Theoretical investigation of pre-symptomatic
SARS-CoV-2 person-to-person transmission in
households
Yehuda Arav
1,*
, Ziv Klausner
1
, and Eyal Fattal
1
1
Department of Applied Mathematics, Israeli Institute for Biological Research, PO Box 19, Ness-Ziona, 7410001
Israel
*
yehudaa@iibr.gov.il
ABSTRACT
Since its emergence, the phenomenon of SARS-CoV-2 transmission by seemingly healthy individuals has become a major
challenge in the effort to achieve control of the pandemic. Identifying the modes of transmission that drive this phenomenon
is a perquisite in devising effective control measures, but to date it is still under debate. To address this problem, we have
formulated a detailed mathematical model of discrete human actions (such as coughs, sneezes, and touching) and the decay
of the virus in the environment. To take into account both discrete and continuous events we have extended the common
modelling approach and employed a hybrid stochastic mathematical framework. This allows us to calculate higher order
statistics which are crucial for the reconstruction of the observed distributions. We focused on transmission within a household,
the venue with the highest risk of infection and validated the model results against the observed secondary attack rate and the
serial interval distribution. Detailed analysis of the model results identified the dominant driver of pre-symptomatic transmission
as the contact route via hand-face transfer and showed that wearing masks and avoiding physical contact are an effective
prevention strategy. These results provide a sound scientific basis to the present recommendations of the WHO and the CDC.
Introduction
The phenomenon of SARS-CoV-2 transmission by pre-symptomatic, otherwise seemingly healthy, individuals poses a major
challenge for policy makers’ efforts to achieve control of the COVID-19 pandemic, as traditional health strategies rely on case
detection through manifestation of symptoms
1
. However, the mechanism that enables this transmission is not fully understood.
Generally, respiratory viruses such as SARS-CoV-2 propagate via four modes of transmission
2
: direct physical contact between
people, indirect physical contact via intermediate object, droplets and droplet nuclei. Transmission by droplets and droplet
nuclei is mediated by virus containing particles that were emitted when a person coughs, sneezes or speaks. The droplets travel
less than 1.5m
3
, due to their size, and settle on the facial membranes of nearby individuals or on surfaces. Droplet nuclei
remain suspended in the air and may infect a susceptible individual once they penetrate the respiratory tract. The commonly
accepted cutoff between droplets and droplet nuclei is 5
µm
2
. However, Xie et al.
3
showed that droplets that are smaller than
approximately 100µm evaporate to their nuclei size before reaching the ground.
The relative contribution of the different modes of transmission in indoor environments is still under debate
4–9
. The
controversy revolves about the relative importance of the droplet nuclei mode of transmission. Several studies have argued that
the transmission of the SARS-CoV-2 virus is mediated primarily by close and unprotected contact (e.g., via physical contact
and droplets)
4–7
, while others have argued that breathing droplet nuclei is the main mode of transmission
8, 9
. The close contact
transmission hypothesis relies on the analysis of COVID-19 cases
6
and the relatively low secondary attack rate (SAR, the
probability of an infected person to infect a susceptible person) of
10%
-
16%
that was observed in households
5, 10–13
. The
droplet nuclei hypothesis relies on several theoretical investigations
8, 9
. The attempts to identify SARS-CoV-2 in air sampling
taken from infection isolation rooms in hospitals and households yield conflicting results. Several studies
14–16
found positive
samples while others
17–19
reported negative air samples ( for example, Dohla et al.,
19
reported negative air samples that were
taken from households).
The aim of this study is to quantify the relative contribution of the different modes of transmission of SARS-CoV-2 to
infection by pre-symptomatic individuals. We focus in this study on the transmission within a household environment, the
venue with the highest risk of infection
5, 10
. The approach taken here is an integrative detailed mechanistic modelling that
describes explicitly the transfer of SARS-CoV-2 between individuals in different modes of transmission, similar to the approach
used by Nicas and Sun
20
and by Atkinson and Wein
21
for quantifying the modes of transmission of respiratory viruses. In
this work we have extended the mathematical framework of Atkinson and Wein
21
to take into account random discrete human
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NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.
actions (such as coughs, sneezes and contact with objects and other people), rather than considering only the mean kinetics.
This was achieved by employing a hybrid stochastic mathematical framework which allows us to explicitly calculate higher
order statistics which are crucial for the reconstruction of the observed distributions. Following this, the model is validated
by reconstruction of observed fundamental attributes of the pandemic, the secondary attack rate (SAR) and the serial interval
distribution. Then, the model is used to assess the contribution of each of the transmission modes as well the effectiveness of
different prevention measures.
Outline of the Mathematical model
The model presented in this study incorporates the processes that influence the number of SARS-CoV-2 virus particles
transferred from a pre-symptomatic infected individual, henceforth the primary, to a susceptible individual, henceforth the
secondary, and the probability to become infected (Figure 1).
Contact transmission begins when the primary touches his facial membranes and, as a result, contaminates his own hands.
Then, the primary transfers the virus either through direct physical contact (Figure 1, mode
1
) or indirectly via small frequently
touched object (fomits), such as a doorknob or a faucet, (Figure 1, mode
2
) to the hands of the secondary. Eventually, the
secondary places his hands into nose, mouth or eyes, which might cause an infection
21, 22
. The droplet and droplet nuclei
modes of transmission (Figure 1, modes
3
and
4
, respectively), begin when the primary coughs, sneezes, or speaks and expels
virus containing droplets. Droplets larger than
100µm
settle by gravity within
1.5m
3
and contaminate large surfaces such
as furniture and table tops (environmental surfaces), while smaller droplets dry out and form droplet nuclei which remain
suspended in the air. As a result, the droplet nuclei may be carried over distances greater than 1.5 m by the air currents of the
room
3
. The deposition of droplets directly on the mocusa of close contacts is a rare event in workplace or household settings
21
.
Therefore, we have considered here only the contamination of environmental surfaces by the droplets after they have settled.
The contaminated areas on the environmental surfaces might also contaminate the hands of the secondary individual when he
touches them. The probability of infection increases with the number of SARS-CoV-2 particles that reach facial membranes of
the secondary individual.
Figure 1. Schematic representation of the modes of transmission from the primary (infector) and secondary (infectee)
individuals. (1) Direct contact (2) Indirect contact via fomites (3) Indirect contact via surfaces (4) droplet nuclei.
The processes described in Figure 1 consist of both discrete random events which are the actions of the individuals, such as
touching each other or touching the facial membranes and continuous events which consist of environmental processes. Hence,
2/15
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we used a hybrid continuous and stochastic-jump framework
23
to describe the dynamics of the transmission and infection
processes using a coupled system of differential equations. The actions of the individuals are described as stochastic jump
Poisson processes, while the environmental processes are described using continuum dynamics. The model explicitly tracks
the dynamics of the concentration on the hands of the individuals (equation 8), the concentration on the fomites (equation
9), concentration on environmental surfaces (equation 11) and the concentration of the droplet nuclei in the air (equation
10). A complete list of the model equations and values of the corresponding parameters are provided in the Methods section.
Since the actions of the individuals are represented as a stochastic process, we have conducted a Monte Carlo simulation in
which multiple realizations were computed to obtain the appropriate ensemble statistics. Using a Monte-Carlo simulation
the probability distributions are embedded in the input parameters directly and allow the comparison of the model results to
observed distributions. Each realization begins when the primary become infected and begins an incubation period whose
duration is drawn from a log-normal distribution with a mean of
5
days and standard deviation (SD) of
0.45
days
24
. The viral
load of the primary increases exponentially with time
21
reaching a maximal level at the end of the incubation period
25
. During
that time, the primary and secondary individuals perform a series of randomized actions such as touching fomites, touching
environmental surfaces, coughing, sneezing, talking, touching each other, or each touching his own face. The probability that
the secondary individual will be infected is determined from his accumulated exposure over a time interval (equation 12) using
the dose-response curve that was reported for SARS-CoV-1
26
(equation 7) and assumed to be similar to SARS-CoV-2. Each
realization ends when the primary develops symptoms, in accordance with the public health policy that isolates the primary at
the onset of symptoms.
We define a reference simulation as a simulation which corresponds to a normal, pre-symptomatic behaviour ( parameters
in Table 1).
Table 1. The hygienic and behavioral parameters of the reference simulation
Parameters Parameter Description Value Unit Reference
τ
social
Person to person physical contact frequency 3 1/d
27
τ
hand− f ace
Rate of face touching 0.2 1/min
22
τ
hand− f omite
Rate of fomite touching 60 1/d
28
τ
hand− f urniture
Rate of furniture touching 1 1/min
22
τ
hand−washing
Rate of hand cleaning 3 1/d
28
τ
f omite−cleaning
Rate of fomite cleaning 2 1/d
28
Results
Validation of the model
A necessary validation criteria for a model such as the one described in this study is to correctly simulate the distribution of the
serial interval and the SAR. The serial interval is the time period between the symptoms’ onset in the primary and the secondary.
Its distribution is closely associated with the estimation of the reproductive number and key transmission variables in epidemic
models and is important for optimization of the length of the obligatory quarantine period and contact tracing strategies
29, 30
.
The serial interval distribution of COVID-19 was estimated in many countries and was usually found to be gamma distributed
with mean between 4.03 to 6.3 days and standard deviation between 3 and 4.2 days (Figure 2A, shaded area)
10, 31–33
.
3/15
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Figure 2.
Model Prediction for the (A) Distribution of the serial index. Shaded area is the bounds of observed data
10, 31–33
(B)
The cumulative SAR over time.
The model prediction for the distribution of the serial interval and the SAR is obtained by conducting a Monte-Carlo
simulation by solving equations 8 to 12 (see the Methods section) for
10000
realizations of the reference simulation (needed for
convergence). Figure 2A compares the model predicted serial interval distribution (red line) with the distributions reported in
the literature. As seen, the model prediction was well between the bounds of the different estimates of this distribution (Figure
2A).
As an additional validation, we compared the model prediction of the SAR to the values reported in the literature. The model
predicts a SAR of
11.5%
in the reference simulation, which is within the reported values ranging between
10 − 16%
5, 10–13
. We
have also analyzed the contagious period of pre-symptomatic patients by examining the cumulative SAR over time (Figure
2B). As seen, the contagious period begins approximately
30
hours before the symptoms’ onset, with increasing probability of
infection as the onset of the symptoms approaches. This result is consistent with the estimation of He, et al.
25
, that inferred
from data of
77
transmission pairs (i.e., primary and secondary) a contagious period of approximately
2
days before symptoms’
onset.
Some of the parameters’ values were obtained from studies that also reported the range of variability of these values.
Therefore, we have performed an extensive sensitivity analysis to check the robustness of the results (see Supplementary
Information). The model’s results remain within the range of the values reported in the literature for the examined range of
parameters.
Modes of transmission in pre-syptomatic cases
Analyzing the realizations of the reference simulation, we have quantified the contribution of the different modes of transmission
to the overall exposure in scenarios where the secondary was infected (Figure 3). Out of the total viral dose that was transmitted
to the secondary,
64.5%
(Inter quartile range, IQR:
55% − 80%
) was received during direct contact events (mode 1) and
26%
(IQR
13% − 32%
) was received during indirect contact via fomites events (mode 2). The contribution of the large droplet route
(mode 3) was negligible while the droplet nuclei transmission (mode 4), contributed
9.5%
(IQR
3.6% − 12%
) of of the total
viral dose. Hence, according to our results, the contact mode of transmission (either direct or indirect) is the dominant mode of
infection, accounting, overall, to the transfer of
90%
of the viral dosage from the primary to the secondary. The main process
that underlies the contact mode of transmission is the hand-face transfer. Therefore, hygienic and behavioral measures that
operate on the elements that constitute the contact processes are expected to significantly reduce the risk of infection. These
will be analyzed in the following section.
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Figure 3.
The Contribution of the different modes of transmission to overall exposure. Box represents the inter-quartile range
(IQR). The whiskers represent the 10th and 90th percentile.
Reducing the risk of infection
The fact that contact transmission is the main route of pre-symptomatic transmission, suggests that the hygienic and behavioral
measures (HBMs) advised to the public should focus on reducing the contamination on the hands or somehow interrupt the
hand-face transfer. We have examined five HBMs: Washing hands, cleaning fomites, avoiding physical contact (i.e., maintaining
social distancing), wearing a mask and gloves. Naturally, conservative precautions measures would be an implementation of all
these measures simultaneously. However, strict adherence to all these HBMs would be difficult to endure and to maintain over
a long period of time. Therefore, we have tried to sort out several combinations of HBMs that should be readily implemented
by the public, while significantly lowering the risk of infection. As the SAR is a proportion, it is appropriate to compare the
HBMs in terms of odds ratio (OR), i.e., the odds that the secondary would be infected when a given combination of HBMs is
taken, compared to the reference scenario in which no HBM is applied. Generally, any HBM that results in OR less than 1
decreases the risk of infection (i.e., provide smaller SAR than the reference)
34
. However, in practice the lower the OR, the
more effective the HBM combination is at lowering the risk. The values brought here are in terms of OR alongside with 95%
confidence interval (95% CI).
Washing hands is known to remove (and also destroy) virus particles from the hands and it is the simplest measure to
implement. Our simulations show that washing hands once every hour rather than
3
times a day, as in the reference simulation
(Table 1), results in OR of 0.72 (95% CI
0.67
-
0.8
) (Figure 4A, column H). This result is consistent with intervention studies
that have shown that increased hand washing decreased respiratory illness by
20%
, albeit different viruses were studied
22
. This
phenomenon seems counter intuitive, as we found that
90%
of the viral dosage is transmitted through the hands and it was
expected that washing it would remove the contamination. In order to understand the reason for the relatively limited effect of
hand hygiene, we have examined the dynamics of the virus concentration on the hands of the secondary individual (Figure 4B).
This concentration exhibits a periodic behaviour, with a period of approximately
30
to
40
minutes, that is governed by contact
events on fomites and the face. Therefore, hand washing is expected to dramatically reduce the risk for infection if it occurs at
at frequency higher than 40 min. Unfortunately, such frequent hand washing is unrealistic.
Cleaning the fomites more frequently reduces the virus repositories that are available. Cleaning of the fomites
10
times a
day rather than twice a day, as in the reference simulation, results in OR of
0.84
(
95%
CI
0.77 −0.92
), similar to washing hands
more frequently (Figure 4A, column F). A combined strategy that consists of frequent hand washing and cleaning fomites does
not decrease the risk considerably and results in OR of 0.70 (95% CI 0.63 − 0.76).
Wearing a surgical mask or a respirator may reduce the hand-face transfer of virus particles
35
as well as the inhalation
exposure to viral particles. Although it is difficult to asses the reduction of the transfer coefficient from hand to facial membranes
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