2.0.1 Preliminary definitions

All the terms we define in this paper are included in Appendix 6. For now, we need the following terms:

By an activity we mean a well-defined set of interactions with clear bounds taking place over a period of time less than a day, for example a trip to a grocery store, or taking an airplane flight, or attending a sporting event as a spectator.

By the participant we mean a person attending the activity, whose probability of becoming infected we wish to model.

A neighbor at an activity is a person not in the participant’s immediate household who, for some part of the activity, is close enough to pose a risk of air-borne infection. We shall say the neighbor is in the participant’s vicinity if they are close enough to be a risk of infection. The precise nature of the vicinity is currently unknown; the CDC asserts that most infections are caused by individuals within 6 feet of each other [10], so a 6 foot radius may be an approximation for vicinity.

2.0.2 Assumptions

Like any model, there are assumptions about the state of the world that are necessary for the model to apply. We state them here with some explanation, and discuss them in more detail in Section 4:

1. A1 Our first assumption is that the probability of infection is additive over sub-activities. This means that if one segments the activity into sub-activities, the probability of getting infected over the whole activity approximately equals the sum of the probabilities over each segment.
   Mathematically, this says that if we break an activity A up into N distinct sub-activities, S₁, …, S_N say, then the probability p of becoming infected during activity A satisfies (1) where x_j is the probability of becoming infected during S_j.
   We cannot actually expect exact equality in (1). Nonetheless—and this is an essential point—we can reasonably expect that the left-hand side and right-hand side of (1) agree with each other to within 10% or less. We give a mathematical proof of this assertion in Appendix 7.
2. A2 For each sub-activity S_j the probability x_j of infection is the sum over the forms of transmission of independent probabilities, each of which has a multiplicative form.
3. A3 If a neighbor is not infectious, there is 0 risk of infection from them.
4. A4 If a neighbor is infectious, the probability that they will infect the participant depends on the distance away, whether they are facing towards or away from the participant, mask usage, viral load in the neighbor, sneeze etiquette, air circulation, and other factors.
5. A5 The probability of infection from a neighbor is linear in the amount of time spent in their vicinity. See Appendix 7 for a justification of this assumption.
6. A6 The probability of infection in each segment is independent of the other segments.

Formally, the model does not need the following assumption, but it will be important when applying the model:

1. A7 There is no increased chance of infection from members of the participant’s immediate household engaging in the same activity, and we will ignore transmission from one’s immediate household members. (For example, sitting beside a household member at an activity will be treated as zero-risk).

2.2 The full model

Putting together all of our assumptions, we wish to model the probability that a participant at an activity contracts SARS-CoV-2. Actual infection is understood to happen in one of three ways [15]:

• Airborne transmission from an infectious neighbor at the activity, either by droplets or aerosols.
• Touching a contaminated surface, and then touching the participant’s face before thoroughly washing the hands.
• Direct physical contact with an infectious person.

Each activity A is broken down into a sequence of segments S_j, j ∈ {1, …, N}, disjoint sub-activities each of which can be thought of as a single uniform event, either as a single event (e.g. going to the restroom) or an event with constant parameters (e.g. sitting for some period of time with one neighbor 3 feet away, 2 neighbors 6 feet away, and no other neighbors within 10 feet).

Following A3, A4, and A5, for each segment S_j, the probability that the participant becomes infected by air-borne transmission is the sum over every neighbor of [the probability the neighbor is infected] times [the probability the neighbor will cause the participant to be infected per unit time] times [the time spent in their vicinity]: Here is the probability per unit time that given the configuration (distance away, orientation, mask-wearing or not, etc.) that if neighbor n is infected, they will infect the participant by air-borne transmission.

Similarly, the probability that the participant becomes infected by surface-born transmission from a surface they touch is [probability the surface is contaminated] times [probability they touch their face before washing their hands] times [probability that the touching leads to an infection]: where is the probability that the participant will convey the infection from the surface to themselves.

Finally, the probability that the participant becomes infected by direct contact with an infected neighbor is [probability neighbor is infected] times [probability of touching] times [probability of transmission]: where is the probability that if n is infected and the participant touches n, then infection will be transmitted.

Combining the above, we have (using A1, A2, and A6): and

Our contention is that this model is strategically valuable even without knowledge of the parameters . Even with no or highly uncertain knowledge of their values, the model allows one to estimate which activities (and which sub-activities) pose the most risk of infection. As we argue in Section 4, combined with knowledge of costs and benefits, that would allow for policymaking to decide on a set of activities to allow, and where adjustments can be made to lower the risk of those activities (e.g., choosing among competing seating configurations for an event venue).

In the next section, we outline how the model could be applied to idealized versions of airplane travel, attending a sporting event, sitting in a classroom, going to a restaurant, and attending a religious service. These examples are meant to give an idea of the value of knowing relative risks, but they do not contain the kind of fine-grained detail known by subject-matter experts that would be required for anything more than a first-order analysis.

3.1 Example: Air travel

Let us take travel on an airplane as an activity, as defined in Appendix 6. To use the model, we need to enumerate sub-activities S_j that together make up the air travel activity, A. The sub-activities S_j are:

1. boarding the plane
2. moving to and entering one’s seat
3. sitting on the plane for the duration of the flight
4. leaving one’s seat, and deboarding the plane.

The relevant parameters for this question, with sample values which can be changed, are:

1. Position of seats in the plane. We employ the seating arrangement used by United Airlines for the Boeing 737, which is available on United’s website [16]. Fig 1 shows the seating plan, with 50% occupancy.
2. Seating arrangement.This can differ between scenarios, but for this example we will assume the plane is full.
3. Time spent boarding, and traveling to one’s seat while on the plane. We will assume the passenger stands in line for 10 minutes boarding, and takes 20 seconds to sit down at their seat once they reach the correct row on the plane. These values are estimates, and will vary according to airline boarding protocols.
4. Order of seating. We shall assume that the plane fills back to front, so that while walking to one’s seat, one does not pass already seated passengers.
5. The distance apart people stand while boarding. This can vary based on preventative measures taken by airlines; for a first approximation, we will use 1.5 foot spacing.
6. Duration of the flight. This example will take flight duration as 180 minutes.
7. Deboarding, which will be modeled the same way as boarding for this example.
8. How risk decays with distance. There is much discrepancy in the literature as to this decay [5, 17]. Let us assume risk is inversely proportional to the square of distance from the source.

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Fig 1. An example seating arrangement of a Boeing 737 at 50% capacity.

Red cells (that are also labeled with 0s) are empty seats. Orange cells represent business class, blue cells represent premium economy, and yellow economy class. Cells with 1s are occupied.

https://doi.org/10.1371/journal.pone.0245381.g001

We shall also assume for this example that there is no direct physical contact between participants and that all surfaces are disinfected.

For each sub-activity, a participant is exposed to some amount of risk from their neighbors. As we do not know absolute risks, we will quantify the risks of the various sub-activities using the hazard × exposure model described in the previous section. Given the analysis is one of relative risk, we do not have an absolute unit to use in the quantification of risk; thus, as a basic risk unit, we will use the risk of spending one minute at a distance of one foot from a stranger. Appendix 9 and the online Supplemental Information contains more detail on the calculations presented below.

3.1.4 Changing parameters.

Given the varying practices of the major airlines [18], the percentage of occupied seats is one parameter for which we are already seeing wide variation The results for similar scenarios are:

1. Case 1 Airplane full, 1.5ft distancing while boarding
   Risk: 359
2. Case 2 Middle seats empty, 3ft distancing while boarding
   Risk: 146
3. Case 3 Airplane half full, 6ft distancing while boarding
   Risk: 100

Although the numbers 359, 146 and 100 are not in absolute units, they do show the relative effect of different possible mitigation strategies. For example, removing roughly one-third of the passengers by keeping the middle seats empty and increasing social distancing while boarding (Case 2) more than halves the risk compared to the full airplane. Other scenarios could be modeled similarly.

In each of the above cases, we used the inverse square decay function to model the change in risk as distance to others changes. While the other parameters used in the model are measurable, the rate of decay of risk with distance has not been experimentally verified, and is quite uncertain. To see how sensitive to the decay function our conclusions are, we will try two very different decay functions of risk with distance. The first has a very slow exponential decay [5], and was based on averaging over many different studies; it concluded that each additional meter of distance decreased risk by a factor of 2.02. The second has a very rapid decay, based on simulations of droplet dispersion [17]. We shall refer to these as the Chu model and the Chen model, respectively. To normalize, we multiply the output of each function by a constant such that the risk for a full flight of three hours is the same for each risk decay model. The risk values of each case with each other decay model are listed below in Table 1.

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Table 1. Relative risks with different decay assumptions for a three-hour flight.

https://doi.org/10.1371/journal.pone.0245381.t001

The relative risk at 2 meters compared to 1 meter is 25% in the inverse square model, 49.5% in the Chu model, and 4.2% in the Chen model.

Despite the great differences between the three model’s decay rates, there are many similarities in the relative risks for the three scenarios. In all cases, the vast majority of risk is incurred while sitting, and Case 3 is about 2/3 the risk of Case 2 no matter the decay assumption. However, in the Chen model, the risk for Case 2 relative to Case 1 is much more reduced than it is under either of the other two decay assumptions. Fig 2 summarizes how the risk score changes as a function of flight time, the cases considered above, and the decay model assumption. The values at t = 0 show the contributions of the boarding/deboarding risk to each scenario, and the relatively larger risk in the Chen and Inverse Square models compared to the Chu model. The risk scores are normalized so that the risk at the three-hour mark is the same across decay models, revealing that, under the Chu model, removing passengers from the flight (Cases 2 and 3) has less of an effect than under the other two models.

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Fig 2. Risk score as a function of flight duration, decay model, and seating/social distancing plan.

The three plots show how the risk score changes as a function of flight duration, decay model (inverse square, Chen, and Chu models), as well as seating plans with social distancing assumptions that are described in the main text. Note that the y-axis in each plot is a normalized score. As a result, absolute values cannot be compared between plots, but other characteristics can (e.g., rate-of-change, difference between scenarios etc.).

https://doi.org/10.1371/journal.pone.0245381.g002

The very large discrepancy in the relative risk in Case 1 to Case 2 is explained by the extreme sensitivity of the Chen model at close ranges. According to this model, moving from 0.2 meters to 0.3 meters reduces risk by a factor of 88. If a sensitive model such as the one presented by Chen et al. is most accurate, it will be important to be aware of the interval or intervals where risk drops rapidly.

Using the idealized example above of the airplane analysis, one can see the relative benefits of different mitigation strategies. Given the contribution of the time spent seated to the total risk score and what is currently known about mask-wearing, making masks mandatory could be a (cost-)effective strategy [19, 20]. Our analysis shows that keeping the middle seat vacant unless there is a party of three travelling together at least halves the risk, under a very wide range of decay assumptions. Managing boarding is likely less costly than leaving seats empty, but our analysis finds that the total impact will be lower that adjustments to the seating plan because it takes up a small part of the total flight time. Ongoing research will likely lead to increased knowledge about mitigation strategies specific to air travel that could be effective [21].

In addition to considering variation in parameters, there are other structural elements of the scenario that could affect the analysis. For example, it is possible that passenger compliance could vary. The above analyses have assumed that passengers follow a sequence of steps, such as boarding the plane a certain way, wearing a mask for nearly the entire flight, etc. [22]. The extent to which people comply with specific norms about protective actions varies by group and by hazard [23]. Social norms, social influence, and social comparisons all play a role in determining what people will do [24–26]. Planning for aberrant responses to airline (or other) policies about protective actions should be part of any private or public entity’s analyses of the possible outcomes of activities during the pandemic. Incorporating uncooperative members of the public into an analysis could demonstrate their possible negative effects, bolstering arguments for policymaking to mitigate those effects (e.g., by empowering staff with the regulatory powers to deny services to such persons when they do not have such powers already).

3.2 Example: Attending a sporting event at a stadium

We base our analysis on the TD Garden Stadium in Boston, MA, in the United States.

Here, the relevant sub-activities are

1. entering the stadium
2. moving to and entering one’s seat
3. sitting in one’s seat
4. getting food and or drink at a concession stand
5. eating in one’s seat
6. going to the bathroom
7. leaving the stadium

The relevant parameters for this question are:

1. Mask protocol. We assume masks are required except when eating.
2. Seating arrangement. For different scenarios, one can map the stadium, with certain seats kept empty, and others sold in small groups to trusted cohorts. By a “trusted cohort”, we mean somebody like a household member with whom one spends so much time in close contact that their presence at the event does not constitute an added risk. This is significant, because having multiple members of the same cohort sit together, while other cohorts sitting at a distance, means that there will be little droplet risk (though long-range aerosol transmission between different cohorts must still be considerd. For each person in the stadium, the distance from their seat to all other seats occupied by strangers (up to some cut-off distance, say 6 feet) is known.
3. Time spent entering. This depends on the number of entrance turnstiles, what the protocol is, and whether arrival times are deliberately staggered. We will assume 0.25 minutes.
4. Time spent walking to one’s seat. This depends on the lay-out of the stadium, walking speed, and density of people. Assuming staggered entrances, we estimate this at 8.3 minutes.
5. Social distancing requirement in corridors. This can be set by policy; we assume 3 feet.
6. Duration of game. Assume 190 minutes.
7. Concessions. This has two parts: ordering and eating. The latter is much more significant, as without a mask, and with the potential for one’s food to be contaminated, the risk of both infecting and being infected during eating is much higher. We will assume that a person eats for 15 minutes, that they are 4 times more likely to transmit infection without their mask, and that while eating they are 3.5 times as likely to be infected. (The factors of 4 and 3.5 are just guesses; as research is conducted, more accurate figures can be substituted).
8. How risk decays with distance. Let us assume risk is inversely proportional to the square of distance from the source.
9. Aerosol risk. One of the thorniest questions in devising a relative risk model is how to account for both aerosol and droplet risk. The relative risk of long-range aerosol transmission compared to short-range transmission from immediate neighbors is not known, though considered significant [6]. There are calculators that estimate the aerosol risk, such as the CU Boulder COVID-19 Aerosol transmitter tool [27]. However, this tool explicitly excludes droplet transmission, and assums that 6 foot social distancing is always maintained. Despite increasing awareness of the significance of aerosol transmission, there are no published studies that numerically compare aerosol risk to droplet risk. So to make a model that incorporates both, we have to make some assumption about their relative risks. This assumption can of course be changed as more information emerges.
   In the paper [28], it is argued that the aerosol risk is approximately c/V, where V is the the ventilation, measured in cubic foot per minute per person, and c is an empirical constant. We will set c = 1, meaning the droplet risk at 1 foot is the same as the aerosol risk if the ventilation is 1 cubic foot per person per minute.
10. Air volume of the stadium. If the stadium starts out empty and clean, the aerosol risk will be reduced somewhat.
11. Bathroom design and constraints. The assumptions we make, based on TD Garden stadium, are that at most 4 people will be in the bathroom at any one time, with an average visit of 4 minutes. The ventilation in the bathrooms is 1075 cfm, which is 127 liters/person/second. This very high ventilation rate leads to a very low aerosol risk. Every other sink and urinal will be blocked off, leaving a gap of 6 feet between users. There will be some passing time as people enter and exit; we estimate an upper bound of 1 minute at one foot, which is the large majority of the risk estimate of going to the bathroom.
12. Presence or absence of screening of attendees, that would catch some percentage of infectious people. For this model, we will assume it is not available.

Using these parameters (included in our supplemental material), we arrive at the following scores:

• Full stadium: 1044 risk units, 696 of which come from the seated portion.
• Half-full stadium: 335 risk units, 219 of which come from the seated portion.
• 21%-full stadium: 125 risk units, 77 of which come from the seated portion.
• 21%-full stadium, no eating or drinking: 83 risk units.

Based on these calculations, concessions pose a larger risk the fuller the stadium. However, many of the relevant parameters are uncertain or unknown, but with better estimates the above set of steps could be used to provide more realistic estimates.

3.3 Example: Classroom

Consider a classroom, with one possible seating arrangement as in Fig 3, where 1 represents an occupied seat, and 0 an empty seat. One of us (JEM) went to a classroom at the Washington University in St. Louis and measured its layout: seats are 3.9 feet apart horizontally, and 4.3 feet apart vertically. For that classroom, its volume is approximately 5834 cubic feet, and the ventilation is 8 liters per person per second when full to capacity of 43, which is 729 cubic feet per minute.

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Fig 3. An example classroom layout.

Red cells (that also contain 0s) indicate empty desks, while green cells (that also contain 1s) indicate occupied desks.

https://doi.org/10.1371/journal.pone.0245381.g003

Let us assume that a class is 60 minutes long, and that the time taken to reach seats is negligible. Moreover, assume that everybody is required to wear a mask at all times. As before, we shall set c = 1, and assume an inverse square decay of droplet risk with distance, and that the droplet risk becomes to 0 at 6 feet.

Using these parameters, there are two different risk scores. One is the steady state score, whereby one assumes that the rate of exhalation equals the amount of exhaled breath removed by the ventilation system. The other takes into account that if the classroom starts out clean and empty, it takes a while for the air in the classroom to reach this equilibrium state. If T is the total duration (here, 60 minutes) and f is the fraction of air in the room removed per minute (here, 0.125), this correction factor reduces the effective time in the room for aerosol exposure (but not droplet exposure) to which in this case is 52 minutes. Table 2 shows the two risk scores given different occupancy levels and configurations, showing a surprisingly slow decay of risk with number of people—it is not much better than linear. The Excel program in the online Supplemental Information contains the calculations and allows the interested reader to adjust the parameters and see the change in risk scores.

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Table 2. Relative risks with different seating assumptions.

https://doi.org/10.1371/journal.pone.0245381.t002

3.4 Further examples

Using similar procedures, one can analyze other events. More detail is provided in the Excel program provided in the online Supplemental Information.

3.5 Summary

In this section, we have endeavoured to outline how one might use the approach described in Section 2.2 to frame an analysis of the risk of SARS-CoV-2 infection during a variety of activities. Of course, decision-makers and analysts in any of the above public or private decisions would need to incorporate more sensitivity analyses and more details specific to those contexts, details that would be known to subject-matter experts. We believe that, when the relevant assumptions hold, the approach we have outlined can be useful in identifying the most dangerous sub-activities of an activity, which could be used to inform policy decisions including strategies to mitigate the risk posed by those sub-activities. In the next section, we discuss the advantages of the approach in more detail, as well as its limitations and the future work that would be required for a responsible application of the approach by policymakers.

4.1 Limitations

Like any model, the model described here depends on its assumptions. In our view, the most problematic assumptions are A2 and A6, which require independence. That could fail, if, for example, infection requires a certain minimum threshold of exposure. Similarly, A7, the assumption that those in one’s household pose no threat, could be problematic. The presence of family members could increase the risk of exposure after the activity when one returns home, given, e.g., their separate trips to the washroom or through the turnstile during the activity. The importance of these extra exposures is an empirical question, and a function of the relative risk of those sub-activities done individually, and protective actions taken after the event (e.g., social distancing, hand-hygiene, proactive testing etc.).

As it stands, our model cannot account for household transmission, given that “living in a household” violates our definition of an “activity” because living in a household lasts much longer than a day, by its nature. Furthermore, decomposing (a day in) a household into its constituent subactivities would be difficult given the variation and number of subactivities. In contrast, when attending a sporting event, there are only so many things one can do in the stadium, and, in fact, subactivities can be constrained by those running the activity in order to lower the risk of infection. That kind of control over subactivities is unlikely to be generalizable to households.

Household transmission has been found to be an important mode of transmission [41], but there are strategies to mitigate it [42]. If people begin to partake in more activities outside of the home, finding ways to encourage increased protective actions within the household prophylactically would help counteract the additional risk posed by leaving one’s home.

More importantly, the model assumes that the risk of infection is a function of the background risk in the population of the activity’s jurisdiction. However, that assumes the subpopulation of potential participants does not have more virus prevalence than the community at large. Whether that is true is also an empirical question. For example, are those who would choose to attend a stadium concert during a pandemic more or less likely to participate in protective actions that lower their overall risk of virus infection or of virus transmission? Part of the empirical study of the public’s risk perceptions would need to include an appraisal of that question.

Furthermore, the model has a specific definition of “activity,” and it is crucial that the definition is clear to those who would use the model. For instance, considering a semester on a university campus as ∼90 separate one-day activities would not be an appropriate use of the model, because the population on campus from one day to the next is almost identical. Unless participation in an activity incurs no additional risk for a participant beyond their usual activities, decision-makers would need to consider how to limit individual participation; for example, in limiting the number of sports games one can attend in a given time period. Otherwise, the risk of transmission among participants will surely increase over the average risk in the community as previously-shuttered activities become a part of their day-to-day life but not of the lives of their fellow community members. The feasibility of such controls should be considered during the decision-making process.

For analytical purposes, our examples show how sensitive many analyses will be to uncertainty about how the probability of infection decays with distance, including the threat of long-range exposure [5, 17, 43]. As long as estimates for those values vary widely in the literature, decision-makers may need to be conservative in cases where an analysis is highly sensitive to changes in the relevant parameters.

Finally, of the various modes of transmission—droplet, aerosol, fomite, and direct contact—it is not known how they compare to each other. Even for the two airborne modes, droplets and aerosols, there is disagreement about their relative importance [3, 6, 15, 44, 45]. However, the model is adaptable enough that different estimates of the relative importance of these pathways can be included, and the model can use whatever assumptions the scientific evidence supports best at the time of use.

7.1 Mathematical derivation of approximate additivity

Let us assume that an activity A is decomposed into N segments, called S₁, …, S_N, and each segment S_j has some risk x_j of causing infection. We further assume that these risks are statistically independent of each other (assumption A6). Then the probability p of being infected at some time during A is (3)

Let (4)

Then we claim that where the symbol ≈ means “is approximately equal to”. Indeed we claim that (5)

To see (5), we use the arithmetic-geometric inequality to show which yields the left-hand inequality in (5).

The right-hand inequality follows from (6)

The correction factor between s and p is , in the sense that

The function f(s) is a decreasing function of s, and has a right-hand limit of 1 as s tends to 0. Since s ≤ −ln(1 − p), we have

So our conclusion is that we always have justifying the claim that p ≈ s. For representative values of f(s) and g(p) = −p/ln(1 − p), see Tables 3 and 4. They can be read as saying that if we know the value in the top row is an upper bound on s (respectively p) then the value in the second row is a lower bound on .

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Table 3. as a function of s.

https://doi.org/10.1371/journal.pone.0245381.t003

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Table 4. as a function of p.

https://doi.org/10.1371/journal.pone.0245381.t004

7.2 Estimating an upper bound on p

As we saw in Subsection 7.1, how good the approximation p ≈ s is depends on how close f(s) is to 1. How can we measure this?

We use the assumption that the activities we are considering last less than a day. While some activities are more risky than others, we further assume that all the events will be designed so that the total risk of an infected individual spreading the infection is no greater than it was at the beginning of the pandemic before any social distancing measures were in place. So we get an upper bound (7) where ν is the probability that before social distancing, an infectious individual would infect a susceptible individual over the course of one day. Note that for many activities, it is reasonable to assume that p is much less than ν, thus tightening the estimation in Subsection 7.1.

Since an infected individual can infect multiple susceptibles, the expected number they would infect over the course of the day would be slightly higher, namely if one assumes a Poisson distribution. (This is a small adjustment that will not materially affect our conclusion, so the reader can ignore it.)

Using a standard SIR model at the early stage of an infection, the proportion of the population that is susceptible is close to 1. So the proportion of the population that is infected will grow exponentially, like e^κt for some rate κ, where 1 + ν′ = e^κ. The doubling time τ is the time at which e^κτ = 2, so κ = ln(2)/τ. Thus we get and (8)

What is τ? In [46] they estimate that the doubling times in Chinese provinces in the period January 20—February 9 2020 ranged from 1.4 days (95% CI 1.2-2.0) in Hunan province to 3.1 days (95% CI 2.1-4.8) in Xinjiang province.

In [47], the authors estimate the doubling time in Italy in March 2020 to be 3.4, 5.1 and 9.6 days in the first, second and third ten day periods of the month.

In [11], the authors estimate the probability of infection in a crowded zone (summed over all neighbors in the vicinity) to be 1.8% per hour, and to be 0.18% and 0.018% in moderate and uncrowded zones, using data from the cruise ship Diamond Princess. If we assume at most 12 hours spent in crowded zones per day on the cruise ship, this would yield the estimate which in turn from (8) gives

In [13], the authors use an SEIR model on the data from [14], and take into account the incubation period (6 days) recovery period (14 days) and a mortality rate of 1%. They assume that transmission rates are the same in asymptomatic and symptomatic states, and get a value of ν that is 0.126. This follows from their equation and using their values R₀ = 2.5, ω^R = 1/14 is recovery rate from symptomatic to recovered, and ω^D = .01 is the mortality rate.

The value R₀, the number of new people infected per infectious person, is widely reported by time and geographic region—see e.g. [12] for estimates of R₀ by U.S. state. If one makes the more conservative estimate that only asymptomatic carriers will be circulating, and using the same 6 day incubation period, then one gets the bound

With an estimate of 2.22 for pre-mitigation R₀ in the U.S. [12], this gives the bound

Table 5 gives some values of ν as a function of τ.

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Table 5. ν as a function of τ, the doubling time.

https://doi.org/10.1371/journal.pone.0245381.t005