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\begin{document}

\title{SARS-CoV-2 transmission: The cigarette smoke analogy}



\author[1]{N Ghadiri}%
\author[1]{m.ali}%
\author[1]{M Ghadiri}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{June 10, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Despite the current presumption that SARS-CoV-2 is transmitted by
droplets, the reported evidence of the presence of the virus in air
suggests that droplets dry fast producing clusters of virus, which are
most penetrative through commonly available fabric filters to the
public.~

~We consider droplets corresponding to a typical bimodal distribution of
droplets of sneeze and calculate the time taken for them to evaporate to
their ultimate size for two virus loadings.~ The dry virus clusters
resulting from fine droplets of the first mode have a size distribution
comparable to cigarette smoke. There are currently no masks available to
the public which can filter these aerosols efficiently, and even FFP
grade masks have low filtration efficiency for these particle sizes.

The implications are that safe distance is considerably greater than the
generally-recommended two metres, as the virus remains fully suspended
in the air.~ Secondly, mask filters available to the public are
inefficient in preventing SARS-CoV-2 spreading, as it could penetrate
through them. The rapid spread of the virus is indicative of the ease
with which it spreads through the air, and apart from wearing a high
efficiency mask, the only safe prevention is to self-isolate at home.%
\end{abstract}%



\sloppy


\par\null

\section*{Authors}

{\label{457509}}

Nima Ghadiri\textsuperscript{1}, Muzammil Ali\textsuperscript{2,3},
Mojtaba Ghadiri\textsuperscript{3}

\textsuperscript{1} \emph{Moorfields Eye Hospital, London, UK}

\textsuperscript{2} \emph{Fluid Comp Ltd, Leeds, UK}

\textsuperscript{3} \emph{School of Chemical and Process Engineering,
the University of Leeds, Leeds, UK}

\section*{Introduction}

{\label{984013}}

\subsection*{}

{\label{101027}}

\subsection*{A Historical Approach to
Pandemic}

{\label{101027}}

In 1025, Avicenna (A Persian polymath) completed his Canon of Medicine,
within which he speculated that some diseases are spread by
microorganisms.~He suggested the method of isolating people for 40 days
to prevent transmission of diseases. For this, he was reprimanded by
religious authorities, but his recommendations were taken up by Venetian
traders who were travelling on the Silk Road. They took his knowledge
back to Italy and from this the word ``quarantena'' (``\emph{the forty}
'') was born.~ A similar situation prevails today, as economic and
financial pressures are thwarting science and logic recommendations,
increasing the likelihood of a devastating second phase.~ ~ ~ ~

\subsection*{Transmission of the SARS-CoV-2
Virus}

{\label{986778}}

The transmission of SARS-CoV-2 virus is regarded to take place mainly
through droplets generated when an infected person coughs, sneezes, or
speaks. It is commonly considered that these droplets are too heavy to
remain suspended in the air and fall quickly on to grounds and surfaces.
On this basis, the infection is considered to take place by breathing in
the virus in the vicinity of an infected person (typically within 1
metre) or by contact with contaminated surfaces and then touching eyes,
nose or mouth. However, the recent study of Wuhan
hospitals\textsuperscript{(1)} reports the airborne virus cluster size
is in the range 0.01 to 10 \selectlanguage{greek}µ\selectlanguage{english}m, with the mode of the distribution in the
range 0.25 to 1 \selectlanguage{greek}µ\selectlanguage{english}m, depending on the hospital zone. This strongly
suggests that SARS-CoV-2 virus could spread primarily through
dispersion, as an aerosol which travels through air at a size of less
than a few micrometres.~For the fines fraction corresponding to the mode
of the size distribution reported above, there are currently no masks
commonly available to the public which can filter out these aerosols
efficiently. In laboratories, particle technologists use FFP1, FFP2 and
FFP3 grade filters when handling benign powders. However, all three FFP
grade masks have low filtration efficiency for the particle size range
around 0.09-0.16 \selectlanguage{greek}µ\selectlanguage{english}m\textsuperscript{(2)}.~There is extensive literature
on filter performance evaluation, and as commonly available filters are
all fabric filters, they all suffer from poor performance for the most
penetrative particle size. This depends on the filter material, but is
typically in the sub-micrometre range. So, considering the average size
of a single virus, measured by transmission as well as by scanning
electron microscopy\textsuperscript{(3)}, ranging from 0.08-0.22 \selectlanguage{greek}µ\selectlanguage{english}m,
they could penetrate through filters even when in clusters if they are
suspended in air.

\subsection*{The Cigarette Smoke
Analogy}

{\label{479913}}

Cigarette smoke provides an analogue for spreading of the SARS-CoV-2
virus, as its particle size distribution is in a similar range of
0.1-0.4 \selectlanguage{greek}µ\selectlanguage{english}m\textsuperscript{(4)}. So virus particles can travel in a
similar way when emitted from an infected person by sneezing or
coughing. The current two-metre social distancing is presumably based on
droplet size and principles of droplet spreading. However, droplets
evaporate quickly and it is evident that we are facing a most
penetrating aerosol. The particle size distribution of droplets coming
out of the nose and mouth has been extensively characterised. Both
unimodal and bimodal distributions have been reported with the modes
around 500 \selectlanguage{greek}µ\selectlanguage{english}m for the former and 90 \selectlanguage{greek}µ\selectlanguage{english}m and 600 \selectlanguage{greek}µ\selectlanguage{english}m for the latter, and
with geometric mean values of 72 and 386 \selectlanguage{greek}µ\selectlanguage{english}m\textsuperscript{(3)}.~ An
early report indicates an even much finer distribution in the range 4-8
\selectlanguage{greek}µ\selectlanguage{english}m\textsuperscript{(5)}. So it is of interest to estimate how long it
takes for the droplets to shrink to their ultimate size encasing the
virus.

Considering the uncertainty about the viral load which causes infection,
and the number of copies of virus prevailing in the droplets, we
consider a number of cases and investigate the size of virus clusters on
drying: (1) a load of 1000 virus particles inside each droplets; (2) a
concentration of 10\textsuperscript{7.2}/ml of virus particles in
respiratory liquid which is sprayed out on sneezing or
coughing\textsuperscript{(6)}; (iii) an infectious dose in the range of
1.95 \selectlanguage{ngerman}× 10\textsuperscript{3}~and 3.0 × 10\textsuperscript{3}~viral
particles, based on Influenza A virus\textsuperscript{(7)}.

\section*{Results}

{\label{767344}}\par\null

\subsection*{\texorpdfstring{\emph{(i) Constant load of 1000 virus
particles per
droplet}}{(i) Constant load of 1000 virus particles per droplet}}

{\label{450823}}

We consider two droplets corresponding to the bimodal distribution with
the corresponding geometric mean sizes of 72 \selectlanguage{greek}µ\selectlanguage{english}m and 386 \selectlanguage{greek}µ\selectlanguage{english}m for the two
modes\textsuperscript{(3)}, with 1000 virus particles inside. The
droplets are ejected into quiescent air at 50 m/s (as commonly reported
for sneezing) at an average relative humidity of 65\%, common in the UK
and an ambient temperature of 50\selectlanguage{ngerman}°F (10°C). The time taken for the
droplets~to evaporate to their ultimate size is
calculated\textsuperscript{(8)}, considering no internal resistance to
mass transfer and drying rate controlled by the external mass transfer
film. Water is considered as the evaporating material and assuming that
the presence of the virus does not hinder the drying of droplets. The
drying rate of the two droplets sizes is shown in Figure 1. The
evaporation time is 20 and 510 s for the two droplet sizes,
respectively, and the ultimate cluster sizes in both cases are around
1\selectlanguage{greek}µ\selectlanguage{english}m. Considering a smaller virus loading of 100 viruses, the ultimate
size will be obviously smaller and is around 0.46 \selectlanguage{greek}µ\selectlanguage{english}m.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure-1/Figure-1}
\caption{{\emph{Drying profile of two droplets each containing 1000 viruses.}
{\label{182412}}%
}}
\end{center}
\end{figure}

\subsection*{}

{\label{574462}}

\subsection*{\texorpdfstring{\emph{(ii) Constant concentration of
10\textsuperscript{7.2}/ml of virus particles in respiratory
liquid}}{(ii) Constant concentration of 107.2/ml of virus particles in respiratory liquid}}

{\label{574462}}

As the droplets originate from a pool of liquid inside the body, this
concentration corresponds to viral load of an infected person with
common flu. In this case a constant volume fraction of the virus is
considered for all droplets. The number of viruses in each droplet then
depends on the droplet size. Taking the above virus concentration
(corresponding to a virus volume fraction of 4.2\selectlanguage{ngerman}×10\textsuperscript{-9})
and the bimodal droplet size distribution reported by Han et
al.\textsuperscript{(3)}, as shown in Figure 2, this represents about 1
virus in the lower droplet size (50 \selectlanguage{greek}µ\selectlanguage{english}m) and 6900 in the largest droplet
size. Following the drying time shown in Figure 1, the size distribution
of the dried virus clusters shifts drastically to the left and is in the
range 0.1 to 1.8 \selectlanguage{greek}µ\selectlanguage{english}m, as shown in Figure 2. Reports on the droplet size
distribution produced by sneezing, coughing, shouting and speaking show
a great variation. If we consider an earlier study by
Duguid\textsuperscript{(5)} in which he reports the most common droplet
size is in the range 4 to 8 \selectlanguage{greek}µ\selectlanguage{english}m, then the time taken to reach the same
ultimate size is even much shorter, around 0.05 to 0.22 s.~ So the
calculated size range of the virus clusters formed by the smaller
droplets is very similar to that of the size distribution of cigarette
smoke particles.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure-2/Figure-2}
\caption{{~\emph{Measured droplet size distribution of sample sneeze reported by
Han et al.}\textsuperscript{\emph{(3)}} \emph{and calculated dry
particle size distribution considering different virus loadings in
droplets.} ~
{\label{403751}}%
}}
\end{center}
\end{figure}

\par\null

\selectlanguage{ngerman}\subsection*{\texorpdfstring{\emph{(iii) Infectious dose in the range
1.95 × 10\textsuperscript{3}~- 3.0 × 10\textsuperscript{3}~viral
particles based on Influenza
A}}{(iii) Infectious dose in the range 1.95 × 103~- 3.0 × 103~viral particles based on Influenza A}}

{\label{428966}}

Here we consider the upper limit, i.e. 3000 particles in 72 \selectlanguage{greek}µ\selectlanguage{english}m droplet
(giving rise to 10\textsuperscript{10.18} virus copies per ml), as this
produces larger clusters of virus and hence less penetrating through
filters. Applying the drying kinetic model to droplet size distributions
given by Han et al.\textsuperscript{(3)}, the resulting particle size
distribution of the dried particles are predicted to be in the range
0.84-18 \selectlanguage{greek}µ\selectlanguage{english}m, as also shown in Figure 2.

These particles are actually so fine they remain airborne due to ambient
air turbulence. The trajectory of an initial 72 \selectlanguage{greek}µ\selectlanguage{english}m droplet is shown in
Figure 3, considering gravity settling under quiescent air conditions.
The droplet is ejected horizontally at 50 m/s at 1.6 m above the ground
level into the stationary air. The terminal velocity of the particle
after reaching a virus packing volume fraction of 0.6 is
8\selectlanguage{ngerman}×10\textsuperscript{-5} m/s and at 20 s the dry particle is 0.38 m
above the ground, having a diameter of 1.36 \selectlanguage{greek}µ\selectlanguage{english}m. It reaches the ground
asymptotically with time, and will require further 1.3 hours before
settling onto the ground. In contrast, a droplet with an initial size of
386 \selectlanguage{greek}µ\selectlanguage{english}m (corresponding to the larger mode of the droplet size
distribution shown in Figure 2) settles onto the ground after 1.2 s of
ejection from 1.6 m height and under the same conditions as above, and
travels a horizontal distance of about 2.3 m. The diameter of the
droplet when reaching the ground is 384.3 \selectlanguage{greek}µ\selectlanguage{english}m, i.e. very little drying
takes place for droplets of this size before settling.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Figure-3/Figure-3}
\caption{{~\emph{Plot of height above the ground versus time for an initial
droplet diameter of 72 \selectlanguage{greek}µ\selectlanguage{english}m with a load of 3000 viruses.}
{\label{543952}}%
}}
\end{center}
\end{figure}

\section*{Conclusion}

{\label{180240}}

In summary, large droplets settle quickly in the vicinity of the
ejection point with little drying, whilst small droplets, less than
about 100 \selectlanguage{greek}µ\selectlanguage{english}m, quickly dry and remain airborne. This has two major
implications for public health: firstly, safe distance is considerably
longer than the generally-recommended two metres, as the virus remains
fully suspended in air. Secondly, mask filters available to the public
are inefficient in~preventing SARS-CoV-2 spreading, as it is most
penetrative through fabric filters. They can retain the droplets from
nasal and throat secretions on coughing and sneezing, but cannot protect
a person from inhaling nearby viruses released into the atmosphere. The
deadliness of this virus has drawn deservedly great attention of the
scientific community on its spreading mechanism, with the latest
re-evaluation of the basis adopted by the World Health Organization and
other agencies~\textsuperscript{(9)}.

\section*{Declarations}

{\label{713874}}

\textbf{Acknowledgements}

The authors are thankful to Dr Paul Meyer and Dr Poul Bach for their
valuable comments.

\textbf{Contributorship statement}

NG and MG first conceived the study. MG was in charge of overall
planning. MA carried out the calculations and data visualization. MG and
MA analysed the data. NG and MG wrote the manuscript with input from all
authors.

\textbf{Conflict of Interest Disclosures} : None reported.

\textbf{Funding/Support:} No funding reported.

\textbf{References:}

\begin{enumerate}
\tightlist
\item
  Liu Y, Ning Z, Chen Y et al. Aerodynamic Characteristics and RNA
  Concentration of SARS-CoV-2 Aerosol in Wuhan Hospitals during COVID-19
  Outbreak. \emph{bioRxiv.org preprint. Published Online First: 5 March
  2020} . doi: https://doi.org/10.1101/2020.03.08.982637
\item
  Lee S, Hwang D, Li H et al. Particle Size-Selective Assessment of
  Protection of European Standard FFP Respirators and Surgical Masks
  against Particles-Tested with Human Subjects. \emph{Journal of
  Healthcare Engineering} 2016: pp. 1-12. doi:
  https://doi.org/10.1155/2016/8572493
\item
  Han ZY, Weng WG, Huang QY. 2013 Characterizations of particle size
  distribution of the droplets exhaled by sneeze. \emph{J R Soc
  Interface} 2013;10: 20130560. doi:
  https://doi.org/10.1098/rsif.2013.0560
\item
  Sahu S, Tiwari M. Bhangare R et al. Particle Size Distribution of
  Mainstream and Exhaled Cigarette Smoke and Predictive Deposition in
  Human Respiratory Tract\emph{. Aerosol Air Qual. Res} . 2013;13:
  324-332. doi: http://www.aaqr.org/doi/10.4209/aaqr.2012.02.0041
\item
  Duguid JP. The size and the duration of air-carriage of respiratory
  droplets and droplet-nuclei \emph{Epidemiology and Infection,}
  1946:44(6), pp. 471-479. doi:
  https://doi.org/10.1017/S0022172400019288
\item
  Oh M. Viral Load Kinetics of MERS Coronavirus Infection \emph{New
  England Journal of Medicine}, 2016: 29 (375) pp.1303-1305. doi:
  10.1056/NEJMc1511695
\item
  Nikitin N, Petrova E, Trifonova E et al. Influenza Virus Aerosols in
  the Air and Their Infectiousness. \emph{Advances in Virology} 2014:
  859090. doi: 10.1155/2014/859090.
\item
  Ali M, Mahmud T, Heggs PJ et al. Modelling drying and crystallisation
  in a single droplet. EuroDrying 2017, 19-21 June, Liege, Belgium.
\item
  Bourouiba L. Turbulent Gas Clouds and Respiratory Pathogen Emissions,
  JAMA Published Online First: \emph{20 March 2020. doi} :
  https://doi:10.1001/jama.2020.4756
\end{enumerate}

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