The Effect of Molar Weight on Airborne Infectiousness of Coronavirus
N. KHAJOHNSAKSUMETH
Department of Mathematics, Faculty of Science, Mahidol University,
Rama 6 Rd., Bangkok 10400,
THAILAND
also with
Centre of Excellence in Mathematics, MHESI,
Bangkok, 10400,
THAILAND
Abstract: - To design an effective ventilation system in healthcare settings, understanding the ventilation
pattern is necessary. In this research, we have investigated the effect of the weight of airborne coronavirus on
the spread of the COVID-19 infection. We have mathematically modeled the distribution of the virus as a
transport of concentration, including the Navier-Stoke equation and continuity equation. The finite element
method was applied to drive the simulations. The numerical results have been obtained and analyzed in this
report.
Key-Words: - Air Borne Infection, Mathematical Modelling, Finite Element, COVID-19, Numerical
Simulation, SARS-CoV-2, Ventilation Flow, Transmission, Airborne Pathogens.
Received: April 11, 2023. Revised: November 27, 2023. Accepted: January 2, 2024. Published: March 2, 2024.
1 Introduction
In late December 2019, the coronavirus disease
(COVID-19), caused by SARS-CoV-2, began to
infect the population and spread around the world.
By March 2020, there were more than 210000
patients infected with the virus in 168 countries,
including nearly 9000 deaths, [1]. Because of the
official recognition of its global rapid spread and
impact of the virus, COVID-19 was declared a
pandemic by the World Health Organization
(WHO), [2]. Several researchers have reported on
their findings regarding different aspects of the
pandemic, [3], [4], [5], [6] and, [7].
COVID-19 is reportedly transmitted through
infectious airborne particles and droplets, [8].
Infected persons release respiratory fluids carrying
the SARS-CoV-2 virus into the air when they
breathe, speak, cough, or sneeze. These particles
vary in size, covering a wide range of sizes so that
the infection can be easily transmitted from one
patient to another. If a susceptible person inhales
infectious airborne particles and those particles are
lodged in a suitable location within the respiratory
tract, that person is at risk of contracting the disease.
Particularly in indoor settings such as patients'
rooms in hospitals, very fine droplets and particles
easily spread through the air throughout the room or
living space, [9]. Moreover, droplets remain
airborne and allow the virus to accumulate, leading
to a higher concentration of infected droplets in the
room. When doctors or nurses need to enter the
patients' room to follow up the patient’s condition,
they must practice greater care for themselves;
otherwise, they risk easy infection from these
droplets or particles. To protect against infection
with COVID-19, or any airborne infectious diseases,
it is very important to gain more information about
the dynamic behavior and characteristics of the
spread of the virus.
Many researchers have examined the relationship
between ventilation and control of airflow directions
in buildings, and the transmission and spread of
airborne disease, [8], [10]. Additionally, researchers
have studied the effect of saliva and mucus particle
size on pathogen generation through the action of
breathing, coughing, and sneezing by an infected
person, [11], [12]. The majority of human exhaled
droplets are within the sub-micrometer size range,
[13]. Furthermore, researchers have investigated the
evaporation process within indoor environments and
the influencing factors, including the ventilation
rate, indoor air relative humidity, indoor air
temperature, and so on [14], [15], [16], [17].
Recent research, [18], provided valuable insights
into the characterization of SARS-CoV-2,
enhancing our understanding of COVID-19 and its
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behavior. Based on this knowledge, our study
focuses on investigating the effect of the weight of
airborne infectious particles on dispersion patterns,
which, to our knowledge, has not been carried out
before. This knowledge is essential for designing
the current ventilation systems in healthcare settings
effectively.
2 Mathematical Model
To describe the transport of SARS-CoV-2 in a
patient's room in a hospital, a transport model for
the dynamical system is developed. In what follows,
the state variables and parameters are as defined in
Table 1. The incompressible laminar airflow and
propagation of SARS-CoV-2 in the patients' room
region
are governed by the following initial
boundary value problem (IBVP) in the 3-
dimensional - space.
The Navier-Stokes equation (1) along with the
continuity equation (2) are applied to describe the
movement of fluid in laminar flow. Additionally, in
studying gaseous and liquid mixtures, it is necessary
to consider the molecular interactions between all
species, upon which the properties of mixtures rely.
This includes incorporating the transport equation
for concentrated species (3).
Laminar Flow
T
p
u u I u u F
, (1)
0
u
. (2)
Transport of Concentrated Species
i i i
R
ju
. (3)
Based on the relative mass flux due to
molecular diffusion using a Fick’s law
approximation, the diffusive flux can be described
as the mixture-averaged diffusion as follows:
,
m m T
n
i m i i i i c i i
n
MT
D D D
MT
jj
, (4)
where
1
mi
ik
ki ik
Dx
D
,
1
i
nii
MM
,
,m
i
c i i k k
kn
MDx
M
j
.
and
denotes the gradient operator. The subscript
i, equaling 1 or 2, corresponds to species 1 or
species 2, respectively, for which air is species 1
and infectious particles constitute species 2, and
similarly for the subscript k being 1, 2 in the above
equations.
Wall Conditions
Under a no-slip condition,
wall 0u
m/s (5)
Inlet Condition
We apply a normal air inflow velocity for each inlet
on the ceiling by
inlet 5u
m/s (6)
Table 1. Definitions of state variables and
parameters in the IBVP described in the text
Variable/
Parameter
Definition
,,x y z
Spatial coordinates
t
Time
, 1,2
k
xk
The molar fraction of species k,
where k = 1 corresponds to air, and
k = 2 corresponds to infectious
particles
T
Temperature
( , , , )x y z tuu
velocity vector at
( , , , )x y z t
u
The norm of u
Air density
m
Mixture density
p
Pressure
Dynamic viscosity
F
External force
i
j
Diffusive flux of species i, where i
= 1 corresponds to air, and i = 2
corresponds to infectious particles
i
R
Rate expression describing
production or consumption of
species i
i
Mass fraction of species i
i
M
The molar mass of species i
n
M
Mean of molar mass
,ci
j
Relative mass flux of species i
ik
D
Multicomponent Fick diffusiveness
i
Mass fraction of species i
1, 2ww
J
Mass flow rate
0, 2w
Initial value of the mass fraction of
infectious particle
Outlet Condition
At the outlet, we assume that the pressure is
constant, that is,
outlet 5p
atm (7)
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Boundary Conditions
On the boundary
, we define
5u
m/s (8)
Under the no-flux assumption,
.0
i
nj
(9)
The nostril inflow from a patient breathing out is
defined as a periodic sine function as follows:
1, 2 4sin( ) 4
ww
Jt
kg/s (10)
For the outflow, we assume that
.0
m
ii
D
n
(11)
Initial Values
Initially, the mass flow rate of infectious particles is
0, 2 0.1
w
(12)
The mixture specification is given in Table 2.
Table 2. Mixture parameter values for coronavirus
simulation based on [19], [20], [21]
Parameter
Description and Unit
Value(s)
m
Mixture density (kg/m3)
1
𝑀𝑤1
The molar mass of air
(kg/mol)
28.9647
𝑀𝑤2
The molar mass of
infectious
particle (kg/mol)
1 and 5
3 Domains of Computation
In this work, we establish the study scenario in
which the patients' room, measuring 10 × 4 × 2.5
meters, is shared by four individuals. The room is
equipped with a closed door measuring 1 × 2
meters, four air inlets on the ceiling at the end of the
bed, each measuring 0.5 × 0.5 meters, and four
outlets on the wall above each patient's head, each
measuring 0.25 × 0.25 meters, designed to release
airborne pathogens from the room.
Moreover, each patient is provided a bed
partition to separate them individually, and the
nostril is designed on each patient’s head to release
the mixture of fluid flow inlet containing infectious
droplets, as shown in Figure 1 (a).
In Figure 1 (b), the meshing of the
computational domain, necessary for the application
of the finite element approach, is shown. In
addition, the locations where data points for
collecting computational values from the finite
element analysis are situated and labeled as points
A, B, C, D, and E, respectively, may be seen in
Figure 2 and Figure 3.
Point A is situated above the leftmost patient’s
breast as shown in Figure 2 (a). Point B is
positioned above the leftmost patient’s feet as
shown in Figure 2 (b). Point C is located below,
near the outlet flow of the leftmost patient as shown
in Figure 3 (a). Point D is placed at the door to
determine if a doctor or nurse enters the room as
shown in Figure 3 (b). Lastly, point E is placed at
the inlet flow of the second leftmost patient in the
room as shown in Figure 3 (c).
(a)
(b)
Fig. 1: Domain of computation: (a) computational
domain, showing the locations of flow inlets and
outlets, and (b) meshing domain. In (a), the patient’s
head has been enlarged to show the position of the
patient’s nostrils which is the flow inlet of the
mixture fluid
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(a)
(b)
Fig. 2: Locations of points, in red, for collected
computation values labeled as (a) point A, and (b)
point B.
4 Numerical Results and Discussion
In this section, the influence of the molar mass of
airborne pathogens on the molar concentration of air
and airborne pathogens at various locations in the
patients' room is numerically investigated. In Figure
4, Figure 5, Figure 6, Figure 7 and Figure 8 the
numerical results are displayed for a scenario in
which the molar mass of airborne pathogens is 1
kg/mol.
Here, the concentrations of both air and airborne
pathogens at five different locations are presented.
We observe that at point A, located near the inflow
of airborne pathogens, the air concentration is very
low to begin with, while the airborne pathogen
concentration is initially equal to 1. The air
concentration then gradually increases until it
reaches approximately 35.5, while the airborne
pathogen concentration decreases to nearly zero.
(a)
(b)
(c)
Fig. 3: Locations of points, in red, for collected
computation values labeled as (a) point C, (b) point
D, and (c) point E
In Figure 9, Figure10, Figure11, Figure12 and
Figure 13, the molar concentration graph is shown,
A
B
C
D
E
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depicting the scenario in which the molar mass of
the airborne pathogens is equal to 5 kg/mol.
Fig. 4: Graph of molar concentration between air
and airborne pathogens (molar mass = 1 kg/mol) at
point A
Figure 4 and Figure 9 depict the molar
concentration of air and airborne pathogens at Point
A, located on the breast of the leftmost patient, with
molar masses of 1 kg/mol and 5 kg/mol,
respectively. We observe that the concentration of
airborne pathogens with a molar mass of 1 kg/mol is
initially higher than the concentration of airborne
pathogens with a molar mass of 5 kg/mol.
Subsequently, their concentrations drop to zero
around t = 400 seconds, while the concentration of
airborne pathogens with a molar mass of 5 kg/mol
oscillates before also dropping to zero. Meanwhile,
we observe that the molar concentration of air in
both cases is initially 31 mol/m3, increasing to
approximately 34.5 mol/m3.
Fig. 5: Graph of molar concentration between air
and airborne pathogens (molar mass = 1 kg/mol) at
point B
Figure 5 and Figure 10 depict the molar
concentration of air and airborne pathogens at Point
B, located at the end of the bed of the leftmost
patient, with molar masses of 1 kg/mol and 5
kg/mol, respectively. We observe that the molar
concentration of airborne pathogens with a molar
mass of 1 kg/mol is initially five times higher than
the concentration of airborne pathogens with a
molar mass of 5 kg/mol. The concentrations of both
molar masses oscillate around their initial values up
to approximately 1600 seconds in time before the
concentrations of these airborne pathogens drop,
indicating that the droplets are suspended in the air
longer than at Point A.
Figure 6 and Figure 11 display the molar
concentrations of air and airborne pathogens at
Point C, situated below and near the outlet flow of
the leftmost patient, with molar masses of 1kg/mol
and 5 kg/mol, respectively. The results reveal
similar concentration trends to those observed at
Point A. However, the airborne pathogens persist in
the air for approximately 600 seconds in time,
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indicating a longer duration of suspension compared
to Point A.
Fig. 6: Graph of molar concentration between air
and airborne pathogens (molar mass = 1 kg/mol) at
point C
Comparing the concentration of airborne
pathogens at Point D, located at the door for
entrance into the room, as shown in Figure 7 and
Figure 12 for molar masses of 1 kg/mol and 5
kg/mol, respectively, reveals the following trends.
For the molar mass of 1 kg/mol, the concentration
slightly increases above the initial concentration
value for approximately 1000 seconds and then
gradually decreases below the initial concentration
thereafter.
However, for the molar mass of 5 kg/mol, the
concentration of airborne pathogens fluctuates and
increases over time. This indicates that pathogens
accumulate and persist at this point.
Next, we consider the concentration of air and
airborne pathogens, with molar masses of 1 kg/mol
and 5 kg/mol respectively, at Point E, located at the
pure air inlet flow. The results show that for the
molar mass of 1 kg/mol, the concentration of
airborne pathogens fluctuates, decreasing until
approximately 1000 seconds in time, and then
increasing afterward. However, for the molar mass
of 5 kg/mol, the concentration of airborne pathogens
at this point continues to increase. This indicates
that pathogens accumulate and persist at this point.
Fig 7: Graph of molar concentration between air and
airborne pathogens (molar mass = 1 kg/mol) at point
D
In Figure 14, the pattern of the normalized vector
field presents the steady-state airflow. The air flows
from each inlet on the ceiling towards the patients
and exits through the outlets positioned above their
heads.
Finally, Figure 15 and Figure 16 show the
patterns of the normalized mass fraction gradient, at
time t =1800, of airborne pathogens, simulated with
a molar mass of 1 kg/mol and 5 kg/mol,
respectively. We observe that the mass fraction
gradient of airborne pathogens exhibits random
directions and spreads throughout the entire
patient’s room.
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Fig. 8: Graph of molar concentration between air
and airborne pathogens (molar mass = 1 kg/mol) at
point E
Fig. 9: Graph of molar concentration between air
and airborne pathogens (molar mass = 5 kg/mol) at
point A
Fig. 10: Graph of molar concentration between air
and airborne pathogens (molar mass = 5 kg/mol) at
point B
Fig. 11: Graph of molar concentration between air
and airborne pathogens (molar mass = 5 kg/mol) at
point C
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Fig. 12: Graph of molar concentration between air
and airborne pathogens (molar mass = 5 kg/mol) at
point D
Fig. 13: Graph of molar concentration between air
and airborne pathogens (molar mass = 5 kg/mol) at
point E
Fig. 14: Graph of the normalized vector field at the
steady state of the air
Fig. 15: Graph of the normalized mass fraction
gradient of airborne pathogens, simulated with
molar mass = 1 kg/mol, at time t =1800
Fig. 16: Graph of the normalized mass fraction
gradient of airborne pathogens, simulated with
molar mass = 5 kg/mol, at time t =1800
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5 Conclusion
In this work, we have examined the impact of the
molar mass of airborne pathogens on the molar
concentrations of air and airborne pathogens. We
have observed that a greater molar mass of airborne
pathogens leads to reduced dispersion of these
pathogens. Specifically, our work has treated the air
movement carrying the airborne virus as a
particulate fluid flow which we believe was able to
simulate the real scenarios more closely.
For possible future research endeavors, we can,
for example, investigate how different arrangements
of furniture or room designs, or different locations
of the points of outlet or inlet flow would lead to
different distribution patterns and how we can
utilize such information to discover the optimum
utility of space at minimum risk of infection. We
can also study the impacts of variations in the air
inflow velocity. The methodology utilized in this
study can be applied to gain valuable information
regarding other settings of enclosed regions under
the influx of other infectious airborne pathogens.
Acknowledgement:
The author would like to thank Emeritus Professor
Dr. Yongwimon Lenbury of the Centre of
Excellence in Mathematics, Thailand, and Associate
Professor Benchawan Wiwatanapataphee of the
Department of Mathematics and Statistics, Curtin
University, Australia, for their valuable suggestions
and advice.
References:
[1] Shaylika Chauhan, Comprehensive review of
coronavirus disease 2019 (COVID-19),
Biomed J. 2020 Aug; 43(4): 334-340.
[2] WHO Director-General’s opening remarks at
the media briefing on COVID19 -March
2020, [Online]. https://www.who.int/director-
general/speeches/detail/who-director-general-
s-opening-remarks-at-the-media-briefing-on-
covid-19---11-march-2020 (Accessed Date:
February 20, 2024).
[3] Babashov, S., Predicting the Dynamics of
Covid-19 Propagation in Azerbaijan based on
Time Series Models. WSEAS Transactions on
Environment and Development, Vol. 18,
2022, pp. 1036-1048,
https://doi.org/10.37394/232015.2022.18.99.
[4] Makanda, G., A Mathematical Model for the
Prediction of the Impact of
Coronavirus(COVID19) and Social
Distancing Effect, WSEAS Transactions on
Systems and Control, Vol.15, 2020, pp. 601-
613,
https://doi.org/10.37394/23203.2020.15.60.
[5] Smith, C., A Dangerous New Coronavirus
Complication was Discovered-and it never
goes away if you get it.
http://bgr.com/science/ coronavirus-symtoms
complications-diabetes-onset-after-covid-
19/(Last Accessed Dates: 06.03.2022)
[6] Rattanakul, C., Lenbury, Y.,
Khajohnsaksumeth, N. Modchang, C.,
Geometric Singular Perturbation Analysis of a
Multiple Time-scale Model for Diabetes and
COVID-19 Comorbidity, WSEAS
Transactions on Biology and Biomedicine,
Vol. 19, 2022,
https://doi.org/10.37394/23208.2022.19.20.
[7] De Luca, C., Olefsky, J.M., Inflammation and
Insulin Resistance, FEBS Lett., Vol. 582(1),
pp. 97-105. doi:10.1016/j.feslet.2007.11.057.
[8] Y Li, G M Leung, J W Tang, X Yang, C Y H
Chao, J Z Lin, J W Lu, P V Nielsen, J Niu, H
Qian, A C Sleigh, H-J JSu, J Sundell, T W
Wong, P L Yuen, Role of ventilation in
airborne transmission of infectious agents in
the built environment – a multidisciplinary
systematic review, Indoor, 2007 Feb; 17(1):2-
18. doi: 10.1111/j.1600-0668.2006.00445.x.
[9] EPA, Indoor Air and Coronavirus (COVID-
19), [Online].
https://www.epa.gov/coronavirus/indoor-air-
and-coronavirus-covid-19 (Accessed Date:
February 20, 2024).
[10] Melanie D Nembhard, D Jeff Burton, and Joel
M Cohen, Ventilation use in nonmedical
settings during COVID-19: Cleaning protocol,
maintenance, and recommendations.
Toxicology and Industrial Health, 2020, Vol
36(9) 644-653.
[11] Nicas, M., Nazaroff, W.W. and Hubbard, A.
toward understanding the risk of secondary
airborne infection: emission of respirable
pathogens.J. Occup. Environ. Hyg.s, Vol.2,
2005, 143-154.
[12] Morawska, L., Johnson, G.R., Ristovski, Z.D.,
Hargreaves, M., Mengersen, K., Corbett, S.,
Chao, C.Y.H., Li, Y. and Katoshevski, D. Size
distribution and sites of origin of droplets
expelled from the human respiratory tract
during expiratory activities. J.Aerosol Sci,
Vol.40, 2009, 256-269.
[13] Yang, S., Lee, G.W.M., Chen, C.M., Wu,
C.C. and Yu, K.P. The size and concentration
of droplets generated by coughing in human
subjects. J.Aerosol Med.,2007, 484-494.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.9
N. Khajohnsaksumeth
E-ISSN: 2224-2902
91
Volume 21, 2024
[14] Wang, B., Zhang, A., Sun, J., Liu, H., Hu, J.
and Xu, L. Study of SARS transmission via
liquid droplets in air. Trans.ASME, 2005,
Vol.127, 32-38A.
[15] Xie, X., Li, Y., Chwang, A.T.Y., Ho, P.L. and
Seto, W.H. How far droplets can move in
indoor environments – revisiting the Wells
evaporation–falling curve. Indoor, Vol.127,
2007, 211-225.
[16] Wan, M.P. and Chao, C.Y.H. Transport
characteristics of expiratory droplets and
droplet nuclei in indoor environments with
different ventilation air flow patterns.
J.Biomech. Eng. T-ASME, 2007, Vol.129,
341-353.
[17] C.Chen, B.Zhao. Some questions on
dispersion of human exhaled droplets in
ventilation room: answers from numerical
investigation. Indoor Air, 2010, Vol.20, 95-
111.
[18] Liu, J., Chen, X., Liu, Y., Lin, J., Shen, J.,
Zhang, H., Yin, J., Pu, R., Ding, Y., Cao, G.
(2021). Characterization of SARS-CoV-2
worldwide transmission based on evolutionary
dynamics and specific viral mutations in the
spike protein. Infectious Diseases of Poverty,
10(1), 89, https://doi.org/10.1186/s40249-021-
00895-4.
[19] Dërmaku-Sopjani M, Sopjani M. Molecular
Characterization of SARS-CoV-2. Curr Mol
Med. 2021; 21(7):589-595. doi:
10.2174/1566524020999201203213037.
PMID: 33272175.
[20] Unacademy. (2024). What is the molecular
weight of air? [Question & Answer], [Online].
https://unacademy.com/content/question-
answer/chemistry/what-is-the-molecular-
weight-of-air/ (Accessed Date: February 21,
2024).
[21] Rahmstorf, L., Emanuel, K., & Ebi, K. L.
(2021). Global weight of all active SARS-
CoV-2 viruses is between 0.1 and 10
kilograms, [Online].
https://phys.org/news/2021-06-global-weight-
sars-cov-viruses-kilograms.html (Accessed
Date: February 21, 2024).
Contribution of Individual Authors to the
Creation of a Scientific Article
Nathnarong Khajohnsaksumeth: problem
formulation, model selection, and boundary
conditions identification, choice of the numerical
scheme, numerical simulations, writing, and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research has received funding support from the
NSRF via the Program Management Unit for
Human Resource & Institutional Development,
Research and Innovation (grant number
B05F640231). Also, this research is partially
supported by the Centre of Excellence in
Mathematics, Ministry of Higher Education,
Science, Research and Innovation, Thailand (grant
number RG-01-65-01-1).
Conflict of Interest
There is no conflict of interest.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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