# Stochasticity and heterogeneity in the transmission dynamics of SARS-CoV-2

Benjamin M. Althouse  Institute for Disease Modeling, Bellevue, WA  University
of Washington, Seattle, WA  New Mexico State University, Las Cruces, NM
<EMAIL_ADDRESS>these authors contributed equally to this work  Edward A.
Wenger  Institute for Disease Modeling, Bellevue, WA  these authors
contributed equally to this work  Joel C. Miller  School of Engineering and
Mathematical Sciences, La Trobe University, Bundoora, VIC, Australia  Samuel
V. Scarpino  Network Science Institute, Northeastern University, Boston, MA
Department of Marine & Environmental Sciences, Northeastern University,
Boston, MA  Department of Physics, Northeastern University, Boston, MA
Department of Health Sciences, Northeastern University, Boston, MA  ISI
Foundation, Turin, Italy  Antoine Allard  Département de physique, de génie
physique et d’optique, Université Laval, Québec (Québec), Canada  Centre
interdisciplinaire en modélisation mathématique, Université Laval, Québec
(Québec), Canada  Laurent Hébert-Dufresne  Département de physique, de génie
physique et d’optique, Université Laval, Québec (Québec), Canada  Vermont
Complex Systems Center, University of Vermont, Burlington, VT  Department of
Computer Science, University of Vermont, Burlington, VT  Hao Hu  Bill &
Melinda Gates Foundation, Seattle, WA  these authors contributed equally to
this work

###### Abstract

SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe,
infecting millions and killing hundreds of thousands. The basic reproduction
number, which has been widely used and misused to characterize the
transmissibility of the virus, hides the fact that transmission is stochastic,
is dominated by a small number of individuals, and is driven by super-
spreading events (SSEs). The distinct transmission features, such as high
stochasticity under low prevalence, and the central role played by SSEs on
transmission dynamics, should not be overlooked. Many explosive SSEs have
occurred in indoor settings stoking the pandemic and shaping its spread, such
as long-term care facilities, prisons, meat-packing plants, fish factories,
cruise ships, family gatherings, parties and night clubs. These SSEs
demonstrate the urgent need to understand routes of transmission, while posing
an opportunity that outbreak can be effectively contained with targeted
interventions to eliminate SSEs. Here, we describe the potential types of
SSEs, how they influence transmission, and give recommendations for control of
SARS-CoV-2.

Keywords: SARS-CoV-2, COVID-19, basic reproduction number, super-spreading
events

Corresponding author:
Benjamin M Althouse
Institute for Disease Modeling
3150 139th Ave SE
Bellevue, WA, 98005
Phone: (425) 777-9615
Email<EMAIL_ADDRESS>

## Significance statement

SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe,
infecting millions and killing hundreds of thousands. The basic reproduction
number, which has been widely used and misused to characterize the
transmissibility of the virus, hides the fact that transmission is stochastic,
is dominated by a small number of individuals, and is driven by super-
spreading events (SSEs). Many explosive SSEs have occurred in indoor settings
such as long-term care facilities, prisons, meat-packing plants, fish
factories, cruise ships, family gatherings, parties and night clubs. These
SSEs demonstrate the urgent need to understand routes of transmission, while
posing an opportunity that outbreak can be effectively contained with targeted
interventions to eliminate SSEs.

## Introduction

While Severe Acute Respiratory Syndrome coronavirus 2 (SARS-CoV-2) has moved
swiftly around the globe, causing millions of Coronavirus Disease 2019
(COVID-19) cases, much attention has been given to the basic reproduction
number ($\textrm{R}_{0}$), estimated to be roughly between 1.5 and 4 [1]. As
the virus has spread it has become clear that relying on a single value to
characterize the number of secondary infections — and thus estimates of the
transmissibility of this virus — is inadequate to capture the true
transmission dynamics and subsequent risk to humanity [2]. Indeed, a litany of
official reports and anecdotes have identified key superspreading events
(SSEs) which have propelled transmission and infected many.

Earliest notable examples include a Briton who returned from a Singapore
conference and infected 13 other people in a ski resort in the Alps [3]; more
than 70 cases were linked to a Boston Biogen conference within 2 weeks [4];
and the most extreme example so far, South Korea “Patient 31” started a
cluster with more than 5,000 cases in Daegu [5]. A Hong Kong resident visited
the Diamond Princess cruise ship on January 25 and later tested positive. The
number of positive cases on the cruise quickly rose to about 700 people, or
17% of all passengers within 20 days [6]. In Chicago, 15 cases stemmed from
one person at multiple family gatherings [7]. An average of 35% secondary
attack rate was found in multiple reported clusters of 2 to 48 people having
meals together [8].

In March, multiple European countries simultaneously reported unusually large
numbers of imported cases from Ischgl, Austria, a popular ski town.
Epidemiological investigations found infections might have been circulating
since late February, and the individual causing the SSEs might have been a
bartender working in an après ski bar who subsequently was diagnosed positive
[9]. In New York, a lawyer was infected and spread to at least 50 others in
New Rochelle [10]. In Mount Vernon, WA after a choir rehearsal on March 10th,
45 out of 60 choir members fell ill, and 28 of those 45 tested positive for
SARS-CoV-2. No one appeared to be sick during the rehearsal [11].
Investigations revealed that the index patient in this case directly infected
52 others, with an the attack rate of 53.3% to 86.7% [12]. An Indian preacher
died after returning from a trip to Italy and Germany, and attending a large
gathering to celebrate the Sikh Festival of Hola Mohalla. A week later, at
least 19 of his relatives were infected, and it resulted in a quarantine of
40,000 residents in Punjab [13].

In April and May, universal testing at a Boston homeless shelter found 36% of
residents were tested PCR positive [14], and an 87-year-old in Harbin, China
directly infected more than 78 individuals within a few days at home and two
hospitals [15]. Singapore has seen a sharp rise in cases, with the vast
majority (88%) being linked to dormitories. S11, a 10,000 person capacity
dormitory, has the largest cluster in Singapore, with more than 2,200
infections [16]. More than 100 cases were traced back to nightclubs in Seoul
that were visited by a young man who was later tested positive [17]. In
Chennai, India, the Koyambedu vegetable market has emerged as a hotspot, more
than 2000 cases have been traced to the market [18].

These accounts in addition to the many examples of SSEs in long-term care
facilities [19], prisons [20, 21], meat processing facilities [22], and fish
factories [23] demonstrate the central role played by SSEs on the transmission
dynamics – and subsequent epidemiological control of – SARS-CoV-2. Here, we
describe the potential types of SSEs, how they influence transmission, and
give recommendations for control of SARS-CoV-2.

## Superspreading events, stochasticity, and negative binomially-distributed
secondary infections

Many classical models in epidemiology either assume or result in a Poisson
distribution of secondary infections per infected individual. Because Poisson
distributions have the same mean and variance, they often fail to capture the
relevant features of SSEs. As a result, SSEs are now commonly modeled using a
negative binomial (NB) distribution of secondary infections per infected
individual [24]. NB distributions can be parameterized with a mean (thought of
as $\textrm{R}_{0}$) and a dispersion parameter, $k$, where smaller values of
$k$ indicate a longer tail (over-dispersion). When $k$ is close to zero, even
with a high $\textrm{R}_{0}$, most individuals give rise to zero or one
secondary infections, and few give rise to many infections – a so-called ‘long
tail’ of infections. However, as $k$ grows larger, a NB distribution
approaches a Poisson distribution (becoming exactly Poisson when $k\to\infty$)
and the effect of SSEs on the epidemic decreases. In Figure 1 we show a
schematic of a NB branching process, and distribution of NB and Poisson
distributions of secondary infections under the same $\textrm{R}_{0}$. Earlier
transmission modeling studies suggest the offspring distribution of SARS-
CoV-2, the etiological agent of COVID-19, is highly over-dispersed with $k$
comparable to SARS-CoV-1 [25].

An outbreak dominated by SSEs and a NB distribution of secondary infections
with small $k$ has distinct transmission features as compared to a Poisson
distributed, ‘mean-field’ outbreak with the same $\textrm{R}_{0}$. First,
secondary infections will be overdispersed, such that early dynamics are more
stochastic and it is less likely that an outbreak will grow large. In Figure 2
we show an example by utilizing a stochastic branching process model with both
Poisson and SARS-CoV-1 like NB distribution ($k=0.16$) under the same mean
$R_{0}=2.6$ [26], with different population sizes ranging from small clusters
of $10$ like households to large ones of $10^{6}$ like city-wide. We observe
the difference of transmission dynamics in 6 generations – 24-36 days using a
4-6 day serial interval [27, 28, 29, 30]. In all cases where population size
$\geq 100$, 63% of outbreaks have no secondary infections, and 77% of
outbreaks have less than 10 total infections and do not establish ongoing
transmission, compared with 7% with no secondary infections and 11% with less
than 10 total infections of simulations for a Poisson model. This is due to
the over-dispersion of the SARS-CoV-1 like NB distribution: as the probability
of giving rise to zero or one additional case increases, onward transmission
becomes less likely to establish. Second, if the outbreak takes off and
increases beyond one of two SSEs, i.e., often only a few dozen cases, the
dynamics begin to show stable exponential growth, with a growth rate
approaching that of a model with the same $\textrm{R}_{0}$, but a Poisson
distribution of secondary infections per infected individual, i.e., a NB with
$k\gg 0$. Third, once the outbreak takes off, it will appear more explosive
(represented in estimated $\textrm{R}_{0}$ based on case counts) in the first
few generations when SSEs will generate the vast majority of secondary
infections, making it possible to spin out large infection clusters in few
numbers of generations, whereas a Poisson model cannot. An example is shown in
Figure 2 where the outbreak appears more explosive (purple line vs blue line)
if we only consider outbreaks that took off, also consider the outbreak aboard
the Diamond Princess Cruise where 135 cases were seen within 5 days [6].
Fourth, and importantly, to establish and stabilize exponential growth of an
outbreak, a continuous fuel of SSEs is necessary and an outbreak can be
brought largely under control – the effective reproduction number
$\textrm{R}_{\textrm{eff}}$ is significantly reduced when SSEs which drive
transmission are eliminated.

Figure 1: Poisson and negative binomially-distributed (NB) secondary
infections. Figure shows a schematic of a NB branching process, and example
distributions of NB and Poisson distributions of secondary infections. Both
distributions have $\textrm{R}_{0}$ = 2.6 [26]and the NB has dispersion
parameter $k=0.16$ [25] with 100,000 draws. Figure 2: Example trajectories of
NB and Poisson branching processes. Figure shows example trajectories (in
number of active infections vs generation) of NB and Poisson branching
processes, and cumulative infection sizes after 6 generations of spread. Both
simulations start with 1 infection, have the same $\textrm{R}_{0}$ = 2.6. For
NB branching process we assume dispersion parameter $k$ = 0.16, same as SARS-
CoV-1 [24, 31]. We run all simulations 10,000 times. Dashed red lines
represent theoretical values $I=R_{0}^{n}$ where $I$ is number of active
infections and $n$ is number of generations. Solid blue lines are the mean
values of all simulations including those that have not taken off, which
overlap with the theoretical values when the susceptibles are not depleted.
Solid purple lines are the mean value for simulations that took off, and the
outbreaks appear more explosive in the first few generations in the NB
simulations. Both number of active cases and cumulative infections are in
$\log_{10}$ scale.

## Characterizing super spreading events

The total number of secondary infections from an infected individual is based
on two variables: number of susceptible contacts (potential infectees) and
probability of infection per contact. SSEs can occur from either an infectious
individual having close contacts with many susceptible individuals and/or
having a higher probability of transmission per contact, or it can be an
opportunistic phenomena where number of contacts and/or probability of contact
is high in an unusual way, like festivals, bars, or social gatherings. There
are at least four types of SSEs:

  * •

Biological. Individuals with a higher probability of transmitting per contact.
These may be hard to identify a priori; for infections with most pathogens,
pathogen loads may vary over many orders of magnitude across individuals [32,
33]. For SARS-CoV-2 individual-level viral loads are dependent on time since
onset, and might be associated with demographics like age and disease severity
[34]. The temporal profile of SARS-CoV-2 viral load peaks at or just before
the onset of symptoms and decreases quickly to near the PCR detection
threshold within a week [29, 34, 35]. Large heterogeneities in viral load – up
to 8 $\log_{10}$ differences – exist between individuals [36].

  * •

Behavioral / social. Individuals causing SSEs may have a higher number of
susceptible contacts per person. Numerous studies have demonstrated marked
differences in individual contacts by profession and over time.

  * •

High-risk facilities and places such as meat-packing plants, workers’
dormitories, prisons, long term care facilities, or health care settings. The
nature of interactions in these places seem to repeatedly place individuals at
higher risk of acquiring and transmitting infection. Importantly, this driver
of SSEs can lead to spillover into the larger community. Controlling a broader
outbreak will be very difficult if there is a focal hotspot which is
continually seeding new transmission chains, as seen in other respiratory
pathogens such as tuberculosis [37, 38].

  * •

‘Opportunistic’ scenarios. The first scenario is when larger numbers of
individuals temporarily cluster, and even with an average probability of
transmission per contact, people are briefly far above their ‘average’ number
of susceptible contacts. The second scenario is probability of transmission
per contact is temporarily increased in an unusual way, such as singing or
frequent loud speaking. These two opportunistic scenarios are more frequently
seen in outbreaks at night clubs, cruise ships, crowded public transportation,
parties, choirs, or other mass gathering events.

In those cases where an individual’s connectivity plays a role, it is
generally expected that these individuals will also be disproportionately
likely to become infected as well. These may be easier to identify in advance,
but the other cases may be more difficult to prevent.

The stochastic characteristics of early transmission resulting in a dichotomy
of frequent extinctions with rare, but explosive outbreaks is reflected in a
few examples. As of May 16, 2020, in Ohio, only 3 out of 16 jails with
positive cases had large outbreaks of attack rate larger than 10% [39]. In
King County, WA, a total of 105 long-term care facilities reported positive
cases, but only 19 of them has 5 or more deaths indicating potentially large
outbreaks within facilities [40]. Among a total of 47 cruise ships reporting
positive cases or COVID-19 related deaths, only a few had notable large
outbreaks, i.e. Diamond Princess, Ruby Princess, Grand Princess, Celebrity
Apex, Horizon, Greg Mortimer, and Costa Atlantica [41].

In January and February, initial clusters of explosive outbreaks did not take
off first at the expected dense metropolitan areas and air transit hubs (such
as Seoul, Tehran, Paris, London, or Frankfurt), but instead started in
neighboring small cities: Daegu in South Korea, Gangelt in Germany, Qom in
Iran, and Lombardy in Europe. This can partially be explained by considering
that outbreaks are much more likely to occur when seeded by an SSE, and while
large population centers will have more introductions than any given smaller
city, there are many small cities, and the initial large outbreaks occur in
whichever locations the first SSEs occur. Therefore, incidence data can look
more random and relevantly unpredictable than expected if we only use
population density and air traffic data as sole predictors.

Besides stochastic SSEs, there are other factors that might contribute to such
patterns. For example, a recent study found epidemic intensity of COVID-19 is
strongly shaped by crowding, suggesting that there might be a spatial
hierarchy of multiple population clusters in large cities, and transmission
might need to try getting past a few of these clusters to start a large
outbreak, which is therefore more difficult [42]. In contrast, in small cities
people are more well-mixed and outbreaks might be easier to start.

This perspective suggests that we need to respond to all identified outbreak
locales under the assumption that an SSE is occurring. If we only find the
ones well established, then we miss out on potentially controlling outbreaks
before they get out of control. With a strong surveillance system, we may be
able to find some outbreaks early enough and might underestimate their future
potential if we assume the growth will be average. Those outbreaks that do not
go extinct on their own also tend to create larger epidemics. Altogether, the
impact of heterogeneity highlights the importance of early outbreak detection
and of local interventions.

## Where do super spreading events take place?

Since the beginning of the outbreak many types of SSEs have been reported with
increasing frequency. Although the data are still scarce, some patterns have
started to emerge on where and under what circumstances SSEs occur: closed
environments, environments with poor ventilation, crowded places, and long
durations of potential exposure all correlate with emergence of SSEs. In
Japan, looking at clusters of transmission, researchers found closed
environments facilitate secondary transmission of SARS-CoV-2 and promote SSEs;
the odds ratio of SSEs in closed environments was as high as 32.6 compared
with those in open spaces [43].

These SSE ‘hotspots’ are both a source of infection for the community as well
as potential targets for intervention. Hotspots are emerging as seeding
infection in small metro areas as large outbreaks have occurred in meat
packing plants and prisons. Many clusters were linked to a wide range of
mostly indoor settings, like households, public transport, hospitals, parties,
bars, elderly care centers, and schools [44, 45].

Because they play an important role in the spread of infection, hotspots pose
an opportunity for surveillance and control: focusing on facilities and
activities known to sustain hotspots, such as healthcare facilities, nursing
homes, prisons, meat-packing plants, homeless shelters, schools, mass
gatherings, as well as those places with closed, poorly circulated
environments, can provide efficient ways to identify potential SSEs before
they happen, therefore, potentially reducing a substantial amount of
transmission in the population.

Figure 3: Controlling outbreaks and the effect of ‘cutting the tail’. (A)
Values of $\textrm{R}_{\textrm{eff}}$ under different thresholds of maximum
number of secondary infections, and the probability of keeping the maximum
number of secondary infection thresholds. We compare three scenarios with
$\textrm{R}_{0}$ = 1.5, 2.0 and 3.0, where lower $\textrm{R}_{0}$ represents
the consequence of population-wide mild to moderate non-pharmaceutical
interventions. The NB distribution is truncated at the max number of secondary
cases, and the distribution is re-normalized after the truncation. Contours
highlight $\textrm{R}_{\textrm{eff}}$ = 1 boundary. Lower $\textrm{R}_{0}$
facilitates the extinction of outbreak, and increases the probability of
success controlling the outbreak, especially when the probability of reducing
hotspot transmission is low. If the vast majority of transmission inside
hotspots can be eliminated, under all $\textrm{R}_{0}$ scenarios a control
target of less than 10 secondary infections can bring
$\textrm{R}_{\textrm{eff}}$ to close to unity. (B) Probability of an outbreak
successfully taking off with one seed infection under NB and Poisson
distributions, as well as a NB distribution with cutoff number of secondary
infections of 10 and 100% efficiency. The stochastic branching process
simulations are run 10,000 times with $\textrm{R}_{0}$ ranging from 0.8 to
3.0. The success of outbreak is defined as having more than 20 infections at
generation 6. Dispersion parameter $k$ for NB is 0.16.

## Controlling outbreaks with targeted interventions: ’cutting the tail’

As seemingly rare SSEs drive the growth trajectory of an epidemic, trimming
the long tail of large secondary infections is important for outbreak control
and suppression (whether through targetted interventions or broad social-
distancing). We demonstrate this effect in the contour plots in Figure 3(A) by
randomly drawing the values of secondary infections from a NB distribution but
if the value exceeds a cutoff threshold, then an efficiency parameter (from
0.5 to 1) gives the probability that the value is redrawn until it does not
exceed the threshold. Thus we assume that the behavior of most individuals
does not change but we remove a fraction of the SSEs determined by our control
efficiency. Using this intervention model under $\textrm{R}_{0}$ = 3, and only
targeting a small number of individuals with larger than 10 secondary
infections (about 10% of all secondary infections) with 50% to 100%
efficiency, the mean $\textrm{R}_{\textrm{eff}}$ is reduced to 2.06 and 1.09,
respectively.

We also present two scenarios assuming a mild to moderate non-pharmaceutical
interventions (NPIs) or behavioral changes bringing population-wide
$\textrm{R}_{0}$ down to 2 or 1.5, as well as with the targeted intervention
on SSEs truncating number of secondary infections. With $\textrm{R}_{0}$ =
1.5, and a cutoff of $10$ secondary infections the overall
$\textrm{R}_{\textrm{eff}}$ is reduced to 0.85 and 1.21 under 100% and 50%
efficiency. It appears the ‘cutting the tail’ interventions are more effective
and impactful than reducing population-wide $\textrm{R}_{\textrm{eff}}$: if
probability of success is large, $\textrm{R}_{\textrm{eff}}$ is under or
around unity if the maximum secondary infections are truncated at 10. These
results suggest stakeholders of different types of hotspots should strive to
achieve the goal of controlling number of secondary infections to at least
less than 10 with industry-specific mitigation efforts. This effect is also
seen in Figure 3(B), looking at the probability of launching large outbreaks
by branching process simulations similar as before. Results show how the
probability of successfully launching a large outbreak – defined as having
more than 20 infections at generation 6 – decreases with
$\textrm{R}_{\textrm{eff}}$ ranging from 0.8 to 3.0. With the same
$\textrm{R}_{0}$, by ‘cutting the tail’ the probabilities of starting large
outbreaks are much lower. These initial results suggest that the collapse of a
transmission chain is happening more frequently once the long tail is gone,
because there are not enough SSEs to fuel its continuous growth.

Efforts should be spent to understand how to reduce transmission in the four
types of SSEs. For the biological type, it is important to understand the
characteristics of individuals causing SSEs. For example, what individual-
level attributes and behaviors lead to SSEs, and how to identify them quickly?
As viral loads decrease over time and most transmissions are front-loaded,
timely identification and/or isolation becomes crucial. If future studies
reveal meaningful viral load differences by demographics – for example a
difference by age – and that there exists meaningful differences in
transmission across viral loads, a risk classification algorithm can be used
together with individual contact networks to stratify individuals, and remind
them the potential of causing SSEs as they are reported sick and need to be
tested. PCR test results could potentially include a rough category of
transmissibility by cycle threshold values, for example, reporting a high
transmission risk for those with cycle thresholds less than 20. This might
help people avoid crowded events and reduce their contacts. Environmental
surveillance might play an important role in identifying the hotspots if viral
load in stools from individuals causing SSEs are also orders of magnitudes
greater [46]. For the behavioral/social type, a complete contact network for
an individual can be estimated using surveys or cell phone tracking and used
to identify two kinds of people – those with a high degree meaning lots of
close contact per day, and those with high centrality, meaning they are
crucial connecting different population clusters. Personalized messages or
tests could be targeted at these individuals stressing the need for better
infection control. For both high-risk facilities and ‘opportunistic’
scenarios, to effectively prevent transmission in hotspots, it is important to
understand the indoor transmission routes of SARS-CoV-2, which might include
droplets and aerosols through coughing, sneezing, and speaking [47, 48, 49],
fecal-oral [50, 51, 52] or indirectly through surfaces [53]. A recent study
estimates 1 minute of loud speaking generates at least 1,000 virion-containing
droplet nuclei and could remain airborne for more than 8 minutes [49],
suggesting a possible mechanism of creating large exposure and starting SSEs.
Contact tracing is a useful tool to reveal the relative importance of
different transmission routes in a variety of hotspots, for example, in
restaurants [54], call centers [55], choirs [12], and churches [56]. More of
these real-world examples will allow policymakers to evaluate the
effectiveness of different control measures – for example, reducing crowd
density, fever screening, rearranging seating layout, requiring masks, good
ventilation – in eliminating major transmission routes in confined spaces.

## Stochasticity, superspreading events and the future of SARS-CoV-2

Multiple lines of evidence at the individual- and population-level strongly
indicate the role of SSEs in the transmission dynamics of SARS-CoV-2 and that
we should not overlook the heterogeneity in numbers of secondary infections
[57]. Our mental picture should not be that most people transmit to two or
three other people, but instead a small number of infections dominate the
transmission while most others fail to have secondary infections. The
distribution of $\textrm{R}_{0}$ is over-dispersed with a high probability of
extinction on the lower end, and a long tail on the higher end.

Outbreaks will be less likely to take off because of the high probability of
extinction. At the early stage we will see more randomness and stochasticity,
and it seems more explosive with huge number of cases reported in the first
few generations. But once it takes off, it still becomes a stable exponential
as per classic deterministic models. Behavioral change, mild to moderate non-
pharmaceutical interventions to lower population-wide
$\textrm{R}_{\textrm{eff}}$, as well as other factors such as population
density [58] and climate [59] might be more impactful, as a lower population
$\textrm{R}_{\textrm{eff}}$ increases the extinction probability and increases
the success probability of ‘cutting the tail’. Therefore, all measures that
could potentially reduce $\textrm{R}_{0}$ should be considered and
implemented, even if some only have a minor impact.

While SSEs are fueling this outbreak, we have an opportunity to take advantage
of this heterogeneity in transmission, and use it to risk-stratify populations
and locations for public health interventions and interrupt future SSEs. Novel
methods are needed to quickly predict, identify, or isolate individuals /
hotspots with the potential for causing SSEs. It may prove difficult to
identify and isolate individuals with the potential for causing SSEs and
infectious individuals still transmit. This is compounded with the sizeable
proportion of pre-symptomatic, as well as asymptomatic or minimally
symptomatic individuals who can actively transmit [60, 61, 62, 63]. If SSE
predictors cannot be identified, then every infectious individual has the
opportunity to cause a SSE if exposed to sizable susceptible populations.
Therefore, it is crucial to understand types of hotspots and patterns of
transmission for each type, as interventions might have to focus on all
hotspots at high risk for SSEs and limiting gatherings at these places, and/or
through rapid-and-extensive testing and contact tracing (both traditional and
digital) to identify pre-symptomatic and asymptomatic people [64]. Contact
tracing efforts should have an explicit goal to understand types of
transmission and hotspots, so that the characterization of transmission could
be used to adapt and prioritize other recommendations such as masks and mass
gatherings. Further, it thus remains important to be cautious in reopening
populations undergoing cordons sanitaires until transmission routes in
different types of hotspots are well understood, or when safe and effective
COVID-19 treatments and vaccines are available.

## Acknowledgements

The authors thank Philip Welkhoff, Mike Famulare, Anna Bershteyn, and Lowell
Wood for insightful discussions and comments. B.M.A. and E.A.W. are supported
by Bill and Melinda Gates through the Global Good Fund. J.C.M. is supported by
startup funding from La Trobe University. S.V.S. is supported by startup funds
provided by Northeastern University. A.A. acknowledges financial support from
the Sentinelle Nord initiative of the Canada First Research Excellence Fund
and from the Natural Sciences and Engineering Research Council of Canada
(project 2019-05183). L.H.-D. acknowledges support from the National
Institutes of Health 1P20 GM125498-01 Centers of Biomedical Research
Excellence Award. The funders had no role in study design, data collection,
data analysis, the decision to publish, or preparation of the manuscript.

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