverbank walkway. This is not surprising because in these situations close contacts mostly occur between group members.

To better understand the observed disparities, we need to identify the key variables that determine the level of risk. Figure 4 confirms the intuition that the instantaneous pedestrian density $\rho(t)$ is a major determinant of the rate of viral transmission $\nu(t)\equiv\left\langle\sum_{j\notin G_{i}}\nu_{ij}(t)\right\rangle_{i}$ (where the average is taken over all pedestrians observed at time t), in that it controls how close each pedestrian is to their counterparts. (Note that all time-dependent variables have been averaged over intervals of two seconds, to reduce the statistical noise.) The variation of $\nu$ with $\rho$ looks similar across scenarios, but is not strictly identical, which indicates that other scenario-dependent variables affect the transmission rate $\nu$. Furthermore, these variations become more muddled as one turns to more optimistic parameter sets, which is consistent with the idea that one then probes the configuration of the crowd at finer length-scales, owing to the shorter transmission range. The total pedestrian flow rate could in principle play a role; however, we found that $\nu$ does not follow any clear trend with this flow rate at fixed density $\rho$ (Fig. S9).

On the other hand, head orientations naturally have some bearing on the risks of infection, as evinced by the failure of isotropic transmission models to reproduce our results⁴⁴4We think that this is largely due to the discrepancy between the face-to-face orientations at cafés (which facilitate transmission) and the more or less random orientations e.g. at a market., but we now show that in non-static scenarios these orientations can be practically inferred using only the trajectories. To do so, we notice that the head orientations of walking pedestrians (speed $v>0.3\,\mathrm{m\cdot s^{-1}}$) are approximately normally distributed around their walking direction, with a standard deviation around $26^{\circ}$, and decorrelate over one second. Therefore, we choose to ascribe angular orientations randomly drawn from this normal distribution to walkers, while their stationary counterparts ($v\leqslant 0.3\,\mathrm{m\cdot s^{-1}}$) are considered purely randomly oriented; the random values are refreshed every second. Quite interestingly, this simple reconstruction of head orientations yields mean infection rates $C$ per hour that agree very well with the values computed with the bona fide orientations, with a relative difference generally lower than 15%, regardless of the transmission model. The correspondence between the individual $C_{i}$ values for each pedestrian is of course imperfect with this method, but overall the differences are not extremely large (Fig. S10). These observations are particularly relevant to bolster risk assessments based on observed or (reliably) simulated pedestrian datasets in which head orientations are missing, as they most often are.

The spatial resolution of our empirical data and models provides deeper insight into the circumstances of infection in the above scenarios; it can contribute to the debate about what physical distance should be prescribed between pedestrians in non-confined environments, and whether 2 meters or ’6 feet’ are enough from an epidemiological perspective [29, 28]. Admittedly, the answer will heavily rest on the transmission model (which was here established in an ad hoc way), but the statistics of inter-pedestrian contacts in the scenario also play a large role. Using the cumulative rate of transmission $\int\nu(t)\,dt$ as a proxy for the incurred risk, we find that its dominant contribution comes from interactions within a distance of 1 meter (for instance, 70% at Bellecour metro station, with ModOpt₃), whereas transmission beyond 2 meters, albeit possible, accounts for only a few percent of the risk (5% at Bellecour), at most. Importantly, here and throughout the paper, risks have been quantified by the number of new cases expected in each setting; this choice is relevant at the collective scale, for policy-making, but not for the evaluation of the risks incurred by an individual in the crowd.

Beyond this debate, the framework introduced here opens the door to evaluating the mitigation efficiency of hypothetical redesigns of streets and venues, consisting e.g. in enforcing one-way circulation on footpaths, a sitting plan at cafés, or increased spacing in queues. Since circulation plans have flourished during the pandemic, let us first explore the impact of one-way vs. two-way foot traffic on sidewalks and pedestrian streets.

To avoid potential situational biases, the question is investigated by separating the periods of time (binned in two-second intervals) when the flow was unidirectional from those when there were pedestrians going in both directions, in each given scenario – the market alley in the bottom right inset of Fig. 4. Since the transmission risks were found to depend on density, but not on the total flow rate (i.e., the sum of the directional flow rates across sections perpendicular to the main flow), we perform a comparison at fixed density. Our data (inset of Fig. 4) reveal only little benefit to switching from two-way to one-way traffic in our wide-path scenarios. To further test this somewhat surprising finding, we exploit the controlled experiments performed a few years ago by the German BaSiGo team⁵⁵5These extensive datasets are openly available under: https://ped.fz-juelich.de/db/doku.php to study unidirectional and bidirectional pedestrian flows in 4 to 5-meter-wide corridors [58]; head orientations are reconstructed as explained above. The results, shown in the top left inset of Fig. 4, are in line with the aforementioned finding: In wide walkways, switching from two-way flow to one-way flow seems to (at best) reduce the risks only moderately, probably because head-on ’collisions’ are rare in these self-organized flows.

Next, we turn to queues and study how their arrangement affects transmission risks. On the basis of our observations at a testing site, we modeled a queue as a line of more or less equally spaced people, swaying in a $50\,\mathrm{cm}\times 50\,\mathrm{cm}$ rectangle around their central spot and whose head orientations are normally distributed (with a standard deviation of $22^{\circ}$) around the queuing axis 75% of the time and purely random for the remaining 25% of the time, due to people turning around or having a look around. Both the positions and orientations are refreshed every second. Albeit simplistic ⁶⁶6The observed scenario is significantly more complex than its reconstruction: It actually features two different, not strictly linear queues, one outdoors and one indoors, as well as a few people who are not queuing., the reconstructed queuing scenario is comparable with our actual observations as far as the estimated infection rates are concerned; these estimates with and without reconstruction differ by less than 50% for any of our models – except the most optimistic ones, which bestow special importance to rare contact events that are overlooked in the reconstitution.

Figures 5 and S7 illustrate the extent to which predicted infection rates vary when the spacing between queuing people or the queuing geometry are modified. Naturally, the risks are minimal in the case of the linear queue with the largest spacing between individuals, whereas they are maximized for winding, S-shaped queues with very close rows and individuals standing right behind each other within each row. Since people cannot be expected to keep their heads strictly in the direction of the queue, there appears to be no smart solution to have at the same time minimal risks and a very compact queue; given the results of Fig. 5 and S7, a simple practical recommendation for S-shaped queues is to keep a little more space between rows than the actual spacing between individuals in each row.

In summary, the foregoing risk assessments in non-confined environments can guide public decisions in times of pandemic, in that (irrespective of the transmission parameters that are used) they confirm the risks of infection incurred at cafés [3] and underline the key role of pedestrian density in determining the rate of viral transmission in moving crowds without masks. Fairly busy streets, with densities up to $\rho\simeq 0.1\,\mathrm{ped/m^{2}}$, are found to present risks that are not completely negligible, but comparatively quite low, and these risks will manifestly reach even lower values for less busy streets. This suggests that the scant reports of outbreaks in outdoor walking crowds are not only due to the intricacy of tracing back these infections (due to unidentified contacts, recall biases, etc.), but also to the limited transmission of the virus in these conditions, even without face coverings. Nevertheless, this remark does not apply to very crowded settings such as markets or metro and train stations, which deserve particular attention. Furthermore, our model-based approach has enabled us to explore the efficiency of street and venue redesigns in mitigating the viral spread. For wide walkways, we have not found clear benefits to switching from two-way foot traffic to one-way traffic. For queues, increasing the space between individuals naturally reduces transmission risks; in the case of an S-shaped queue, a simple rule that could be enforced in practice is to keep a little more space between rows than between individuals in each row.

Given that our study pioneers the coupling of empirical crowd data to spatial models of viral transmission at mesoscales, it undoubtedly suffers from some limitations. To start with, the empirical data could be extended to include more scenarios and longer footage.

Perhaps more crucially, the transmission model should be refined. The models used in this study are admittedly overly simple, even though this problem was partly warded off by ensuring the robustness of our qualitative conclusions across diverse model variants, including a spatio-temporal model. More sophisticated models, which may differentiate transmission rates as a function of people’s activity (reflecting known variations in droplet emission [18, 20]) and account for the effect of the wind [23] and ambient air flows, will afford more accurate estimation of the rate of new infections. In addition, fluid dynamics simulations of long-range aerosol propagation would make it possible to study enclosed spaces with poor ventilation, where our current models that discard the airborne transmission route can only provide lower bounds on infection risks.

Another task is to generalize the transmission models to people who are (adequately or inadequately) wearing a mask [13], in order to determine how serious an issue very crowded streets really are in current times. It would be straightforward to account for the particle filtration efficiency of masks in the present framework, by simply multiplying the transmission rates by a reduction factor (say, $\sim 20\%$ for cloth masks, $\sim 10$% for surgical masks and $\ll 5\%$ for N95 masks [41, 47], if only the emitters have their face covered), but masks are probably even more efficient, because they also reduce the reach of the exhaled puff [14, 40], thereby probably shortening the range of transmission of droplets [41].