key: cord-0744573-37cx8dds
authors: Rowe, B. R.; Mitchell, J. B. A.; Canosa, A.
title: Outdoor long-range transmission of COVID-19 and patient zero
date: 2022-03-18
journal: nan
DOI: 10.1101/2022.03.16.22272493
sha: 3b8cc5faa29d9a0a82d92404f05d2baabc617b09
doc_id: 744573
cord_uid: 37cx8dds

Following the outdoor model of risk assessment developed in one of our previous studies, we demonstrate in the present work that long-range transport of infectious aerosols can initiate patient "zero" creation at distances downwind beyond one hundred kilometers. The very low probability of this outdoor transmission can be compensated by a high number and density of susceptibles such as it occurs in large cities.

Although it was originally discredited by governments and even health agencies, it is now well accepted that COVID-19 is mainly transmitted via aerosols (Wang et al., 2021) . This has brought focus to the need for ventilation in interior spaces and the need for mask wearing, amongst general measures not specific to this way of contamination. In a recent paper (Rowe et al., 2021) , we have shown via simple modelling based on air flows, that the outdoor risk of being contaminated is generally several orders of magnitude less than indoors. Indeed, our paper uses concepts developed by Wells (Wells, 1955) and results in an outdoor model very close to the famous Wells-Riley model (Riley et al., 1978) for the probability of being infected by virions present in breathed air.

Another unknown in this epidemic is where does the infection originate from? In the beginning the question focused on the Chinese city of Wuhan which has been the epicentre for the outbreak of COVID-19. Despite extensive investigation, in particular focused on the Virus Research Laboratory there or on the culprit animal that allowed the strain to jump over to the human host, no so-called "smoking gun" has emerged from these studies. Many countries then focused on the so-called "patient zero", in order to prevent or circumvent an epidemic outbreak due to the cross border transit of a sick individual.

More recently we have seen the emergence of the more infectious Delta variant, apparently with an origin in India and sometime later, the even more infectious Omicron strain first identified in South Africa, but which quickly became rampant in the United Kingdom. Again we saw borders closing to try to contain this outbreak but with little hope of success. As discussed in another of our recent papers (Rowe et al., 2022) , the higher viral load of these new variants results in an even higher infectious risk by the aerosol route.

Based on our previous work (Rowe et al., 2021) , what we wish to investigate in the present paper, is the possibility that there is a long-distance airborne route for the passage of the virus from one region to another, leading to the creation of a few "patient zeros" who could serve as starting igniters of the epidemic. Of course, we insist on the fact that, in a given region, the epidemic itself cannot spread by this process alone, due to the extremely low probability of being infected in this way for a single individual (see Table 1 ). However, when the target is a territory of a fraction of a million of individuals or more, our model and calculation show that the emergence of patient zero is possible in this way.

The present paper is organized as follow: first we comment on some wellknown cases of aerosol transport over very long distance, then we establish our model of outdoor transmission in a slightly different manner than in our previous paper (Rowe et al., 2021) and in section IV we apply the model to a specific case: the possible contamination of Northern France by Southern England and London, especially with new variants of high viral load such as Delta or Omicron. Finally, we discuss our results in the light of the problem of contamination by very low dose, as related to single hit models (Teunis and Havelaar, 2000; Zwart et al., 2009; Haas et al., 2014; Brouwer et al., 2017; Rowe et al., 2022) .

The first thing to remember in the following discussion is that the SARS-CoV-2 virus is, first and foremost, a nanoparticle (diameter on the order of 100 nm). It can be exhaled by an infected person (hereafter infector) when included in microdroplets of micron and submicron size (Greenhalgh et al., 2021; Tang et al., 2021) . These microdroplets originate from respiratory fluids which, besides water as the main component, include a variety of other minor components: proteins, salt, etc. (Nicas et al., 2005) . Water can evaporate leading to the creation of "dry nuclei" which include these minor components together with the virus. Due to the presence of non-volatile components, the reduction in size of the microdroplets cannot exceed a factor of around 0.4. Whether in aerosols or on surfaces, the virus is fragile. Its activity is influenced by temperature, humidity and especially ultraviolet (UV) light. We shall discuss this aspect below but the fact of being included in a microparticle means that it can float in the air for extended periods, driven by air currents both thermal and mechanical. The questions, therefore, are: can an infectious aerosol micro-particle travel over large distances? and if so, can it have a biological effect after travelling?

The second question centres around the survivability of viruses in an open environment and ultimately the statistics of the sources and receivers leading to a probability of "patient zero" creation. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The first question, concerning the long-distance transmission of micron and sub-micron sized particles, can be answered by experience and modelling. We can cite as an example, the experience gathered from the 11 December 2005 fire in the Buncefield fuel depot, North of London (Vautard et al., 2007) , which was the largest fire in Europe since the second World War, in which 58,000 tonnes of oil burned. It has been estimated that 8000 tonnes of PM 10 (<10µm) and 5000 tonnes of PM 2.5 (<2.5µm) were released in this incident. Simulations of the smoke plume from this fire indicate that it travelled more than one hundred kilometres passing over Normandy and Brittany in France and eventually in other locations throughout Europe though highly diluted. Since the high temperature buoyancy of the plume wafted it high into the troposphere and the winter conditions kept it there, measurements on the ground indicated little localised deposition in mainland Europe. Vautard et al. (Vautard et al., 2007) Other examples of long-distance transmission of particulate matter include the transport of Saharan sand (Francis et al., 2022) , plastic microparticles (Allen et al., 2021) , soot particles (PM 2.5 ) from biomass burnings (Martins et al., 2018) , and pollen transport from eastern North America to Greenland ( (Rousseau et al., 2003) and references therein). These phenomena are well studied and documented and their importance evaluated. With the exception of pollen, these examples refer to nonbiological, inert matter and are cited from the point of view of the coupling of observation and simulation to understand the modes and parameters associated with their transmission. It is well understood that long-distance travel can give rise to physical effects from these particles.

Does this long-distance transmission of microparticles have relevance to the spread of the COVID-19 virus? What is the likelihood of biological matter and in particular viruses inducing illness through long-distance transmission? Here, there is much to learn from animal studies and indeed this is a transmission route that is taken very seriously by researchers worldwide. For example, Gloster et al. (Gloster et al., 2010) presents the findings of a workshop held at the Institute for Animal health in is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 18, 2022. ;  the UK in 2008 that brought together researchers from the UK, US, Canada, Denmark, Australia and New Zealand to compare models for wind-borne transmission and infection of Foot and Mouth Disease. What was clear from this workshop was that, under favourable meteorological conditions, the risk of longdistance infection was far from negligible though input parameters to the models (virus release, environmental fate, and subsequent infection) are clearly sources of considerable uncertainty ( (Gloster et al., 2010) and references therein). Other studies have highlighted the long-distance transmission of the bird flu virus (Zhao et al., 2019) between farms in different states of the United States. It is significant that these studies take as a basis, that a single virus (or at least very few) can induce an infection (Sutmoller and Vose, 1997; Cannon and Garner, 1999) .

Could a similar effect occur with the SARS-CoV-2 virus leading to outbreaks of COVID-19 without the necessity for a cross border transit of a sick individual? Let us look at the statistics and probabilities to see what they have to tell us.

Since the dawn of humanity, mankind has suffered of infectious diseases due to a variety of pathogens. In the recent decades, epidemiology has focused more on nontransmissible illnesses, such as heart disease, cancer, or obesity. However, the COVID-19 pandemic reminds us of the burden of infectious illnesses.

Infectious diseases can be classified considering their target organs and the route of transmission following the path taken by the pathogen as it enters the body.

As the target organs of the coronaviruses responsible of the COVID-19 are located in the respiratory tract, this disease can be classified as respiratory. As discussed in (Rowe et al., 2021) , the route of transmission has been a matter of intense debate but, as stated in the introduction, it is nowadays largely recognized that the major transmission path is through airborne exchange i.e., by inhalation of an aerosol that has been exhaled by an infected person. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10. 1101 /2022 In matters of infectious disease and epidemiology, a key problem is to assess the dose-response relationship i.e., what is the probability of infection resulting from a given level of exposure (the dose) to a pathogen (Brouwer et al., 2017) . The dose is clearly linked to a number of pathogens. A dose-response function ( ) relates the dose to a probability of infection. It is clear that ( ) must be a monotonic increasing function of the dose, starting from zero at zero dose and increasing toward an asymptote = 1 for large values of . There are several probability laws that can be used for ( ) as discussed in (Brouwer et al., 2017) , one of the most widely used being the exponential form:

where Π is a numerical factor which depends on the choice of the dose counting unit reference. In fact, one of the recognized difficulties in the dose-response model is first to define the dose. It is beyond the scope of the present paper to examine this question in detail and the reader is referred to the book of Haas (Haas et al., 2014) and to (Brouwer et al., 2017) . For an airborne disease, Wells (Wells, 1955) , using the exponential law, defined a quantum of contagium as a hypothetical quantity that has been inhaled per susceptible individual when 63.2% (corresponding to 1 − exp (−1)) of these individuals display symptoms of infection. It is linked to a probability of infection which follows a Poisson law:

(2)

The quantum has no dimension but is a counting unit (as dozens versus unity, or moles compared to molecules) which is clearly linked to a choice of Π = 1 in eq. (1). Of course, and as discussed in (Brouwer et al., 2017) and (Rowe et al., 2022) , its value, in terms of the number of pathogens, depends on a variety of mechanisms: inhalation of airborne particles, pathogen inhibition by host defenses or losses by some other processes, before any replication will start in an infected cell.

Obviously, a quantum corresponds statistically to a number of pathogens greater than one.

Considering the concentration of quanta in space (in m -3 units), ( , ��⃗ ) the inhaled dose during a time of exposure t can be written as:

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The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 being the pulmonary ventilation rate (taken as 0.5 m 3 /h in the present investigation). Note that this definition of the dose does not require a homogeneous distribution of quanta in space. Only ( , ��⃗ ) at the inhaled location (mouth and nostrils) has to be considered. Note also that due to the extremely low concentration of quanta in air, ( , ��⃗ ) is not really a continuous function of , ��⃗ (since a number of viruses is of course an integer) but can be treated as such due to the statistical aspect of the problem.

As shown in our previous papers (Rowe et al., 2021; Rowe et al., 2022) , indoor and in the case of a room with well-mixed air, it is possible to write an equation of conservation for the quanta, which, together with eq. (3), leads directly to the stationary state dose value and to the famous Wells-Riley probability:

where is the probability for a susceptible person to be infected, a quantum production rate per infector and per unit time, the pulmonary ventilation rate, the ventilation rate of the room, the number of infectors in the room, and the time of exposure.

In one of our previous papers (Rowe et al., 2021) , we have developed models of indoor and outdoor transmissions considering infectious microdroplets and make the link with the notion of the quantum. In the following, we develop in some detail, an identical outdoor model using the Wells notion of quantum. We consider an outdoor volume (airshed box) as illustrated in figure 1 with the wind blowing along the x axis and, as in our previous work, we consider that there are no quanta escaping the volume above a height H along z. it is also assumed that the quantum density does not change across the wind:

It results that the quantum concentration is considered as a function of along the wind only. In fact, it is clear that, at least at low values of , is a function of . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101/2022.03.16.22272493 doi: medRxiv preprint but assuming that eq. (5) holds everywhere it does not change anything for the concentration balance between what is produced in the bulk of the airshed box and what emerges at the downwind border of the airshed at large x, where everything has been mixed by the turbulent dispersion. It can therefore, safely be concluded that it has no influence on the quantum concentration at this border.

Then, assuming stationary state i.e., = 0, an equation of conservation for the quanta can be written as:

where is a density of infectors per unit surface (assumed homogeneous and therefore constant), ∞ is the wind velocity and is the virus lifetime (inactivation time). Note that the infectors are located at the bottom of the airshed box (which can include houses as we will discuss in section IV-3) but this has no influence since we admit an homogeneous dispersion in the vertical dimension of the airshed, as discussed in previous paragraph.

With a quantum concentration (0) at = 0, we can derive the following value for the quantum concentration:

In an area where there is no infector = 0 eq. (7) leads to a simple downwind exponential decay of the quantum concentration:

On the other hand and for a virus lifetime much longer than the hydrodynamic time ℎ (i.e., ≫ ℎ = ∞ ) ⁄ eq. (7) leads to the following value for ( ):

This is analogous to the equation derived in (Rowe et al., 2021) for (0) = 0, and which expresses the conservation of quanta in the airshed box shown in figure 1 when there is no decay due to viral inactivation. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint Using equation (2) and (3) it is then possible to calculate the probability of infection at a distance . In equations (7) and (9), a key parameter is the value of and therefore , as discussed at length by Rowe et al. (Rowe et al., 2021) in their supplementary materials. For strong winds i.e. ∞ > 6m/s at 10 m height and night time or low solar insolation, the atmosphere can be considered as neutral in the socalled Pasquill-Gifford-Turner classification (Pasquill, 1961; Gifford, 1961; Turner, 1994) , which means no tendency for air to rise (unstable) or drop (stable) by the buoyancy effect. In fact, it is admitted that the airborne pollutants emitted locally are transported and dispersed within the so-called atmospheric boundary layer (ABL: the tropospheric bottom layer), whose thickness is lower than one thousand meters (Sáez de Cámara Oleaga, 2016), excepted for strongly unstable atmospheres. At any distance from the source, an order of magnitude of H versus x can be given by the vertical dispersion length used for Gaussian plumes and shown in figure 2 (Turner, 1994; Seinfeld and Pandis, 2016) . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint Note that, to be realistic, the condition = 0 clearly requires that there is no gradient in infector density along i.e., across the wind. If we assume a value of for the width of the source, then, for (0) = 0 eq. (9) also reads:

where ( ) = is now the total population in the area × .

The wind itself depends on the altitude but its variations above ten meters are rather small within the ABL (Hsu et al., 1994) . Therefore, meteorological values of the . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101/2022.03.16.22272493 doi: medRxiv preprint wind are recorded for a ten-meter altitude. In the next section it will be assumed independent of altitude and taken as the ten-meter value.

III-3. The question of the virus lifetime in aerosol form.

The virus lifetime can be defined by the analog in time of eq. (8):

Note that with this definition, it is slightly different from the so-called half-life which is the time required to decrease the virus concentration by a factor of two (instead of (−1) = 0.37).

As shown by previous equations, the effect of virus lifetime is critical for the possibility of virus transmission over long distances. If, for ≫ ℎ , the decrease of quantum concentration downstream of a laterally extended source is mainly due to vertical atmospheric dispersion, for ≪ ℎ the quantum concentration will drop to an even much lower value, with a ratio of ℎ ⁄ . This condition corresponds to a distance to the source ≫ × ∞ where the transmission probability drops essentially to zero.

Viruses are inactivated by a variety of factors, but it is recognized that the principal ones are temperature, humidity, and UV radiation. Although solar UV radiation is very efficient for virus inactivation (Lytle and Sagripanti, 2005) , we shall not consider it here, restricting ourselves to the case of mean and high latitude in winter, where nighttime is much longer than daytime, and where the sky is often overcast during the day, making UV inactivation negligible outdoors during atmospheric virus transport.

The effect of humidity on lifetime is rather difficult to assess (Ijaz et al., 1985; Yang and Marr, 2012) . After some discussion (Marr et al., 2019) about the parameter to consider, either absolute humidity (AH) or relative humidity (RH), it is generally admitted that RH drives the virus lifetime with a U-shaped curve for = ( ). This behavior is due to the variation of the solute/solvent concentrations in the aqueous solution constituting the infectious microdroplet. This U-shape has been rationalized by Morris et al. (Morris et al., 2021) considering the efflorescence relative humidity . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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(ERH) which corresponds to the RH below which a spontaneous evaporation of a salty water solution initiates a crystallization called efflorescence (leading then to what is called dry nuclei for viral aerosols). Note that the inverse process, when the crystal starts to disappear with the increase of humidity, is called deliquescence (Horst et al., 2019) .

The inverse of the lifetime can be considered as a rate of inactivation (unit = time -1 ) and it is generally admitted (Yap et al., 2020; Morris et al., 2021) that, for a given value of RH, it follows an Arrhenius law with temperature:

On the basis of numerous experimental results for a variety of viruses, this behavior has been rationalized by Yap et al. (Yap et al., 2020) (Fears et al., 2020) found a much higher value (up to sixteen hours). Most of the reported values in fact concern microdroplets deposited on surfaces. This can be understood due to the extreme difficulty of virus concentration measurements in air and, worse, of their characterization whether as active or not (Haas et al., 2014) .

Based on the above studies (Haas et al., 2014; van Doremalen et al., 2020; Yap et al., 2020; Morris et al., 2021) however, it can be easily shown that at low temperatures (<5°C) and large RH (around 80%), SARS-CoV-2 has a lifetime at least of several tens of hours, an important conclusion for the following discussion.

IV-Possible airborne creation of "patient zeros" by long-range transmission IV-1. General considerations Imagine two strongly populated areas, designated as the "source" and the "target" respectively, separated by an unpopulated area (no man's land). At the downwind . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 border of the source, it is possible to quantify the quantum concentration following the above equations. Its evolution then in the "no man's land" will be ruled only by the virus lifetime following eq. (6) with = 0. This will lead to its new value at the upstream border of the target area with a possible evolution downstream. Assuming no evolution of the quantum concentration downwind (consistent with the virus lifetime discussion of section III-3) or across the wind (which will be the case if the width of the target is smaller than the width of the source) then the calculation of a probability of infection in the target area of population is straightforward.

It follows that the statistical number of contaminated susceptible people is:

The value of will most often be extremely small, which shows a quasi-zero risk at the individual level. However, if the target is composed of a very high number of individuals, then a few people ≥ 1 could be infected. Of course, this process alone cannot sustain an epidemic but creates a few infectors ("patient zero") which will ignite it. Note that is the total number of individuals in the target who are assumed healthy and therefore susceptible.

In winter and early spring, the strongest winds blow most often from west to east in Western Europe. Below we examine a hypothetical case study of the creation of patients zero in Northern France from Southern England in wintertime. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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We can use the equations developed in section III-2 directly to make an estimation of the quantum concentration at the downwind border of this source box.

However, the following problem arises: in wintertime most of the quanta will be emitted indoors, with a room temperature around 20 °C, and a rather low RH (we assume 35 % as a mean), but outdoors they are transported by the wind at low temperature (around 5°C) and rather high humidity (80%) conditions, where the virus lifetime is expected to be much longer than the hydrodynamic time. Therefore, viral inactivation will only occur indoors, via thermal effect at rather low RH. Indoor air is continuously renewed however and contaminated air exhausted outdoors with a characteristic time equal to 1/ where is the air change per hour. Therefore, the effect of viral inactivation indoors prior to exhaust can be taken as a reduction of the quantum emission rate per infector used in section III-2 following the formula:

with the conservative hypothesis of = 2 ℎ −1 and = 2 ℎ, this results in = 0.78 × .

Thus, under winter conditions, the simple formula (11) can be used, with is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 (9) is used in place of 11)), and the quantum production rate of an infector as 10 h -1 .

The numerical application leads then to (45 ) = 7.15 × 10 −6 −3 assuming a proportion of infected persons of r = 0.03 in the Greater London area.

Convection by the wind then will lead quanta at the upstream border of the target area whose width is assumed less than or equal to the width of the source.

Due to meteorological conditions (mean temperature around 5°C, mean RH around 80% (Weather and Climate, 2022), and absence of UV radiation) the virus lifetime is much higher than the convective hydrodynamic time (<10 h, see Table 1 ) for distances up to 250 km and therefore the reduction in quantum concentration is solely due to the increase of the dispersive height . In a conservative way we take for , a value of one thousand meters corresponding to a commonly admitted upper value of ABL thickness for neutral or stable conditions (Sáez de Cámara Oleaga, 2016). It results a numerical value of of 2.15 × 10 −6 −3 at the upstream border of one of our targets.

We consider two plausible targets in France, either the city of Dunkerque or the Lille agglomeration with populations of 0.2 and 1.2 millions of people respectively.

We assume that the wind direction is the same as the direct path between the source and the target, a quite dominant direction in wintertime, which grossly corresponds to a wind direction blowing from the west/northwest (respectively 288 and 294 degree).

We also assume a wind value of 30 km/h as before which is only slightly higher than the mean wind velocity in February/early March (Weather Sparks, 2022) . Note again that both target areas have a width across the wind less than that of the source. Table 1 summarizes the assumed values of various parameters leading to a statistical number of infected "patient zeros". Since this number appears to be a few units in the frame of our assumptions, it clearly reveals the potential possibility of an infection ignition through long-range transportation of airborne viruses.

Note that the purpose of the calculations presented in Table 1 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 18, 2022. ;  and RH effects like assumed for the source box, but again without changing the major conclusion. 2,1 x10 -6 2,1 x10 -6 Dose for 24 hours 2,6 x10 -5 2,6 x10 -5 Probability of infection 2,6 x10 -5 2,6 x10 -5 Number of patients "zero" 5 31

IV-4. Discussion: the very low dose question

The literature on virus transmission very often refers to a quantity named "Minimum Infective Dose" with the acronym MID. As stated by Haas et al. (Haas et al., 2014) the term "Minimum" is very misleading as it seems to imply a minimum number of pathogens needed to start an infection and should be replaced by "Median". A MID would imply a thresholding effect which is not observed experimentally. If such an effect existed, it would prevent the long-distance transmission presented above.

However, there exists a very large literature on the subjects [ (Teunis and Havelaar, 2000; Zwart et al., 2009; Haas et al., 2014; Brouwer et al., 2017) and it is now mostly admitted that a single pathogen can trigger infection (Brouwer et al., 2017) , although with a small probability (single hit models). The exponential dose-risk function used in the present paper is clearly without threshold and used widely elsewhere. Together with the quantum concept, it completely takes care of the statistical and probabilistic aspects of the transmission problem.

In the present paper we have shown, with a rather simple model of conservation equations and of dose-risk function, that creation of "patient zero" at large distance from a densely populated and infected area is possible if the target is a large . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted March 18, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 population. Surprisingly, this is well-known and accepted in veterinary science but the link with human airborne transmission, to our knowledge, has not been made. One of the consequences is that the search for "patient zero" could sometimes be meaningless. It also shows that, at the level of a continent, viruses ignore borders and that there is no need of personal travel, to spread infection downwind of a contaminated region. However, due to the importance of climate on the virus lifetime, it has to be kept in mind that the conclusions and hypotheses presented here apply mainly to mid and high latitudes under winter conditions. Acknowledgments:

Bertrand Rowe wishes to thank Dr. Daniel Preston, of Rice University, for very helpful discussions on virus lifetimes and Melinda Rowe for her help as digital designer.

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