# Triggering of an Epidemic Outbreak via Long-Range Atmospheric Transport of Bio-Aerosols—Application to a Hypothetical Case for COVID-19

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## Abstract

**:**

## 1. Introduction

## 2. Long-Distance Transport of Both Inert and Bio-Aerosols

_{2.5}) from biomass burnings [26], and pollen transport from eastern North America to Greenland ([27] and references therein). These phenomena are well studied and documented, and their importance has been evaluated. Except for pollen, these examples refer to non-biological, 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 and to demonstrate that long-distance travel can give rise to physical effects from these particles. Aerosols containing biological matter either made of living or non-living components are usually named bio-aerosols.

_{50}values (50% tissue culture infective dose) allowing the risk of infection to be assessed.

## 3. Outdoor Airborne Transmission of Pathogens: Extension of a Wells–Riley Type Model

#### 3.1. Basic Concepts in (Indoor and Outdoor) Airborne Transmission

^{−3}units), ${n}_{q}\left(\overrightarrow{r,}t\right)$, the inhaled dose X during a time of exposure t, can be written as:

^{3}/h in the present investigation). Note that this definition of the dose does not require a homogeneous distribution of quanta in space. Only ${n}_{q}\left(\overrightarrow{r,}t\right)$ at the inhaled location (mouth and nostrils) has to be considered. Note also that due to the extremely low concentration of quanta in air, ${n}_{q}\left(\overrightarrow{r,}t\right)$ is not really a continuous function of $\overrightarrow{r,}t$ (since a number of viruses is of course an integer) but can be treated as such due to the stochastic character of the problem and the search for a statistical solution. When the quantum concentration can be considered as being constant during the time of exposure, then the dose X expression simplifies to:

#### 3.2. Box Model of Outdoor Transmission

#### 3.3. Possible Airborne Epidemic Triggering via Long-Range Transmission

_{1}, it is possible to quantify the quantum concentration n

_{1},

_{q}(L

_{1}) following Equation (8) taking into account that at the upstream border of the source n

_{1,q}(x

_{1}= 0) = 0. The downwind border of the source also coincides with the upwind border of the “no man’s land” section (hereafter box 2). However, as explained further in Section 4.2, the dispersive height H in box 2 (H

_{2}) is higher than in box 1 (H

_{1}). This impacts the initial quantum concentration n

_{2,q}(x

_{2}= 0) by a factor of H

_{1}/H

_{2}at the upstream border of box 2 such that:

_{2,q}(x

_{2}= 0) = H

_{1}/H

_{2}× n

_{1,q}(L

_{1})

_{2,q}(x

_{2}) in box 2 is then governed only by the virus lifetime according to Equation (9) since ${D}_{I}=0$ in box 2. This leads to a new value n

_{2},

_{q}(L

_{2}) at the downstream border L

_{2}of box 2. Again, this border coincides with the upstream border of the target area, box 3. At this interface, however, the dispersive height is not modified compared to box 2 (H

_{3}= H

_{2}) since H

_{2}is already taken as an upper limit of the ABL thickness (see Section 4.2). The quantum concentration entering box 3 is then n

_{3,q}(x

_{3}= 0) = n

_{2,q}(L

_{2}). The quantum concentration n

_{3,q}(x

_{3}) can be considered as spatially and temporally constant within box 3 provided that:

- The pathogen lifetime is clearly larger than the hydrodynamic time within the target depth, which is typically around 10–20 km.
- The width of the target is less than the width of the source.
- The emission source rate and meteorology do not change significantly during the time of exposure.

## 4. Results for a Hypothetical Case of Long-Range Transmission of COVID-19 from Southern England to Northern France

#### 4.1. General Considerations

#### 4.2. Details of the Long-Range Model of Transmission for the Present Hypothetical Case

_{1}of 45 km, an area inside which we assume a population N

_{p}(x

_{1}= 45 km) of 11 million people.

_{1},

_{q}(L

_{1}) 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 relative humidity (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 (see discussion in Section 5) is expected to be much longer than the atmospheric transport (hydrodynamic) time. Therefore, viral inactivation, as discussed in Section 5.2, will only occur indoors, via thermal effects at rather low RH. Indoor air is continuously renewed as contaminated air is evacuated outdoors with a characteristic time equal to $1/ACH$ where $ACH$ is the number of times that the total air volume in a room is completely removed and replaced in an hour. Therefore, the effect of viral inactivation indoors prior to evacuation can be taken as a reduction of the quantum emission rate per infector used in Section 3.2 following the formula:

_{1}= 45 km and state D in Figure 2), the population within the source is assumed as $1.1\times {10}^{7}$, the wind velocity taken as 30 km/h, the width of the source as W = 40 km (which influences the density of infectors if Equation (10) is used in place of (12)), and the quantum production rate of an infector as 10 h

^{−1}. The numerical application leads to ${n}_{1,q}\left(45\mathrm{km}\right)=7.15\times {10}^{-6}{\mathrm{m}}^{-3}$ assuming a proportion of infected persons of r = 0.03 in the greater London area.

_{2},

_{q}(L

_{2}) = n

_{2},

_{q}(x

_{2}= 0). Using a conservative estimate for $H$of 1000 m, a value corresponding to a common upper value of ABL thickness for neutral or stable conditions [46] results in a numerical value of ${n}_{3,q}\left({x}_{3}=0\right)$ of $2.15\times {10}^{-6}{\mathrm{m}}^{-3}$ at the upstream border of one of our targets.

^{6}and 1.2 × 10

^{6}people, respectively. We assume that the wind direction is the same as the direct path between the source and the target, a dominant direction in wintertime, which roughly corresponds to a wind direction from the west/northwest (respectively 288 and 294 degrees). As before, we also assume a wind velocity of 30 km/h, which is only slightly higher than the mean wind velocity in February/early March [55]. Note again that both target areas have a width across the wind less than that of the source. Table 1 gathers the main characteristics of the three boxes as depicted in Figure 3. Table 2 summarizes the assumed values of various parameters leading to a statistical number of primary cases. Since this number appears to be a few units in the frame of our assumptions, it clearly reveals the potential for an infection being triggered through long-range transportation of airborne viruses.

## 5. Discussion

#### 5.1. Validity of the Atmospheric Box Model

^{−14}m

^{−3}over northeastern France. Using the effective quantum production rate q

_{eff}value derived from Table 2, the unit emission HYSPLIT values (through the expression: N

_{P}× r × q

_{eff}× 24) yield a quantum concentration of 3.1 × 10

^{−6}quantum m

^{−3}, which is very close to the upstream box model value of 2.1 × 10

^{−6}. Due to the line source configuration, lateral dispersion along the centerline would be negligible, and the concentration results would primarily depend upon the vertical mixing. An examination of the diagnostic vertical mass profile after 12 h (not shown) indicates that 94% of the mass was in the first 1200 m above ground and 99% was within the first 1500 m, consistent with the well-mixed box model assumptions.

#### 5.2. The Question of the Virus Lifetime Indoor and Outdoor in Bio-Aerosol Form

^{−1}), and it is generally admitted [61,63] that, for a given value of RH, it follows an Arrhenius law with temperature:

^{21}min

^{−1}, respectively.

#### 5.3. The Very Low Dose Question

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Vertical dispersion length for Gaussian plumes. Classification of atmosphere state: A: Extremely unstable; B: Moderately unstable; C: slightly unstable; D: neutral; E: slightly stable; F: moderately stable.

**Figure 3.**Model of three boxes between greater London and northern France. Note that the upstream border of box 3 is dependent on the considered target (i.e., either Dunkerque or Lille).

Box 1 | Box 2 | Box 3 | |
---|---|---|---|

Length: L (km) | 45 | 150/230 ^{a} | --- ^{b} |

Width: W (km) | 40 | 40 | <40 |

Dispersive height: H (m) | 300 | 1000 | 1000 |

Wind speed V_{∞} (km/h) | 30 | 30 | 30 |

n_{q} (quanta/m^{3}) | 7.1 × 10^{−6 c} | 2.1 × 10^{−6} | 2.1 × 10^{−6} |

^{a}: Depending on the considered target, either Dunkerque or Lille.

^{b}: This length is not fixed since it is not useful for the present estimation.

^{c}: Value at the downstream end of the box.

**Table 2.**Possible number of primary cases created (per day) via the long-distance transport of aerosols. London area population of 11 million; wind velocity: 30 km/h; exposure of 24 h; proportion of possible infectors in greater London: r = 3%; quantum production rate q = 10 h

^{−1}/infector.

Dunkerque | Lille | |
---|---|---|

Distance from London, center to center (km) | 180 | 244 |

Population (10^{6}) | 0.2 | 1.2 |

Hydrodynamic time (h) | 5.0 | 7.7 |

Upstream quantum concentration (m^{−3}) | 2.1 × 10^{−6} | 2.1 × 10^{−6} |

Dose for 24 h | 2.6 × 10^{−5} | 2.6 × 10^{−5} |

Probability of infection P_{t} | 2.6 × 10^{−5} | 2.6 × 10^{−5} |

Number of primary cases | 5 | 31 |

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## Share and Cite

**MDPI and ACS Style**

Rowe, B.R.; Mitchell, J.B.A.; Canosa, A.; Draxler, R.
Triggering of an Epidemic Outbreak via Long-Range Atmospheric Transport of Bio-Aerosols—Application to a Hypothetical Case for COVID-19. *Atmosphere* **2023**, *14*, 1050.
https://doi.org/10.3390/atmos14061050

**AMA Style**

Rowe BR, Mitchell JBA, Canosa A, Draxler R.
Triggering of an Epidemic Outbreak via Long-Range Atmospheric Transport of Bio-Aerosols—Application to a Hypothetical Case for COVID-19. *Atmosphere*. 2023; 14(6):1050.
https://doi.org/10.3390/atmos14061050

**Chicago/Turabian Style**

Rowe, Bertrand R., J. Brian A. Mitchell, André Canosa, and Roland Draxler.
2023. "Triggering of an Epidemic Outbreak via Long-Range Atmospheric Transport of Bio-Aerosols—Application to a Hypothetical Case for COVID-19" *Atmosphere* 14, no. 6: 1050.
https://doi.org/10.3390/atmos14061050