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Liu et al.

Modeling the Risk of In-Person Instruction during the COVID-19 Pandemic

Modeling the Risk of In-Person Instruction during the COVID-19 Pandemic

Brian Liu††thanks: Equal contribution. Operations Research Center,
Massachusetts Institute of Technology<EMAIL_ADDRESS>Yujia
Zhang††footnotemark:  Center for Applied Mathematics, Cornell University,
<EMAIL_ADDRESS>Shane G. Henderson School of Operations Research &
Information Engineering, Cornell University<EMAIL_ADDRESS>David B. Shmoys
School of Operations Research & Information Engineering, Cornell University,
<EMAIL_ADDRESS>Peter I. Frazier School of Operations Research & Information
Engineering, Cornell University<EMAIL_ADDRESS>
During the COVID-19 pandemic, implementing in-person indoor instruction in a
safe manner was a high priority for universities nationwide. To support this
effort at the University, we developed a mathematical model for estimating the
risk of SARS-CoV-2 transmission in university classrooms. This model was used
to design a safe classroom environment at the University during the COVID-19
pandemic that supported the higher occupancy levels needed to match pre-
pandemic numbers of in-person courses, despite a limited number of large
classrooms. A retrospective analysis at the end of the semester confirmed the
model’s assessment that the proposed classroom configuration would be safe.
Our framework is generalizable and was also used to support reopening
decisions at Stanford University. In addition, our methods are flexible; our
modeling framework was repurposed to plan for large university events and
gatherings. We found that our approach and methods work in a wide range of
indoor settings and could be used to support reopening planning across various
industries, from secondary schools to movie theaters and restaurants.

decision analysis, education systems operations, epidemiology, stochastic
model applications, simulation applications

## Introduction

During the initial period of the COVID-19 pandemic, from March 2020 to May
2021, many universities switched entirely to virtual instruction because of a
fear that a large outbreak in the student population could quickly overwhelm
local healthcare capacity and endanger students, employees, and residents who
live near campus (Walke et al. 2020, Cipriano et al. 2021). These
interventions were not without costs, as they harmed the social well-being and
educational outcomes of college students (Lee et al. 2021, Dorn et al. 2020)
and damaged the local economies of college towns (Payne 2020, Sullivan 2020).
Moreover, prolonged campus shutdowns negatively impact student learning (Dorn
et al. 2020) and the livelihoods of those who work around campus (Sullivan
2020). Therefore, safely reopening college campuses for in-person instruction
is important for universities nationwide for potential future pandemics.

The University in Ithaca, New York (name removed to support blinding) was a
leader in safely re-opening for residential instruction (Frazier et al. 2022).
In the Fall of 2020, over 75% of all students enrolled at the Ithaca campus
returned for in-person instruction (Rosenberg 2020b) and extensive testing,
contact tracing, and classroom de-densification protocols resulted in fewer
than 200 COVID-19 cases throughout the semester out of a population of over
18,000 students (Cornell COVID-19 Modeling Team 2021b). During this semester,
however, while two-thirds of students had at least one in-person class
(Rosenberg 2020b), only 30% of all courses were held in-person (Srivastava and
Rosenberg 2020). A mandated six-feet distancing requirement, set by the New
York State Department of Health (New York State Department of Health 2021b),
constrained the number of students that could be accommodated in each
classroom. For example, a class with 200 students required a classroom that
seated 1600 people. This mandate dramatically reduced the number of rooms on
campus that could accommodate a large class of students. Furthermore, rooms
with poor air circulation were excluded from usage, and only a limited number
of classrooms could be retrofitted with HVAC to enhance ventilation due to
high operational and energy costs. As a result, it was impossible to schedule
many classes in person.

While the Fall 2020 semester proved that the University could safely reopen
campus, and the level of in-person instruction during that semester was
substantially above that offered by many other universities at the time (Patel
and Lee 2022), the number of in-person classes remained significantly below
pre-pandemic levels. This continued in Spring 2021: the number of in-person
courses offered remained lower than pre-pandemic levels and again most
students who returned to campus enrolled in hybrid schedules and took the
majority of their classes virtually (Rosenberg 2020a).

The University started to gauge the possibility of offering the full roster of
courses in-person when planning the Fall 2021 semester, since much of the
community would have been vaccinated at the onset of the semester. However,
the University faced considerable uncertainty about the safety of offering a
full roster of in-person courses. The level of safety associated with using
all classrooms, not just rooms with high-quality ventilation, and filling them
at greater density was not well understood. Adding to this challenge, the
SARS-CoV-2 Delta variant emerged in the summer of 2021 with increased
infectivity compared to the original strain and resistance to vaccines
(Callaway 2021). Figure 1 shows a timeline of how the emergence of the Delta
variant coincided with our planning period for the Fall 2021 semester.

Figure 1: Timeline of significant events during the planning period for the
Fall 2021 Semester.

Note: The Delta variant was declared a variant of concern in the United States
halfway through the planning process and influenced our modeling approach and
decisions.

To respond to this uncertainty, our team developed a modeling framework to
estimate the risk of COVID-19 transmission in classrooms during the Delta wave
of the pandemic. Using this framework, we determined that fully dense
classrooms with mandatory masking and without special ventilation or
restrictive seating plans would result in minimal risk to students, graduate
student instructors, faculty, and teaching staff throughout the semester.
Thus, safe in-person instruction could be offered without further enhancing
ventilation in classrooms or developing fixed seating plans for each class,
interventions that would have been difficult to implement. Following our
recommendations, the University proceeded with dense in-person classes in Fall
2021, and empirical evidence aggregated at the end of the semester suggested
that classroom transmission was extremely rare (Cornell University COVID-19
Response 2021).

Our modeling framework can be used to support the design of interventions
during respiratory disease outbreaks in any context with indoor seating, from
K-12 schools to restaurants and movie theaters. Our framework is flexible and
allows a user to estimate the risk of virus transmission in rooms with various
configurations. In addition, our framework can evaluate the effectiveness of
various interventions, such as vaccines, masking, and ventilation. These
functionalities make our framework a valuable tool for modeling indoor
transmission.

### Contributions to the University

Our modeling framework and analysis guided decision makers at the University
in planning for the difficult task of resuming normal teaching operations for
the Fall 2021 semester. We used our framework to recommend an implementable
classroom configuration that allowed classes to meet in full density while
ensuring safety for students and instructors. We also communicated our
modeling approach with transparency and rigor through published analyses and
town hall meetings (Cornell COVID-19 Modeling Team 2021a, Cornell University
2021a). As a result, the university was able to more effectively communicate
and inform instructors, teaching assistants, students, and the broader
community that returning to normal teaching operations had minimal risk.

The classroom policies that we recommended, namely mandatory masking with no
distancing or additional ventilation requirements, sufficiently prevented
COVID-19 transmission in classrooms. Our retrospective analyses (including
contact tracing of COVID-positive students and employees, adaptive testing of
students in classrooms with positive cases, and genetic sequencing of viral
samples) at the end of the semester found that student travel and social
events were much more influential drivers of COVID-19 spread on campus
compared to classroom transmission (Cornell University 2021b).

Beyond classrooms, we also used our modeling framework to evaluate the risk of
holding and attending other university events, such as Homecoming, concerts,
holiday events, sporting events, and graduation. These analyses informed
executive-level decisions on which events to hold throughout the Fall 2021
semester. As a result, we found our modeling approach to be useful for
evaluating the risk of virus transmission in many indoor settings.

Our modeling framework and analyses were widely distributed and influenced
return-to-campus decisions at other universities. Notably, Stanford University
cited our analyses in their decision to return to on-campus instruction for
Fall 2021 (Stanford University 2021). We believe that our success, along with
the flexibility and generalizability of our modeling approach, makes our
framework a useful tool for managing indoor operations during respiratory
disease outbreaks and a valuable contribution towards mitigating the impacts
of pandemics.

### Related Work

Various universities leveraged mathematical models to support reopening
decisions during the pandemic. In Summer 2020, the Georgia Institute of
Technology used an integer programming framework to assign courses to
instruction modalities (virtual, hybrid, or in-person) and courses to
classrooms under social-distancing constraints (Navabi-Shirazi et al. 2022).
In Fall 2020, Clemson University developed a rotational cohort-based schedule
for courses, which aimed to minimize interactions between students across
cohorts (Gore et al. 2022). Oklahoma State University, that same semester,
developed an optimization tool in Excel to assign courses to classrooms to
maximize the number of in-person courses offered (Johnson and Wilson 2022).

These modeling approaches allowed universities to resume in-person
instruction, in limited capacities, in accordance with social distancing
regulations. Our work is novel in that we assessed the risk of in-person
instruction when social distancing regulations were relaxed. It was important
for a university to understand this risk when returning to pre-pandemic levels
of in-person instruction.

The rest of this paper is organized as follows. First, we describe in detail
the challenges faced by the University when planning for the Fall 2021
semester. We then explain our framework for estimating the risk of COVID-19
transmission in classrooms, which includes mathematical modeling and a
computer simulation. We apply our framework to evaluate different
interventions and develop a strategy to safely operate dense in-person
classrooms that was recommended to university leadership. We conclude with a
retrospective evaluation of our model’s validity and discuss its broader
impact beyond modeling transmission in classrooms. Further details of our
model are presented in the appendix.

## Problem Statement

While planning for the Fall 2021 semester, the University aimed to offer as
many in-person classes as possible while maintaining classroom safety. For the
Fall 2020 and Spring 2021 semesters, the University had held only a limited
number of de-densified classes, where the students were spaced six feet apart.
The constraint of having a finite number of classrooms on campus posed a
challenge, as expanding in-person classes would elevate student density in
classrooms, potentially heightening the risk of indoor COVID-19 transmission.
To mitigate this potential for elevated transmission risk, the University
would need to implement classroom interventions. Interventions under
consideration included requiring masking, improving ventilation, increasing
social distancing, and assigning seats randomly. (Assigning seats randomly
would reduce the risk that unvaccinated students, who are more vulnerable to
infection and have higher transmission when infected, would sit together in
socially connected groups.) At the time, we had a limited understanding of the
effectiveness of these interventions in preventing disease spread, whether
deployed individually or combined. Amidst such uncertainty, one major goal of
our modeling work was to identify a combination of interventions that could
efficiently curb disease transmission within classrooms, all while maintaining
a reasonable cost.

The University also faced additional concerns in the months leading up to the
Fall 2021 semester. The more infectious Delta variant of COVID-19 was
spreading globally and was responsible for a deadly second wave in India
(Tareq et al. 2021). There was concern that the variant would spread to the
United States and quickly become the dominant strain. Though many students
were fully vaccinated, the vaccine’s efficacy against Delta was uncertain. In
addition, there was concern among the broader community, especially from
faculty and student instructors, about the risk from teaching and attending
classes.

Thus, the goal of our modeling work was to understand what classroom
interventions would be needed to safely hold dense in-person classes, and to
assess and communicate how these interventions would address the concerns that
we faced heading into the Fall 2021 semester. We further discuss classroom
density, classroom interventions, the Delta variant, and community concerns
below.

### Classroom Density

Figure 2: Floor plan of Olin 155, a large lecture hall at the University.

Note: The socially distanced seating configuration used in Fall 2020 is marked
with “x”; only 36 seats were used among the 287 seats available.

During the Fall 2020 semester, only 20% of courses were offered in-person.
While the majority of students returned to Ithaca, few students were in
classrooms on any given day during the semester. As such, the University could
aggressively de-densify classrooms to reduce the potential for in-person
transmission. All classrooms were configured to be socially distanced, where
students were seated six feet apart (Cornell University CTRO 2020). Figure 2
shows the floor plan of a socially distanced classroom, Olin Hall 155, that
can normally accommodate 287 students during normal university operations. In
Fall 2020, with social distancing, the hall could only accommodate a maximum
class size of 36 students, a 90% reduction from the pre-pandemic capacity.

Overall, social distancing reduced campus-wide classroom capacity by 87%. This
reduced capacity was sufficient for the Fall 2020 semester, where only a
fraction of courses were offered in-person under reduced schedules. However,
maintaining the same distancing level for the increased in-person course
schedule for Fall 2021 would have required each room to be used for more than
24 hours each day. Thus, further analysis was necessary to assess the safety
of increasing classroom density.

### Classroom Interventions

The potential interventions that the University could take included requiring
masking, improving ventilation, increasing social distancing, and assigning
seats randomly in classrooms, all of which faced implementation difficulties.
Requiring masks in classrooms was the easiest intervention to execute.
Assigning seats randomly would randomly assign students to seats once at the
beginning of the semester and require them to sit in their assigned seats
throughout the semester. This would reduce the chance that unvaccinated
students, who have higher susceptibility and transmissibility when infected,
would sit together in groups. To implement randomly assigned seats, seating
charts would need to be planned in advance for each class. Increasing social
distancing in classrooms would reduce classroom capacity and limit the number
of in-person courses offered. Classes would also need to meet at inconvenient
times (late night or early mornings) to accommodate reduced classroom
capacity. Finally, increasing ventilation in classrooms was the most difficult
intervention to implement, due to the cost of retrofitting all classrooms with
HVAC equipment. Such improvements were only made in Summer 2020 for the
largest classrooms at the University (classrooms with more than 100 seats that
were used for socially-distanced instruction in Fall 2020 and Spring 2021).

In Appendix Section A2, we describe how we modeled classroom interventions to
estimate their efficacy before the start of the semester. This analysis
informed the university on interventions needed to ensure safety.

### Delta Variant Uncertainties

Figure 3 shows daily COVID-19 case counts in New York State in 2021. The dot-
dashed line indicates the first date where the majority of cases in New York
City were determined to be from the Delta variant (New York State Department
of Health 2021a). The total daily case count in the state rose steadily from
that date until the start of the Fall 2021 semester, indicated by the red
dashed line in the figure. In retrospect, it is apparent that the semester
started during the peak of the Delta wave of the pandemic.

Figure 3: Daily COVID-19 cases counts for New York State based on reports from
state and local health agencies (The New York Times 2021).

Notes: The gray dot-dashed line shows when the majority of cases in New York
City were first determined to be from the Delta variant. The University
started its Fall 2021 semester roughly 2 months later, at the peak of the
Delta wave.

The emerging Delta wave presented challenges when planning for the Fall 2021
semester. First, while the Delta variant drove an increase in cases during the
summer of 2021, the exact increase in Delta’s infectivity compared to the
original strain was not well understood at the time when the University needed
to decide on classroom density. In addition, the literature on vaccine
efficacy against the Delta variant was sparse. Preliminary reports from the
United Kingdom and Israel were not encouraging; early studies from the UK NHS
(Andrews et al. 2021) and Israel Health Ministry (Israel Ministry of Health
2021) estimated BNT162b2 Pfizer vaccine efficacy of $88\%$ and $39\%$
respectively against symptomatic illness from the Delta variant. In context,
the BNT162b2 Pfizer vaccine achieved vaccine efficacy of $95\%$ against the
original strain in clinical trials (Polack et al. 2020).

Appendix Section A2 explains how we estimated the Delta variant’s increased
infectivity and decreased vaccine efficacy to produce models of classroom risk
that are robust to uncertainty in both parameters. We used these models to
determine if the University could safely hold in-person classes during the
Delta wave of the pandemic.

### Community Concerns

Prior to the start of the Fall 2021 semester, many in the community expressed
concern about the safety of in-person classes. In August 2021, the University
Chapter of the American Association of University Professors expressed in a
letter to the University President concerns about the risk of teaching in
person due to the increased transmissibility of the Delta variant (Lieberwitz
2021). During a faculty and staff town hall the same month, multiple questions
were asked about the risk of transmission from teaching class and holding
office hours, masking in classrooms, and the efficacy of vaccines against the
Delta variant (Cornell University 2021a). In addition, multiple faculty
members emphasized that the University needed to be more transparent about how
classroom safety was assessed (Cornell Faculty 2021). We developed and
communicated our modeling framework to reassure the community that in-person
instruction was safe using transparent, data-driven methods.

## Modeling Framework

The modeling framework we developed consists of two parts: a mathematical
model used to estimate transmission risk between individuals under different
conditions and a simulation tool used to evaluate overall classroom risk. We
sketch the main ideas here and provide a full description of our methodologies
in the appendix.

### Main Assumptions and Parameters

Our models rely on a set of parameters, the values of which are key to the
predictions. We estimated parameter values from the literature available at
the time of our analysis. For parameters with high uncertainty, we imposed
reasonably chosen prior distributions on their values rather than using point
estimates. Our assumptions were influenced by our previous work on developing
epidemiological models for COVID-19 at the University (Frazier et al. 2022).

We assumed that the Delta variant would be dominant at the University at the
start of the Fall 2021 semester, and would be 2.4 times as transmissible as
the original COVID-19 strain (Washington et al. 2021, Callaway 2021). We
conservatively estimated that $90\%$ of the undergraduate population would be
fully vaccinated at the start of the semester. Among the vaccinated
population, we estimated the distribution of vaccine efficacy (VE) against
infection and VE against transmission to be centered around 52% and 51%
respectively. These estimates were obtained by weighting the results from
several different studies by their sample size. We estimated that if either
the source or the susceptible person were masked, the transmission probability
would be reduced by 50% to 80%. Finally, we assume perfect compliance with any
masking guidelines given by the University, since, in previous semesters,
compliance to COVID-19 regulations was very high.

### Mathematical Model of Transmission

Given an infectious person in a classroom, we decompose the risk of
transmission into a short-range and a long-range component, each representing
a major mode of SARS-CoV-2 transmission (Centers for Disease Control and
Prevention 2021b). The short-range component models transmission due to the
deposition of virus-containing respiratory droplets onto exposed mucous
membranes; the long-range component models transmission due to the inhalation
of virus-containing aerosols or fine droplets. In both components, we use an
exponential dose-response model (Watanabe et al. 2010), where the dose is the
amount of virus a susceptible individual is exposed to. The likelihood a
susceptible individual becomes infected, given dose $D$ and a positive
constant $c$, can be expressed as

$\mathbb{P}(\text{transmission})=1-\exp(-c\cdot D).$ (1)

As the dose increases, the likelihood of transmission increases as well.

#### Short-range Transmission

In short-range transmission, the source exhales virus-containing droplets,
which are large, heavy particles that tend to deposit on the ground or other
surfaces. As the droplets are heavy and cannot travel far, the concentration
of droplets in the air decreases with the distance from the source case
(Mittal et al. 2020).

To model the fact that students mostly face the instructor, who typically
stands in the front of the classroom, we assume the source case emits virus
particles in a cone of directions towards the front; we call this set of
directions the source case’s cone of exposure. We model the transmission
probability in two dimensions, accounting for the distance and angle of the
susceptible individual relative to the source case.

We use maximum likelihood estimation to fit the model parameters (including
the angle of the cone of exposure) based on a large dataset on COVID-19
transmission aboard high-speed trains in China (Hu et al. 2021), assuming that
all secondary infections in the data were due to short-range transmission. The
dataset gives the relative positions between infectious index cases on the
train and nearby susceptible passengers, as well as the subsequent case
incidence rates among the susceptible passengers. The seating configuration of
the train car is similar to a lecture hall, where all individuals face the
same direction and are spaced apart by rows of seats. To the best of our
knowledge, this dataset was the best available at the time we fit our model.

#### Long-range Transmission

We use the model and parameters in Schijven et al. (2021) and model long-range
transmission by quantifying the concentration of virus-containing aerosols or
fine droplets suspended in the air. (Hereafter, we call them “aerosols”.) The
model assumes that aerosols are distributed uniformly across the room. As a
result, the probability of transmission does not depend on distance or angle
from the source and only depends on the rate of aerosol emission from the
infectious source, duration of exposure, room volume, and level of
ventilation.

#### Overall Risk

We combine the estimated short-range and long-range transmission risk by
taking the larger of the two.

When estimating the parameters for the short-range model, we assumed that all
secondary infections in Hu et al. (2021) were due to short-range transmission,
while in reality some cases may have arisen from long-range transmission.
Therefore, the estimates for the short-range model may implicitly include some
effect of long-range transmission. Setting the overall risk to the maximum,
rather than the sum, of the two risks prevents overestimation. In fact, the
simulated short-range risk is usually one to two orders of magnitude larger
than the long-range risk within three meters, so it dominates the overall risk
for those exposed to it. This is consistent with Public Health Ontario (2022),
which finds that shorter distance usually implies higher transmission risk.

We assume that instructors are sufficiently distanced from the students such
that short-range transmission is not possible. In our model, the risk from
short-range transmission is negligible after six feet of distancing and we
assume, based on prior semesters, that most instructors spend the majority of
their time over six feet away from students.

We do not explicitly model an infectious instructor, because case
investigations in the 2020-21 academic year did not reveal any faculty or
student infections that were linked to classroom-based transmission (Cornell
University 2021a), and faculty prevalence was much lower than that of
students. In addition, the number of students in a class is typically much
larger than the number of instructors. Moreover, even if the instructor is
infectious in addition to an infectious student in the classroom, this merely
approximately doubles the risk due to long-range transmission for each
susceptible student. For the susceptible students most at risk, i.e., those
sitting in the proximity of the infectious student, the risk from short-range
transmission dominates that from long-range transmission by two orders of
magnitude. Therefore, the expected number of secondary transmissions remains
almost the same regardless of the instructor’s infection status.

#### Reflections

We developed this modeling framework in Summer 2021 to support reopening
decisions at the University for the Fall 2021 semester. As the body of
COVID-19 related literature expands, we recommend these modifications to our
framework for future use.

  1. 1.

We would evaluate and compare other theoretical models for estimating the risk
of COVID-19 transmission through droplets and aerosols (Mirzaei et al. 2021,
Bazant and Bush 2021). In addition, we would calibrate the model to more
datasets that shed light on COVID-19 transmission in enclosed spaces, such as
in restaurants (Cheng et al. 2022) and theaters (Adzic et al. 2022), as well
as adjust for more recent variants such as Omicron (Ji et al. 2022).

  2. 2.

We would update our estimates of virus transmissibility and vaccine efficacy
based on the most up-to-date findings (Ciotti et al. 2022, Wan et al. 2023).

### Classroom Simulation Tool

In conjunction with our mathematical classroom model, our simulation tool
allows us to estimate the risk of classroom transmission along with the
effectiveness of various interventions, such as masking, social distancing,
and increased ventilation. Figure 4 presents an illustration of the classroom
simulation tool for a large lecture hall.

For each parameter setting (density level, vaccination rate, vaccine
efficacy), we estimate the expected number of secondary infections in the
classroom over a 1-hour period given one infectious source case among 50
students, averaged over 500 trials. (We omit the scenarios where there are two
or more source cases in the same classroom at the same time. Such scenarios
are unlikely compared to scenarios with one source case because prevalence is
low, so they contribute little to overall risk. Further discussion is given in
Appendix A1.) For each trial, we randomly generate a seating configuration and
vaccination statuses among the students and randomly draw a student to be the
source case. We repeat this for all combinations of density level, vaccination
rate, and vaccine efficacy. We assume everyone is unmasked in the simulation.
The effect of masking, modeled as an uncertain parameter with a normal prior,
can be directly imposed on the results above through multiplication.

Figure 4: Example illustration of classroom simulation tool.

Notes: The purple X indicates the infectious source case and the purple cone
indicates the cone of exposure, the set of directions in which the source case
is modeled as emitting virus particles. Unvaccinated and vaccinated students
are represented with red and blue dots respectively. The instructor is located
on a stage sufficiently distanced from the class, far above the top margin of
the illustration.

### Interventions and Scenarios Evaluated

Table 1: Effectiveness of intervention methods at reducing short and long-range transmission. Intervention/Reduction in transmission | Short-range | Long-range
---|---|---
Masking | ✓ | ✓
Seating Policy | ✓ | -
Distancing | ✓ | -
Ventilation | - | ✓

Combining the mathematical model and classroom simulation tool, we evaluate
several interventions (masking, seating policy, distancing, ventilation)
across different scenarios. Table 1 summarizes the possible interventions
along with their effectiveness against short-range and long-range
transmission. We discuss these interventions and scenarios in further detail
below.

#### Masking:

Based on experimental and observational studies, we assume the masking
effectiveness against transmission to range from 50% to 80% if either the
infectious or the susceptible individual is masked (see details in Appendix
A2). If both of them are masked, the risk of transmission is reduced by 75% to
96%.

We evaluate the intervention of masking for the entirety of the Fall 2021
semester, and we assume that there is perfect compliance with the masking
mandate, consistent with the high compliance observed in previous semesters
(Cornell University 2021a).

#### Seating policy:

We consider two different seating policies: (1) randomly assign students to
seats and enforce that students always sit in their assigned seats (fixed
seating); (2) allow students to sit wherever they want (unrestricted seating).

Unrestricted seating could be more risky in that unvaccinated students could
potentially group together. This would lead to a higher expected number of
transmissions since unvaccinated students are more susceptible to COVID-19
infection and have a higher transmissibility if infected (de Gier et al. 2021,
Lopez Bernal et al. 2021).

On the other hand, fixed seating is operationally difficult to implement. Our
initial simulations showed that the fixed and unrestricted seating policies
lead to comparable risk (see Figure 10 in Appendix A1). As a result, the
University adopted the unrestricted seating policy; all simulation results
shown here are thus based on unrestricted seating.

#### Social Distancing:

We evaluate three social distancing options. In fully dense seating, the
default spacing for lecture halls pre-pandemic, students are distanced one
foot apart from each other in the classroom. In moderately dense seating,
students are distanced three feet apart. In distanced seating, students are
seated six feet apart. This last configuration was used in the 2020-2021
academic year during the pandemic.

#### Ventilation:

For an infectious source case, we assume that aerosol viral particles are
emitted continuously over the hour at a constant rate and are immediately
distributed evenly across the room once emitted. We quantify ventilation rate
by measuring how often air is exchanged from the room, in the unit of air
exchanges per hour (ACH), and we assume that air exchanges happen evenly over
time. According to the University’s Facilities Department, most classrooms
have a ventilation rate of 1 ACH. We assume that this rate reduces the amount
of viral aerosols accumulated in the classroom over an hour by half relative
to having no ventilation (American Society of Heating, Refrigerating and Air
Conditioning Engineers 2002).

We evaluate the worst case, where a poorly ventilated room can have 0 ACH and
the risk of aerosol transmission is not reduced at all by ventilation. In
addition, we evaluate the intervention where ventilation is improved to 3 ACH,
which reduces the overall dose of transmission from aerosols by a factor of 4
relative to no ventilation.

#### Class Type:

The type of class determines the intensity of respiratory activity that occurs
in the room, which corresponds to different rates of viral aerosol emission.
We assume breathing is the dominant type of respiratory activity for students
attending lectures and that the effect of occasional speaking (e.g., asking
and answering questions) is negligible. However, our simulation is able to
handle activities such as talking and singing.

### Risk Over the Semester

Given a set of interventions, we adopt the following procedure to assess the
risk of transmission for students and instructors across the entire semester.
More details are given in Appendix A2. We only consider undergraduate
students, but our results easily translate to graduate or professional
students who typically take fewer classes.

We first use our classroom simulation tool to estimate the risk of
transmission per hour spent in the classroom $\eta$, conditioned on the class
having an infectious source case. Given average class size $n_{0}$ and
prevalence of infectious individuals at the University $p$, the probability
that a susceptible student attends a class with an infectious student is given
by

$1-(1-p)^{n_{0}-1}\approx(n_{0}-1)\cdot p.$

This approximation is justified since $p$ is small at the University under our
projections for the Fall 2021 semester (we also impose a prior on $p$ to
incorporate uncertainty in its estimated value). Multiplying the above
expression by $\eta$ gives the unconditional probability of infection per hour
of class. Assuming a student’s average time spent in class per semester to be
$T$ hours, the probability of infection in class over a semester is given by

$1-(1-\eta\cdot(n_{0}-1)\cdot p)^{T}\approx\eta\cdot(n_{0}-1)\cdot p\cdot T.$

Lastly, we generate a distributional estimate using 100,000 samples, with each
sample representing the semester-wise risk associated with a specific
parameter configuration drawn from the priors.

A similar procedure can be applied to faculty and graduate student
instructors. We assume that the instructor is sufficiently distanced from the
students so that risk only arises from transmission over long distances. We
adjust the unconditional probability of infection per hour of class to account
for their population sizes relative to the undergraduate population size. We
also assume the $T$ hours of classes are divided between faculty and graduate
student instructors with a 2:1 ratio.

## Results and Recommendation for the Fall 2021 Semester

Here, we summarize the results and recommendations of our modeling analysis.
Among the combinations of interventions deemed to be safe, enforced masking in
dense classrooms with unrestricted seating is the easiest to implement.
Masking reduces the risk of short-range transmission which allows for dense
seating in classrooms. As a result, more classes can be held in-person given
limited classroom space. Our model results show that fixed and unrestricted
seating policies result in comparable risk, while the latter is easier to
implement. In addition, the model shows that the reduction in transmission
from masking and distancing are similar, and both of these interventions are
more effective than increasing ventilation.

We next present the simulated distributions of infection risk for students and
instructors. We assume 15,000 undergraduate students, 850 faculty instructors,
and 3,120 graduate student instructors (details are given in Appendix A2).

#### Students Classroom Risk:

We project that the median risk of infection per student due to lecture
transmission in Fall 2021 is 0.5% at 90% vaccination rate. (An earlier version
of this analysis (Cornell COVID-19 Modeling Team 2021a) predicted this number
to be 0.4% due to outdated parameters.) Figure 5 shows the estimated
distribution of this risk across 100,000 simulated outcomes; the median is
indicated by the red dashed line. The right tail of the estimated risk
distribution mainly results from the right tail of the log-normal prior over
the prevalence parameter.

Figure 5: Distribution of risk of lecture transmission for a student across
the entire Fall 2021 semester, over $10^{5}$ simulation trials.

Note: Median risk (red dashed line) is 0.51%; the 5% and 95% quantiles (orange
and purple dashed lines) are 0.16% and 1.31% respectively.

#### Instructor Classroom Risk:

We project that the median risk of infection per instructor due to lecture
transmission in Fall 2021 is 0.02% for vaccinated faculty instructors and
0.003% for vaccinated graduate student instructors. The estimated distribution
of risk across simulated outcomes is presented in Figures 6 and 7.

The risk is approximately doubled for an unvaccinated instructor. Since over
99% of professorial faculty had been vaccinated by the start of the semester
(Rosenberg 2021), and those who chose not to be vaccinated would be highly
cautious, we only show the estimated risk for the vaccinated instructors here.

Figure 6: Distribution of risk of lecture transmission for a vaccinated
faculty instructor across the entire Fall 2021 semester, over $10^{5}$
simulation trials.

Note: Median risk (red dashed line) is 0.018%; the 5% and 95% quantiles
(orange and purple dashed lines) are 0.0056% and 0.059% respectively.

The projected risk for instructors is much lower than that for students. This
is mainly due to the modeling choice that they are not subject to short-
distance transmission, based on the natural distancing between instructors and
students in classrooms. In addition, instructors spend less time in class over
a semester compared to students.

Figure 7: Distribution of risk of lecture transmission for a vaccinated
graduate student instructor across the entire Fall 2021 semester, over
$10^{5}$ simulation trials.

Note: Median risk (red dashed line) is 0.0025%; the 5% and 95% quantiles
(orange and purple dashed lines) are 0.0008% and 0.008% respectively.

### Recommendations

Based on our analysis, we believed that fully dense in-person classes, with
masking enforced, could be safely implemented for the Fall 2021 semester. We
estimated the total risk of classroom transmission per student across the
entire semester to be around 0.5%, or roughly 1 in 200. For faculty and
graduate students, the estimated risk of classroom transmission was even
lower, roughly 1 in 5,000 to 40,000 across the entire semester. An
individual’s odds of being struck by lightning in their life is on the order
of 1 in 10,000 (National Weather Service 2019), which is comparable.

Under the assumption that 15,000 students would return to the University for
the Fall 2021 semester, we anticipated an additional 75 cases due to classroom
transmission. We did not expect these additional cases to strain the testing
and quarantine capacity of the university; the University’s testing
infrastructure was able to handle tens of thousands of tests per week and the
university had the capacity to quarantine hundreds of students at a time. In
addition, given the estimate of COVID-19 hospitalization rates for college-age
students of 0.005% (Centers for Disease Control and Prevention 2021a), we did
not expect that any students would be hospitalized from an infection due to
classroom transmission. Finally, assuming 850 faculty members and 3,120
graduate students serve as instructors in the Fall 2021 semester, we did not
expect to observe any instructor cases linked to classroom transmission.

## Evaluation

To evaluate our modeling framework and recommendations, we retrospectively
investigate COVID-19 cases from August 26, 2021 to December 7th, 2021. We
excluded a spike in cases due to importation of the Omicron variant in mid
December for two reasons: it happened after the end of the instruction period
when no classes were in session, and our modeling analyses and recommendations
were specific to the Delta variant.

### Student Transmission

We present the following body of evidence that minimal classroom transmission
occurred among students during the Fall 2021 semester (Cornell University
2021b).

  1. 1.

When a student tested positive during the Fall 2021 semester, the University
tested all students attending the same class to the extent feasible. In
addition, the genetic sequencing of positive cases were compared to determine
if cases were related. These investigations did not yield evidence of
classroom transmission.

  2. 2.

We collected seating data for a class held in a lecture hall that contained
over 1000 students. When a student in that class tested positive, we
investigated to see if any students seated near the infected student
subsequently tested positive. While this data is sparse, there were 20
instances of an infected student sitting within 3 seats of susceptible
students. None of these cases was associated with a susceptible student
testing positive.

  3. 3.

Throughout the semester, the weeks with the highest rates of on-campus
transmission corresponded to breaks when classes were not held. This is
consistent with travel and social gatherings, rather than classes, driving
COVID-19 transmission on campus, as was also observed in previous semesters
(Cornell COVID-19 Modeling Team 2021b). Figure 8 shows the daily count of new
cases for undergraduate students. The outbreaks occurred right after students
returned to campus from breaks.

  4. 4.

Contact tracing revealed that most positive cases can be linked by social
gatherings, cohabitation, or travel.

This collection of evidence strongly suggests that classroom transmission was
rare during the Delta wave of the Fall 2021 semester at the University.

Figure 8: Undergraduate daily case count during the Fall 2021 semester.

Notes: The gray intervals indicate school breaks (classes not in session). The
largest spikes in cases occurred after events that involve significant student
travel (move-in period and Thanksgiving break). December 7th was the last day
of classes for the semester. The Omicron variant was responsible for the
rightmost spike in cases, which occurred after classes ended.

### Instructor Transmission

Throughout the Fall 2021 semester, only a single faculty member tested
positive for COVID-19 at the University. In addition, the prevalence of
positive cases among graduate students was 4 times lower than the prevalence
among undergraduate students. In all, infection rates among faculty and
graduate students were much lower compared to the rest of the University
population, which suggests that in-person teaching did not appreciably
increase the risk of contracting COVID-19 during the Fall 2021 semester
relative to other sources of transmission.

## Extensions

### Beyond the Classroom

While the main focus of our work was to model the risk of COVID-19
transmission in lectures and classrooms, we received many requests from the
University administration to assess the risk of holding extra-curricular
events and gatherings during the Fall 2021 semester. We were able to modify
our modeling framework to accommodate these requests; we used the same model
structure but updated our parameters to model eating, singing, socializing,
and other events that occur in social gatherings. Our modeling analysis
influenced the following decisions. For Homecoming weekend, we determined that
the homecoming football game and the Class of 2020’s belated graduation
ceremony were relatively low-risk events. However, we found that parties and
festivities that occur after formal events incur substantially higher risk of
COVID-19 transmission. As a result, we recommended canceling post-homecoming
festivities such as the fireworks and light shows on-campus; these
recommendations were accepted by the administration. We did not observe a
large spike in cases on campus after homecoming weekend.

We were also asked by executive-level decision-makers at the University to
assess the risk of the campus-sponsored Rosh Hashanah dinner. We determined
that given the high vaccination rate on campus, this event would be safe to
attend. Finally, we were asked to evaluate the risk of indoor physical
education classes and music and choir classes. We modified our model to
account for the increased aerosol emission due to these activities and
determined that it was safe to hold these classes with dense seating
configurations. No cases were linked to these courses at the end of the
semester.

The flexibility of our framework in accommodating these ad hoc situations
indicates that our framework can be applied to other industries besides higher
education, supporting reopening decisions for indoor activities across a wide
range of applications.

### Beyond COVID-19

In addition, by re-fitting the models on short range and long range
transmission and by re-estimating the parameters on vaccine and mask efficacy,
we can easily adapt our framework to model other respiratory diseases or
COVID-19 variants. As such, our modeling framework can be used to assess
infection risk in future pandemics across various settings.

### Conclusion

Our modeling framework for COVID-19 transmission in classrooms allowed the
University to analyze the risk of holding in-person classes and compare the
effectiveness of interventions. Using the recommendations provided by our
modeling framework, the University was able to return to pre-pandemic levels
of in-person instructions for the Fall 2021 semester, improving the
educational experience of students compared to previous semesters while
ensuring safety. Post-hoc analysis at the end of the semester confirmed that
classroom transmission was rare and that teaching in-person classes was a low-
risk activity. Finally, our modeling framework is flexible and can be adapted
to model infections risk for respiratory diseases across a wide range of
applications.

This work was conducted under the support from Cornell University when the
authors served in the Cornell COVID mathematical modeling team. This work was
partially supported by National Science Foundation grant DMS2230023.

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Modeling and Simulation Details

We describe the details of our modeling and simulation for estimating
secondary infections in classrooms and projecting to the risk over the entire
semester. In Section A1, we introduce the mechanism behind our simulation
tool. In Section A2, we state the assumptions and parameter estimates in
different components of the simulation. In Section A3, we develop the
mathematical model for the transmission probability between a source case and
a susceptible person over different distances, which is an important component
in the simulation. Throughout the appendix, we use # to denote “number of” and
% to denote “fraction of”.

## A1. Simulation

We implement a simulation tool in Python to simulate classrooms under
different scenarios. We investigate the effect of different intervention
measures (masking, increasing distancing, ventilation) on transmission risk
under varying simulation parameters (vaccination rate among students, vaccine
efficacy). We describe the simulation mechanism in this section. In Section
A2, we discuss the assumptions and values for these parameters in more detail.

Our simulation has two stages. In the first stage, we estimate the conditional
probability that a student or instructor is infected in a one-hour class given
there is one positive source case in the same class. In the second stage, we
extrapolate the probability that a student or instructor becomes infected due
to attending or teaching classes over the entire semester.

We omit the scenarios where there are two or more source cases in the same
classroom at the same time. Such scenarios are relatively unlikely because
prevalence is low. Moreover, the fact that a few positives occur in the same
classrooms at the same time makes reality slightly more optimistic than our
estimates: in our estimates, everyone else in the classroom is susceptible
while in reality, the other infectious individual cannot be infected again. In
addition, the increase in the risk of infection created by adding a second
positive is smaller than the increase created by adding a first positive111
This follows (1) when the dose resulting from each positive is independent and
identically distributed and (2) from concavity of the probability of infection
as a function of the dose, given in Appendix A3. In particular, let $V_{1}$
and $V_{2}$ be the (strictly positive) dose to a given susceptible person
associated with the first and second positives in a classroom, so that the
dose is $V_{1}$ if there is one positive in the classroom and $V_{1}+V_{2}$ if
there are two. We assume that $V_{1}$ and $V_{2}$ are independent and
identically distributed after marginalizing over the random locations of the
two positive individuals. Then let $P(v)$ be the probability of infection
given a dose $v$, as given in Section A3. Since $P$ is concave, and also using
that $P(0)=0$, then for strictly positive $V_{1}$ and $V_{2}$,
$P(V_{1}+V_{2})-P(V_{1})\leq P(V_{2})-P(0)$. Then, because $V_{1}$ and $V_{2}$
are identically distributed,
$\mathbb{E}[P(V_{1}+V_{2})-P(V_{1})]\leq\mathbb{E}[P(V_{2})-P(0)]=\mathbb{E}[P(V_{2})]=\mathbb{E}[P(V_{1})]$.
The left-hand side $\mathbb{E}[P(V_{1}+V_{2})-P(V_{1})]$ is the increase in
the risk of infection created by adding a second positive, and the right-hand
side $\mathbb{E}[P(V_{1})]$ is the increase in risk from adding the first
positive. . The level of optimism introduced by this fact is extremely small,
however, and our assumption produces nearly the same estimate as one that
allows multiple positives to be in the same classroom.

### Generating Classroom Seating Arrangements

We simulate seating at different density levels by assigning a fixed number of
individuals to classrooms of different sizes. We assume that an average class
contains $n_{0}$ = 50 students and one instructor. From university floor
plans, we identify three representative rooms, namely Hollister 206, Gates
G01, and Rockefeller 201, that correspond to roughly 1’, 3’, and 6’ distancing
respectively for 50 students. The corresponding seating capacities are
presented in Table 2.

Table 2: Seating capacity for Hollister 206, Gates G01, and Rockefeller 201. Room | Pre-COVID capacity (1’ distancing) | COVID capacity (6’ distancing)
---|---|---
Hollister 206 | 52 | 12
Gates G01 | 156 | 22
Rockefeller 201 | 383 | 56

Using the seating plan tool developed by the Committee on Teaching
Reactivation Options (C-TRO) team greenberg2021automated, we identify seats in
these classrooms and assign maximally distanced seats such that approximately
50 students can fit in each room. The generated seating plans are displayed in
Figure 9. The rooms used have extra space above and beyond what is required to
accommodate the social distancing requirements we have assumed. This extra
room does not offer additional benefit in our simulations as it is not used.
We assume the instructor is standing in the front of the classroom, with at
least 6 feet distance from all the students.

Figure 9: Seating plans generated for Hollister 206, Gates G01, and
Rockefeller 201. They correspond to 1’, 3’, and 6’ social distancing,
respectively. The green dots represent available seats (i.e., seats that are
allowed to be occupied) and the empty circles are considered unavailable in
our simulation tool.

We initially consider two seating policies: (1) fixed seating, where students
are randomly assigned to their own seats independent of their vaccination
statuses; (2) unrestricted seating, where students can freely choose among the
allowed seats. For unrestricted seating, we pessimistically assume that
unvaccinated students tend to sit together, as the same demographic factors
that cause students to be unvaccinated upon arrival to Ithaca may also create
social connections. Since fixed seating is operationally difficult to
implement, and initial simulations show that these two seating policies lead
to comparable risk (Figure 10), the University decided to adopt unrestricted
seating. Figure 11 shows examples of unrestricted seating arrangements at
different distancing levels.

Figure 10: Comparison of fixed and unrestricted seating policies. We show the
average number of student secondary infections over a one-hour lecture with a
positive student for different vaccination rates (from 40% to 100%),
distancing levels (1, 3, and 6 feet distancing), and masking rates (0% and
100%). The differences between the solid lines (unrestricted seating) and
dashed lines (fixed seating) are small and further decrease as vaccination
rate increases. The vaccination rate we assume in our subsequent simulations,
90%, is indicated by the grey dotted line.  Figure 11: Example simulated
seating arrangement of 50 students at different levels of social distancing.
At each distancing level, students are placed into available seats shown in
Figure 9. The red dots represent unvaccinated students, the blue dots
represent vaccinated students, and the empty circles represent unavailable
seats due to the distancing requirement or empty available seats. We assume
pessimistically that unvaccinated students tend to sit together.

### Stage 1: Simulating Infections in a Single Classroom

In the first stage, we estimate the conditional probability that a student or
instructor is infected in a one-hour class given there is one positive source
case in the same class. We do this by simulating the expected number of
secondary infections that occur in a one-hour class with $n_{0}=50$ students,
where initially a single student is infected. The simulation depends on the
following configuration parameters: seating density, percentage of students
vaccinated, and vaccine efficacy (VE). In particular, vaccine efficacy
$\texttt{VE}=(v_{\text{source}},v_{\text{susceptible}})$ is a tuple of two
parameters characterizing the reduction in infectivity of a vaccinated source
case and the reduction in infection probability of a vaccinated susceptible
person respectively. We assume each student is vaccinated independently with
probability equal to the vaccination rate among the students.

In this stage, we assume everyone is unmasked. This allows the flexibility to
model varying masking rate among students as well as uncertainty in masking
effectiveness in the next stage of the simulation.

We loop over all the possible combinations of seating density and VE (values
of VE are discussed in detail in Section A2). For each configuration with
fixed parameter values, one run of the simulation proceeds in the following
steps:

  * •

Simulate a seating arrangement of $n_{0}=50$ students in the corresponding
classroom.

  * •

Choose one student uniformly at random as the source case and simulate their
vaccination status.

  * •

For each susceptible student, first simulate their vaccination status, then
compute the probability $p$ that they are infected over the 1-hour lecture
depending on their relative location to the source case. The mathematical
model that produces this probability is derived in Appendix A3.

  * •

Return the sum of the individual infection probabilities in the last step
divided by $(n_{0}-1)$. This is the average probability that a susceptible
student becomes infected in a one-hour class given that there is an infectious
student in the class, denoted
$\eta_{\text{student}}(\text{density},\texttt{VE})$.

  * •

We also compute the infection probability for the instructor, who is assumed
to be subject to only the risk of aerosol transmission due to sufficient
distancing, denoted $\eta_{\text{instructor}}(\text{density},\texttt{VE})$.

For each simulation configuration, we perform 500 replications to obtain the
average conditional probability of infection over 1 hour given a positive case
in the classroom.

### Stage 2: Extrapolating Classroom Transmission Risk over the Semester

In the second stage, we extrapolate the probability of infection over the
entire semester based on results in Stage 1 and additional parameters
(prevalence, masking effectiveness, and fraction of population masked).

To deal with uncertainty in the parameter values, we impose independent prior
distributions on each of them. We sample parameter configurations from the
prior and compute the output at each configuration, which produces a
distribution of the output. In particular, for each sampled parameter
configuration (vaccine efficacy
$\texttt{VE}=(v_{\text{source}},v_{\text{susceptible}})$, masking
effectiveness $m$, prevalence $p$) and a selected seating density, we perform
the following computation for an average student:

  * •

Get $\eta_{\text{student}}(\text{density},\texttt{VE})$ for the seating
density and sampled vaccine efficacy from Stage 1 simulation results.

  * •

Adjust for masking according to $\beta_{\text{masked}}$, the fraction of
students wearing a mask, and masking effectiveness $m$, the reduction in
transmission probability due to mask-wearing. We obtain the transmission
probability adjusted for masking, conditioned on the existence of an
infectious student in the class (we omit the terms (density, VE) for brevity):

$\tilde{\eta}_{\text{student}}=\eta_{\text{student}}(\text{density},\texttt{VE})\cdot[\beta_{\text{masked}}\cdot(1-m)+(1-\beta_{\text{masked}})].$

This expression is an approximation in that we set $m$ to be the reduction in
transmission probability when both the source and susceptible individuals are
masked, and we are not explicitly modeling the case where only one of them is
masked. In practice, we assume $\beta_{\text{masked}}=1$ in our modeling since
masking compliance in previous semesters was very high faculty_townhall.

  * •

Under the assumption that infectious students are uniformly distributed across
classes, the probability that a susceptible student attends class with an
infectious student is given by

$1-(1-p)^{n_{0}-1}\approx(n_{0}-1)\cdot p,$

where the approximation is justified since prevalence, $p$, is small in
practice. Thus, the unconditional infection probability for a student over a
one-hour lecture is given by

$\tilde{\eta}_{\text{student}}\cdot(n_{0}-1)\cdot p.$

  * •

Extrapolate the infection probability over the semester. Suppose an
undergraduate student spends on average $\tau_{\text{UG}}$ hours in class in a
semester. The probability of infection over $\tau_{\text{UG}}$ hours is

$\text{Risk}_{\text{student}}=1-(1-\tilde{\eta}_{\text{student}}\cdot(n_{0}-1)\cdot
p)^{\tau_{\text{UG}}}\approx\tilde{\eta}_{\text{student}}\cdot(n_{0}-1)\cdot
p\cdot\tau_{\text{UG}}.$ (2)

The above procedure does not eliminate already-infected students from the
susceptible pool over time, so the estimated results are more pessimistic than
reality. A similar computation can be done for faculty and graduate student
instructors.

  * •

We adjust for masking for instructors in a similar way as for students:

$\tilde{\eta}_{\text{instructor}}=\eta_{\text{instructor}}(\text{density},\texttt{VE})\cdot[\beta_{\text{masked}}\cdot(1-m)+(1-\beta_{\text{masked}})].$

  * •

We assume faculty instructors teach a fraction $\beta_{\text{faculty}}$ of all
class hours, while graduate students teach the rest. Let $n_{\text{UG}}$,
$n_{\text{faculty}}$, and $n_{\text{graduate}}$ denote the number of
undergraduates, faculty instructors, and graduate student instructors,
respectively. The average hours a faculty instructor spends in class teaching
in a semester is

$\tau_{\text{faculty}}=\frac{n_{\text{UG}}\cdot\tau_{\text{UG}}\cdot\beta_{\text{faculty}}}{n_{0}\cdot
n_{\text{faculty}}},$

where ${n_{\text{UG}}\cdot\tau_{\text{UG}}}/{n_{0}}$ gives the total
undergraduate lecture hours to be taught by all instructors. Similarly, the
average hours a graduate student instructor spends in class teaching in a
semester is

$\tau_{\text{graduate}}=\frac{n_{\text{UG}}\cdot\tau_{\text{UG}}\cdot(1-\beta_{\text{faculty}})}{n_{0}\cdot
n_{\text{graduate}}}.$

  * •

With the same reasoning as in Equation 2, we derive the risk for instructors
over the semester:

$\displaystyle\text{Risk}_{\text{faculty}}$
$\displaystyle=1-(1-\tilde{\eta}_{\text{instructor}}\cdot(1-(1-p)^{n_{0}}))^{\tau_{\text{faculty}}}\approx\tilde{\eta}_{\text{instructor}}\cdot
n_{0}\cdot p\cdot\tau_{\text{faculty}}.$
$\displaystyle\text{Risk}_{\text{graduate}}$
$\displaystyle=1-(1-\tilde{\eta}_{\text{instructor}}\cdot(1-(1-p)^{n_{0}}))^{\tau_{\text{graduate}}}\approx\tilde{\eta}_{\text{instructor}}\cdot
n_{0}\cdot p\cdot\tau_{\text{graduate}}.$

In the next section we give details on how we obtain estimates for
$\tau_{\text{UG}}$, $n_{\text{UG}}$, $n_{\text{graduate}}$,
$n_{\text{faculty}}$, and $\beta_{\text{faculty}}$.

## A2. Assumptions and Parameters

This section presents our assumptions and estimates for the parameters used in
the modeling. A summary is given in Table 3. For parameters with high
uncertainty, we design sensible prior distributions for their values rather
than using a point estimate. From the joint priors on the parameters, we
sample $10^{5}$ parameter configurations and obtain a distribution for
$\text{Risk}_{\text{student}}$, $\text{Risk}_{\text{faculty}}$, and
$\text{Risk}_{\text{graduate}}$ respectively. We treat the median as a main
point estimate and use the 5% and 95% quantiles as optimistic and pessimistic
estimates. Running simulations at a large number of parameter configurations
sampled from the priors enables a better understanding of how the possible
outcomes are distributed.

Table 3: Simulation parameters. Parameter | Value / Prior distribution
---|---
transmissibility of the Delta variant | 2.4 times that of Alpha
respiratory activity | breathing
$\beta_{\text{vaccinated}}$, fraction of students vaccinated | 90%
$v_{\text{source}}$, reduction in a vac’d source’s infectivity | see Table 4
$v_{\text{susceptible}}$, reduction in a vac’d susceptible’s infection prob. | see Table 4
$\beta_{\text{masked}}$, fraction of students masked | 100%
$m$, masking effectiveness |  $\mathcal{N}$(0.855, 0.0536), truncated to [0,1]
$p$, prevalence | LogNormal(-6.157, 0.413)
$\tau_{\text{UG}}$, avg hours an undergrad spends in class in a semester | 315
$n_{0}$, class size | 50
$n_{\text{UG}}$, number of undergraduates | 15,000
$n_{\text{faculty}}$, number of faculty instructors | 850
$n_{\text{graduate}}$, number of graduate instructors | 3,120
$\beta_{\text{faculty}}$, fraction of classes taught by faculty instructors | 2/3

  1. 1.

The Delta variant

    1. (a)

We assume the Delta variant had dominated all infections by the start of the
Fall 2021 semester.

    2. (b)

The Delta variant is 2.4 times more transmissible than the non-variant strain.
The Alpha variant was approximately 50% more transmissible than the original
SARS-CoV-2 washington2021emergence, and Delta is approximately 60% more
transmissible than Alpha callaway2021delta. This gives a multiplicative
increase of 1.5 * 1.6 = 2.4.

  2. 2.

Type of respiratory activity

Respiratory activities of different intensities are associated with varying
transmissibility. We assume breathing is the dominant type of respiratory
activity for students attending lectures and that the effect of occasional
speaking (e.g., asking and answering questions) is negligible. However, our
simulation is able to handle activities such as talking and singing.

  3. 3.

Level of vaccination among undergraduates ($\beta_{\text{vaccinated}}$)

By the start of the Fall 2021 semester, 99% of undergraduate students and
professorial faculty were fully vaccinated dailysun_fall2021. Erring on the
conservative side, we used 90% for the fraction of vaccinated undergraduates
at the time the analysis was performed. We compute the infection probability
for vaccinated and unvaccinated instructors separately.

  4. 4.

Vaccine efficacy ($\texttt{VE}=(v_{\text{source}},v_{\text{susceptible}})$) We
base our estimates of vaccine efficacy on the literature available at the time
of our modeling.

    1. (a)

For $v_{\text{source}}$, we assume vaccination reduces the infectivity of a
source case by 0, 50%, and 71% with probabilities proportional to the sample
size of the corresponding study levine2021initial, harris2021effect,
brown2021outbreak. See Table 4.

    2. (b)

For $v_{\text{susceptible}}$, we assume vaccination reduces the risk of
infection for susceptible individuals by 40%, 42%, 66%, 76%, 79%, and 88% with
probability proportional to the sample size of the corresponding study
puranik2021comparison, sheikh2021sars, lopez2021effectiveness,
pouwels2021effect, fowlkes2021effectiveness. See Table 4.

    3. (c)

We have also applied uniform discrete distributions for both VE parameters
(over three values for the VE in reducing viral load, and over six values for
the VE in protecting against infections) and found the outcome to be of the
same order of magnitude.

    4. (d)

Furthermore, our simulation for a single classroom requires specifying the
vaccination status of the source case. This is simulated using a Bernoulli
random variable with parameter
$\mathbb{P}(\text{vaccinated}\mid\text{infected})$, which in turn can be
deduced from $v_{\text{susceptible}}$ and $\beta_{\text{vaccinated}}$ using
Bayes Rule. By definition,

$\displaystyle\mathbb{P}(\text{infected}\mid\text{vaccinated})=(1-v_{\text{susceptible}})\cdot\mathbb{P}(\text{infected}\mid\text{unvaccinated}).$

By Bayes Rule,

$\displaystyle\mathbb{P}(\text{vaccinated}\mid\text{infected})$
$\displaystyle=\frac{\mathbb{P}(\text{infected}\mid\text{vaccinated})\cdot\mathbb{P}(\text{vaccinated})}{\mathbb{P}(\text{infected}\mid\text{vaccinated})\mathbb{P}(\text{vaccinated})+\mathbb{P}(\text{infected}\mid\text{unvaccinated})\mathbb{P}(\text{unvaccinated})}$
$\displaystyle=\frac{(1-v_{\text{susceptible}})\cdot\beta_{\text{vaccinated}}}{1-v_{\text{susceptible}}\cdot\beta_{\text{vaccinated}}}.$

Table 4: Vaccine efficacy estimates in the literature at the time of the analysis (summer 2021). | Study | Mean | Sample size
---|---|---|---
$v_{\text{susceptible}}$ | Pouwels et al. (2021) | 40% | 199,411
Puranik et al. (2021) | 42% | 22,064
Fowlkes et al. (2021) | 66% | 2,840
Puranik et al. (2021) | 76% | 21,179
Sheikh et al. (2021) | 79% | 53,679
Andrews et al. (2021) | 88% | 15,871
$v_{\text{source}}$ | Brown et al. (2021) | 0% | 469
Harris et al. (2021) | 50% | 96,898
Levine-Tiefenbrun et al. (2021) | 71% | 4,938
  5. 5.

Masking ($\beta_{\text{masked}}$)

Effective July 30, 2021, the University required all individuals, including
fully vaccinated ones, to wear masks indoors cornell_indoor_mask_requirement.
We assume perfect compliance with this mandate. Thus we set
$\beta_{\text{masked}}=\textbf{100\%}$.

jefferson2023physical conducted a meta-analysis on the effectiveness of
masking for reducing the spread of respiratory viruses during the COVID-19
pandemic. The results of the meta-analysis were inconclusive due to the limits
of the primary-source evidence included in the review Soares-Weiser_2023. The
included studies were conducted in various settings (e.g., hospitals,
communities, households), and masking compliance was low and the intensity and
duration of interaction was high in many of these settings. On the other hand,
students are usually fully compliant with masking in the University’s
classrooms. Therefore, the review conducted by jefferson2023physical would not
have altered our decision to enforce masking.

  6. 6.

Masking effectiveness ($m$)

We define masking effectiveness as the reduction in unmasked transmission
probability due to masking. We assume two-way masking effectiveness, where
both the source and susceptible are masked, to follow Normal(0.855, 0.0536),
from studies on one-way masking effectiveness in the literature.

    1. (a)

We set 80% as an optimistic estimate for one-way masking effectiveness.

      1. i.

Masking of infectious individuals: Kumar et al. (2020) used fluid dynamics
simulation and estimated that 12% of the airflow carrying virus particles
leaks around the side of a mask. Wang et al. (2020) observed that masking by
the primary case and family contacts before the primary case developed
symptoms was 79% effective in reducing transmission.

      2. ii.

Masking of susceptible individuals: Konda et al. (2020) experimentally
measured the filtration efficiency of masks made from different materials and
found that many materials could block particles larger than 0.3 micrometers
with at least 96% filtration efficiency. (Morawska et al. (2009) observed that
most human expiratory activities generate droplets or aerosols with size
larger than this.) Doung-Ngern et al. (2020) reported that people wearing a
mask all the time during contact with a COVID-19 patient are 84% less at risk
of infection. Howard et al. (2021) noted that wearing masks provides
additional protection by preventing touching the nose and mouth, which is
another vector of transmission.

    2. (b)

We set 50% as a conservative estimate for one-way masking effectiveness. Konda
et al. (2020) observed that improper mask-wearing (e.g., having a gap between
the face and the mask) can result in a large decrease in filtration
efficiency. Unmasking temporarily to eat or drink would also reduce the
protection.

    3. (c)

Together, these imply that if masking is enforced, where the source and
susceptible are both masked, transmission risk is reduced from the no-masking
scenario by a multiplicative constant of $(1-80\%)^{2}=0.04$ to
$(1-50\%)^{2}=0.25$. Thus, for two-way masking, we set the prior on the risk
reduction factor to be Normal(0.855, 0.0536), truncated to $[0,1]$. The
untruncated distribution is designed such that [0.75, 0.96] is the 95%
symmetric confidence interval.

  7. 7.

Prevalence in the University population ($p$)

We impose a prior distribution on prevalence, which is computed using the
following procedure:

     * •

We sample $N_{\text{infections}}$, the total number of infections over the
entire semester from a distribution (details below).

     * •

We conservatively assume each case is non-isolated and infectious for half a
week, because surveillance testing requires that undergraduate students get
tested twice a week on average. This is conservative since half a week is the
maximum interval between tests, and the university took explicit measures to
ensure consistent high compliance to testing cornell_test_noncompliance.

     * •

There are roughly 14 weeks in a semester. Thus, we approximate the prevalence
at any time point during the semester as the total number of infected student-
days divided by the total number of student-days in the semester:

$\frac{N_{\text{infections}}\cdot 3.5\text{ days}}{n_{\text{UG}}\cdot(14\cdot
7)\text{ days}}.$

This is a simplifying assumption as we do not model the temporal change of
prevalence as the semester proceeds.

Before the start of the Fall 2021 semester, we assumed that
$N_{\text{infections}}$ follows LogNormal(6.791, 0.413). This is derived by
setting the mode to be 750 and the 97.5% quantile to be 2000. Modeling results
in early 2021 feb2021_report show that a variant with more than twice the
transmissibility of the original strain would lead to approximately 500
student infections, under the same masking and social distancing conditions as
the Fall 2020 semester. To account for the relaxation of social distancing and
masking (outside the classroom) in Fall 2021, we increased this number by 50%
and set 750 to be the mode.

By Sep. 19, 2021, we observed 488 student cases since the start of the
semester and 32 student cases in the past week. Assuming the constant rate of
32 cases/week for the remaining 11 weeks of the semester, we would then expect
to see 488 + 32*11 = 840 cases total. We repeated the simulations with
$N_{\text{infections}}$ following the LogNormal(6.872, 0.372) distribution,
derived by setting the mode to 840 and the 97.5% quantile to be 2000, and got
similar results. At mid-October 2021, the University’s public COVID-19
dashboard showed 22 weekly cases cornell_covid_dashboard, so the assumed mode
of 840 student cases was conservative.

     * •

Prevalence is assumed to be proportional to $N_{\text{infections}}$ (which
follows LogNormal(6.791, 0.413)), so it follows LogNormal(-6.157, 0.413),
where the mean is shifted from that of $N_{\text{infections}}$ by the log of
the proportionality constant.

  8. 8.

Average hours an undergraduate spends in class in a semester
($\tau_{\text{UG}}$)

On average, students at the University are enrolled for 15 credits with 45
hours of course-related work per week, including both lecture, non-lecture
time in classrooms (e.g., recitation), and time spent outside of class on
homework and other coursework. We assume half of the 45 hours is spent in the
classroom. Over a 14-week semester, a student spends $45/2\times 14=315$ hours
in the classroom. Thus we set $\tau_{\text{UG}}=\textbf{315}$.

  9. 9.

Population sizes

    1. (a)

$n_{\text{UG}}=15,000$ university_facts

    2. (b)

$n_{\text{faculty}}=850$

The Ithaca campus has roughly 1,700 faculty members university_facts. We
assume half of them teach undergraduate classes in a semester.

    3. (c)

$n_{\text{graduate}}=3,120$

In Fall 2020, there were 6,239 graduate students IRP_fall_2020. We assume half
of them work as TAs for classes or recitations. Note that some of these TAs
may not interact face-to-face with students, e.g., if their primary
responsibility is grading, and that the risk calculated is the average risk
across all TAs, including these individuals.

    4. (d)

$\beta_{\text{faculty}}=2/3$

On average, a course at the University has three meetings per week with two
being lectures taught by faculty and one being a recitation led by a graduate
student. Thus, we set this parameter to be 2/3.

  10. 10.

Sufficient distancing of instructors We assume that faculty and graduate
student instructors are sufficiently distanced from the students during
lecture that their risk of infection only arises from long-distance
transmission. We assume social distancing is maintained during one-on-one
discussions between instructors and students and that they do not add
significantly to the total interaction time between instructors and students.

## A3. Mathematical Model for the Risk of Transmission over Short and Long
Ranges

In this section, we present the mathematical model for the transmission
probability of COVID-19 depending on the relative location of the source case
and susceptible individual. This is an important component in simulating
infections in a single classroom, as described in Stage 1 in Section A1.

Exposure to respiratory fluids is a major mode of transmission of SARS-CoV-2.
An infectious source case releases the virus through exhalation of virus-
containing respiratory fluids (e.g., through speaking, coughing, or sneezing).
A susceptible person becomes exposed to the virus if they inhale the virus-
containing aerosols or fine droplets, if the virus particles deposit on their
mucous membranes via larger droplets, or if they touch a contaminated surface
and then touch their mucous membranes cdc_transmission_modes. We consider the
first two possibilities here. (Hand sanitizers and disinfecting wipes are
provided in all classrooms, which reduces the risk from the third mode of
transmission.) In particular, we model (1) long-range transmission via
aerosols and fine droplets that suspend in the air and (2) short-range
transmission via large droplets that eventually deposit after being emitted.
Under the assumption that instructors are at least 6 feet away from all
students, instructors are only at the risk of long-range transmission, while
students are subject to the risk of both. The modeling of both types of
transmission relies on the exponential dose-response model, introduced below.

### Exponential Dose-response Model

A dose-response model calculates the transmission probability as a function of
dose, the amount of virus particles a susceptible person is exposed to. In the
exponential dose-response model watanabe2010development, the transmission
probability given dose $D$ takes the form

$\mathbb{P}(\text{transmission})=1-\exp(-c\cdot D),$ (3)

where $c$ is a positive constant. Observe that
$\mathbb{P}(\text{transmission})$ is concave in the dose $D$, a fact used in
Section A1 (Simulation Tool) to observe that the increase in risk created by
adding a second positive in the classroom is smaller than the increase in risk
created by the first positive.

### Long-range Transmission

We base our analysis of long-range transmission on the results from Schijven
et al. (2021), who developed a model for predicting the transmission risk in
an enclosed space due to aerosols only, under the assumption that emitted
aerosols are dispersed across the entire room. The estimated risk depends on
the aerosol-emitting activity (such as breathing, speaking, and singing), the
level of ventilation of the room, the duration of interaction, and the virus
concentration of the source case (measured in the number of virus particles
per unit volume). One key property is that the risk due to aerosol
transmission is uniform across all locations of the room, since the aerosols
disperse quickly across space once emitted.

The risk of aerosol transmission over time $T$ is estimated by the exponential
dose-response model:

$\displaystyle\mathbb{P}_{\text{aerosol}}(\text{transmission},T)=1-\exp\left(-\frac{D(T)}{1440}\right),$

where $D$ denotes the dose, i.e., the amount of virus particles that a
susceptible person receives from the infectious person over time, and 1,440 is
the estimated average number of virus copies to cause illness, according to
schijven2021quantitative. The dose depends on the number of virus particles
emitted over time $T$, which we denote $N(T)$, as follows

$\displaystyle D(T)=N(T)\cdot\frac{\text{inhalation rate of the susceptible
(volume / time)}}{\text{volume of the room}},$

where $N(T)$ depends on the type of aerosol-emitting activity, the viral load
of the infectious person, the ventilation condition of the room, and the
duration of interaction $T$. It is estimated that breathing emits aerosols
containing 3,300 virus RNA copies per hour (assuming a nominal viral load
$10^{8}$ copies per milliliter), while the value is higher for speaking and
singing. We refer the readers to Equations (4) - (14) in Schijven et al.
(2021) for details of the calculation.

We implement a few additional calculations when deploying this model for the
classroom simulation:

  * •

We average the risk over the distribution of the source viral load, which
Schijven et al. (2021) estimated to be log-normal. In particular, we compute
the weighted average of transmission risk given that the source case viral
load is $10^{k}$ copies per milliliter, for $k=5,6,\ldots,11$, with weights
0.12, 0.22, 0.3, 0.23, 0.103, 0.0236, 0.0034 respectively.

  * •

We take into account the effect of vaccination and masking, as described in
the previous assumptions.

  * •

We implement different ventilation conditions, namely no ventilation (a
conservative estimate of the amount of ventilation in naturally-ventilated
spaces), 1 air exchange per hour, and 3 air exchanges per hour. We model the
amount of aerosols present in the room per hour as being reduced by a factor
of two and four under the latter two conditions, respectively.

### Short-range Transmission

Table 5: Notation for functions and parameters in the model for transmission over short distances. Notation | Meaning
---|---
$\phi(r)$ |  | fraction of droplets that deposit at distance $\geq r$ meters
---
from their location of emission in 1D
$\gamma_{1D}(r)$ |  | fraction of droplets deposited at distance $r$ away from the source
---
in 1D
$\gamma_{2D}(r,\theta)$ |  | fraction of droplets deposited at $(r,\theta)$ away from the source
---
in the 2D model
$\alpha$ | parameter defining cone of exposure over $[-\alpha,\pi+\alpha]$
$\phi_{2D,ind}(r,\theta;\alpha)$ |  | fraction of droplets emitted by the source case that reach an
---
individual at $(r,\theta)$ assuming $\alpha$-cone of exposure
$D(r,\theta,T)$ |  | dose of virus that a susceptible person at $(r,\theta)$ away
---
from the source receives throughout interaction duration $T$
$N(x,y)$ |  | number of close contacts seated $x$ rows and $y$ columns
---
away from an index case
$Y(x,y)$ |  | number of close contacts seated $x$ rows and $y$ columns
---
away from an index case that were later confirmed as positive
$d(x,y)$ |  | distance between an index case and a close contact at
---
relative location $(x,y)$, computed from row-wise distance
$d_{r}(x,y)$ and column-wise separation $d_{c}(x,y)$
$q_{\alpha}(x,y)$ |  | expected fraction of close contacts counted at $(x,y)$
---
that are in the $\alpha$-cone of exposure
| $p_{c_{2}}((x,y),T\mid\text{in cone})$
---
| probability that a close contact in the cone of exposure
---
at $(x,y)$ is infected over duration $T$

In this section, we first derive a mechanical model for the deposition of
droplets over two-dimensional space over short distances. Based on the
mechanical model, we derive an expression for the amount of droplets that a
susceptible person at a certain location relative to the source receives
(equivalently, the amount of droplets that reach the susceptible person
spatially). We then model the susceptible person’s risk of infection due to
exposure to viral droplets using the exponential dose-response model (Equation
3). Finally, we estimate the model parameters from a dataset of transmissions
on high speed trains in China hu2021risk. Table 5 summarizes the notation for
the functions and parameters used in developing the model.

We first make a fundamental assumption for model tractability. The
concentration of virus particles in droplets exhaled by a source case is
uniform across all droplets of different sizes.

Assumption 5 allows us to use the volume of viral droplets as a proxy for the
amount of virus that a susceptible person is exposed to. This simplifies the
calculation and allows us to better leverage existing results from the fluid
dynamics literature. We next assume that the transmission of droplets is not
blocked by any obstacles. There are no obstacles between a source and a
susceptible person, regardless of their locations. A susceptible person at
distance $r$ from the source receives all droplets that would deposit at
distance $r$ or further.

Using fluid dynamics modeling and experimental data, Sun and Zhai (2020)
estimated a function for the expected fraction of droplets that deposit at
distance no less than $r$ meters from their location of emission, assuming all
droplets travel in the same direction, over a distribution of droplet sizes
from a typical cough:

$\phi(r)=-0.1819\cdot\ln(r)+0.43276.$ (4)

This formula is valid for $r\in[0.04,10.8]$ meters, at the two ends of which
$\phi(r)$ is equal to 1 and 0 respectively. We let $r_{\min}=0.04$ and
$r_{\max}=10.8$.

If the source case exhaled all droplets in one direction, then a susceptible
person at distance $r$ away in that exact direction would receive a fraction
$\phi(r)$ of the droplets, while a susceptible person in all other directions
would not receive any. In reality, however, the droplets emitted by an
infectious individual may travel in multiple directions xie2009exhaled,
putting surrounding neighbors at different angles at risk. As the droplets are
being spread out in space, the one-dimensional model in Equation 4 does not
accurately capture the amount of droplets that reach one susceptible person in
the vicinity.

Thus, we extend this one-dimensional model to two dimensions. We design our 2D
model of droplet deposition such that it depends on both the distance and
angle of the susceptible person with respect to the source. To ensure our 2D
model is consistent with the 1D model, we make the following assumption:

The fraction of droplets traveling beyond the entire radius-$r$ circle around
the source is the same as $\phi(r)$, the fraction beyond distance $r$ in the
1D model.

Now we derive the 2D model under Assumption Short-range Transmission. Let
$\gamma_{1D}(r^{\prime})$ denote the fraction of droplets deposited at
distance $r^{\prime}$ away from the source in the 1D model. By definition,

$\phi(r)=\int_{r}^{r_{\max}}\gamma_{1D}(r^{\prime})dr^{\prime}.$ (5)

Let $\gamma_{2D}(r^{\prime},\theta)$ denote the fraction of droplets deposited
at distance $r^{\prime}$ and angle $\theta$ away from the source in the 2D
model. By definition and Assumption Short-range Transmission,

$\phi(r)=\int_{r}^{r_{\max}}\int_{0}^{2\pi}\gamma_{2D}(r^{\prime},\theta)r^{\prime}d\theta
dr^{\prime}=\int_{r}^{r_{\max}}\left[\int_{0}^{2\pi}\gamma_{2D}(r^{\prime},\theta)d\theta\right]r^{\prime}dr^{\prime}.$
(6)

The term in the bracket is the fraction of droplets that deposit over the
entire circle of radius $r^{\prime}$. From Equations 5 and 6, the 1D and 2D
models should satisfy the following consistency condition:

$\int_{0}^{2\pi}\gamma_{2D}(r^{\prime},\theta)d\theta=\frac{\gamma_{1D}(r^{\prime})}{r^{\prime}}.$
(7)

Our goal is to model the transmission risk that one susceptible person at
distance $r$ and angle $\theta$ is subject to. Going from the fraction of
droplets depositing over the entire circle, we next explicitly model the
dependence of $\gamma_{2D}(r^{\prime},\theta)$ on $\theta$.

In realistic settings like classrooms or buses, a susceptible person could be
seated in different directions from the source. We would naturally expect some
directions to be riskier and others to be safer. For example, we would think
of seats right behind the source as relatively safe, because the source is
most likely facing and exhaling forward and the chair backs may block the
droplets. On the other hand, it is possible that the source may turn their
head from left to right when seated, so that droplets may potentially even
reach someone sitting in rows behind them.

Based on these observations, we set up a “cone of exposure” model that
quantifies the dependence of risk on angle $\theta$. The cone of exposure,
with parameter $\alpha$ (ranging from 0 to $\pi/2$), covers an arc of
$(\pi/2+\alpha)$ degrees on both sides of the direction that the source case
is facing. Hereafter, we call this an ”$\alpha$-cone of exposure”. An
illustration is given in Figure 12. The source case (purple) is facing up and
emits droplets in directions ranging from angle $\alpha$ behind on their left
to angle $\alpha$ behind on their right. Susceptible cases (blue) sitting
within this cone are at the risk of receiving the droplets; susceptible cases
sitting outside of this cone are not at risk.

Figure 12: Cone of exposure model. The source case is represented with a
purple “X” and the susceptible person is represented with a blue dot. The
source emits virus uniformly over the cone extending from $-\alpha$ to
$\pi+\alpha$. The susceptible person is located at distance $r$ and angle
$\theta$ away from the source. They occupy an angle of $\Delta\theta$ that
scales inversely with $r$.

We choose the right-hand direction of the source to be of angle 0 and measure
$\theta$ counterclockwise. We let $\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}$ be an indicator of whether $\theta$ is in the $\alpha$-cone
of exposure, i.e., $\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}=\mathbbm{1}\\{\theta\in[-\alpha,\pi+\alpha]\\}$. The amount of
droplets depositing is positive for those in the cone and zero for those
outside the cone. Next, we make a further assumption about droplet
distribution within the cone.

At the same distance, droplets are distributed uniformly over all angles in
the cone of exposure.

This simplification aids the analysis, but in reality the distribution of
droplets within the cone may depend on the angle in a more complicated way. We
lack sufficient data to accurately estimate a complex angle-dependent droplet
distribution model, and so a complex model might degrade accuracy rather than
improve it. We thus adopt Assumption Short-range Transmission in the spirit of
regularization, understanding that the model we adopt approximates a more
complex angle-dependent distribution model by replacing small droplet
densities by 0 and large droplet densities by a constant.

With this cone-of-exposure model set up, we can model how the transmission
risk depends on $\theta$. We assume all individuals have comparable width.
Then, for an individual at distance $r$ and angle $\theta$ from the source,
they occupy an arc whose central angle $\Delta\theta(r)$ is approximately
inversely proportional to $r$ and they receive all the droplets that would
land in the sector across $[\theta,\theta+\Delta\theta(r)]$ and
$[r,r_{\max}]$. Based on this insight, we derive
$\phi_{2D,ind}(r,\theta;\alpha)$, the fraction of droplets emitted by the
source case that reach such an individual. We call this quantity the droplet
reception factor.

$\displaystyle\phi_{2D,ind}(r,\theta;\alpha)=~{}$
$\displaystyle\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}\int_{r}^{r_{\max}}\int_{\theta}^{\theta+\Delta\theta(r)}\gamma_{2D}(r^{\prime},\theta)r^{\prime}d\theta
dr^{\prime}$ $\displaystyle=~{}$ $\displaystyle\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}\int_{r}^{r_{\max}}\int_{0}^{2\pi}\gamma_{2D}(r^{\prime},\theta)r^{\prime}d\theta
dr^{\prime}\cdot\frac{\Delta\theta(r)}{\pi+2\alpha}$ $\displaystyle=~{}$
$\displaystyle\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}\int_{r}^{r_{\max}}\gamma_{1D}(r^{\prime})dr^{\prime}\cdot\frac{\Delta\theta(r)}{\pi+2\alpha}$
$\displaystyle=~{}$ $\displaystyle\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}\cdot\phi(r)\cdot\frac{\Delta\theta(r)}{\pi+2\alpha}$
$\displaystyle\propto~{}$ $\displaystyle\mathbbm{1}\\{\theta\text{ in
cone};\alpha\\}\cdot\phi(r)\cdot\frac{1}{r},$ (8)

where the second equality follows from Assumption Short-range Transmission,
the third equality follows from Equation 7, and the fourth equality follows
from Equation 5. Later, we will show that having an undetermined
proportionality constant does not affect our results as long as the dependence
on $r$ and $\theta$ is modeled correctly.

##### Transmission probability calculation

Given the expression for the droplet reception factor, we translate this to
the probability of transmission using the exponential dose-response model in
Equation 3. The dose that a susceptible individual at $(r,\theta)$ from the
source receives is proportional to the fraction of the source’s virus
particles that they receive. By Assumption 5, this in turn is proportional to
the droplet reception factor $\phi_{2D,ind}(r,\theta;\alpha)$.

Next, we observe that the dose is larger if the source case and susceptible
person maintain the same relative location longer. We call this the duration
of interaction, denoted $T$. For example, $T$ is roughly one hour for a
lecture. We make the following assumption about the droplet emission rate over
time. The amount of droplets a source case emits per unit time is constant.

Under Assumption Transmission probability calculation, the amount of droplets
emitted, and hence the dose, is proportional to the duration of interaction
$T$. Thus, the dose of virus particles that a susceptible person at
$(r,\theta)$ away from the source receives can be expressed as

$\displaystyle D(r,\theta,T)=c_{1}\cdot\phi_{2D,ind}(r,\theta;\alpha)\cdot T,$

where $c_{1}$ captures the proportionality of the dose to the droplet
reception factor and the duration of interaction. We further absorb into
$c_{1}$ the proportionality relation within the droplet reception factor
(Equation 8), yielding another constant $c_{2}$, and derive the final
expression of the transmission probability for a susceptible person at
$(r,\theta)$ away from a source case for a duration of interaction $T$:

$\mathbb{P}_{\text{droplet}}(\text{transmission},r,\theta,T;\alpha,c_{2})=1-\exp\left(-c_{2}\cdot\mathbbm{1}\\{\theta\text{
in cone};\alpha\\}\cdot\frac{\phi(r)}{r}\cdot T\right).$ (9)

As a sanity check, we can see that if a susceptible person at $(r,\theta)$ is
not in the cone of exposure, the transmission probability is 0.

##### Parameter estimation

The goal of this section is to first derive the likelihood for an empirically
observed dataset based on the model above, and then find values of $\alpha$
and $c_{2}$ that maximize the likelihood.

Hu et al. (2021) studies 2,334 confirmed positive cases (“index cases”) and
72,093 close contacts who had co-travel times of 0-8 hours from 12/19/2019
through 3/6/2020 on high-speed trains in China. They examine the association
of attack rate with the spatial distance between pairs of index cases and
close contacts. Here, a “close contact” was defined as a person who had co-
traveled on a train within a 3-row seat distance of an index case within 14
days before symptom onset. We treat a close contact as equivalent to a
susceptible individual in our model. Table 6 reports the number of close
contacts, and, among them, those that were later confirmed as positive, at
different seat locations with respect to an index case, within the period of
study. We would like to derive the likelihood of observing the data in Table
6.

We now introduce additional notation for formalizing our likelihood model. Let
$(x,y)$ denote the seat at $x$ rows and $y$ columns away from an index case,
where $x$ ranges from 0 to 3 and $y$ ranges from 0 to 5. Let $N(x,y)$ denote
the number of close contacts that are seated $x$ rows and $y$ columns away
from an index case (hereafter we call this “at relative location $(x,y)$” for
abbreviation). Let $Y(x,y)$ denote the number of close contacts at relative
location $(x,y)$ from an index case that were later confirmed as positive.
Based on the train cabin layout give in Figure 1 in Hu et al. (2021), we
calculate the separation between an index and a close contact at relative
location $(x,y)$. In particular, let $d_{r}(x,y)$ and $d_{c}(x,y)$ denote the
row-wise and column-wise distance in meters. For all $y$, $d_{r}(x,y)=0.9x$;
for all $x$, $d_{c}(x,y)$ is equal to 0, 0.5, 1.05, 1.6, 2.1, and 2.6 for
$y=0,\ldots,5$ respectively. We then calculate
$d(x,y)=\sqrt{d_{r}(x,y)^{2}+d_{c}(x,y)^{2}}$ (Table 7).

Table 6: Cases number and total number of passengers co-traveled with an index patient (Hu et al. (2021), Table S1). Entries are $Y(x,y)/N(x,y)$, where $x$ is the number of rows apart and $y$ is the number of columns apart. Rows apart | Columns apart
---|---
0 | 1 | 2 | 3 | 4 | 5
0 | – | 92/2605 | 33/1996 | 7/1845 | 7/1825 | 3/1028
1 | 10/4791 | 12/5084 | 5/3664 | 3/3464 | 1/3525 | 1/1872
2 | 11/4386 | 8/4751 | 8/3429 | 5/3212 | 3/3250 | 3/1769
3 | 2/4026 | 2/4395 | 4/3110 | 3/2945 | 3/2970 | 1/1589
Table 7: Distance $d(x,y)$ in meters, where $x$ is the number of rows apart and $y$ is the number of columns apart. Rows apart | Columns apart
---|---
0 | 1 | 2 | 3 | 4 | 5
0 | – | 0.5 | 1.05 | 1.6 | 2.1 | 2.6
1 | 0.9 | 1.03 | 1.38 | 1.84 | 2.28 | 2.75
2 | 1.8 | 1.87 | 2.08 | 2.41 | 2.77 | 3.16
3 | 2.7 | 2.75 | 2.90 | 3.14 | 3.42 | 3.75

The data does not contain information about which direction the close contacts
were seated with respect to the index cases. However, directionality
information is crucial for our modeling. Thus, we make the following
assumption about the symmetry of distribution of close contacts. A close
contact counted at $(x,y)$ is equally likely to have been seated in all
possible directions at location $(x,y)$ away from the source.

With slight abuse of notation, we let $(+x,+y)$ and $(+x,-y)$ denote the seats
$x$ rows in front of the index case and $y$ columns to the right and left,
respectively. Similarly, we let $(-x,\pm y)$ denote the seats $x$ rows behind
the index case and $y$ columns to the right or left. Because we assume the
cone of exposure has an angle larger than $\pi$, the seats $(+x,\pm y)$ are
always in the cone of exposure. The seats $(-x,\pm y)$ are in the cone of
exposure if and only if
$\arctan\left({d_{r}(x,y)}/{d_{c}(x,y)}\right)\leq\alpha$.

Let $q(x,y;\alpha)$ denote the expected fraction of close contacts counted at
$(x,y)$ (which could be at $(\pm x,\pm y)$) that are in the $\alpha$-cone of
exposure. Based on Assumption 7,

$\displaystyle
q(x,y;\alpha)=\frac{1}{2}+\frac{1}{2}\mathbbm{1}\\{\arctan\left({d_{r}(x,y)}/{d_{c}(x,y)}\right)\leq\alpha\\}.$
(10)

We can calculate $q(x,y;\alpha)$ for all possible $(x,y)$ pairs using Table 7.

Next, let $p((x,y),T\mid\text{in cone};c_{2})$ denote the probability that a
close contact in the cone of exposure at location $(x,y)$ is infected. Based
on Equation 9, we model this as

$\displaystyle p((x,y),T\mid\text{in
cone};c_{2})=1-\exp\left(-c_{2}\cdot\frac{\phi(d(x,y))}{d(x,y)}\cdot\kappa_{\text{mask}}\cdot
T\right),$ (11)

where we include an additional factor of masking effectiveness
$\kappa_{\text{mask}}$, due to the fact that wearing a mask can reduce the
amount of virus that a susceptible person is actually exposed to (compared to
the virus in the droplets that reach where the person is), and that mask-
wearing had been quite prevalent since late January of 2020 in China.
konda2020aerosol estimated that a poorly fitting mask made of cotton or silk
will reduce virus dose by approximately 30%. We heuristically select
$\kappa_{\text{mask}}=0.8$, assuming that the dataset involved a mix of both
masked and unmasked passengers222We constructed this model prior to conducting
the simulations described in Appendix A1 and A2, so we used a point estimate
for masking effectiveness. Nevertheless, the point estimate of 0.8 is well-
aligned with the prior choice for masking effectiveness parameter $m$,
discussed in Appendix A2.. We keep this constant separate from $c_{2}$ so that
$c_{2}$ solely captures the way transmission probability depends on the
unreduced dose. We set $T$ to be 2.1 hours. This is the mean co-travel time
over all pairs of index cases and close contacts in the data. Unfortunately,
no information is given about the co-travel time for each individual pair.

We let $r(x,y;\alpha,c_{2})$ denote the overall transmission probability at
relative location $(x,y)$ under parameters $\alpha,c_{2}$. This is the product
of the probability that a close contact at $(x,y)$ is in the cone of exposure
(Equation 10) and the conditional probability that they become infected given
that they are in the cone (Equation 11):

$\displaystyle r(x,y;\alpha,c_{2})=q(x,y;\alpha)\cdot p((x,y),T\mid\text{in
cone};c_{2}).$

For the likelihood model, we assume the number of transmissions at each
$(x,y)$ independently follows a binomial distribution:

$\displaystyle Y(x,y)\sim\text{Binomial}(N(x,y),r(x,y;\alpha,c_{2})).$

Assuming the numbers of transmissions at each $(x,y)$ are independent, the
likelihood of all observations
$\mathcal{D}:=\\{N(x,y),Y(x,y)\\}_{\begin{subarray}{c}x=0,\ldots,3\\\
y=0,\ldots,5\end{subarray}}$ is given by:

$\displaystyle\mathcal{L}(\mathcal{D})=\prod_{(x,y)}\binom{N(x,y)}{Y(x,y)}\cdot
r(x,y;\alpha,c_{2})^{Y(x,y)}\cdot(1-r(x,y;\alpha,c_{2}))^{N(x,y)-Y(x,y)}.$

We compute the log-likelihood and let $c_{3}$ denote the constant term that
does not depend on $\alpha$ or $c_{2}$:

$\displaystyle\mathcal{\ell}(\mathcal{D})=\displaystyle\sum_{(x,y)}\left[Y(x,y)\log(r(x,y;\alpha,c_{2}))+(N(x,y)-Y(x,y))\log(1-r(x,y;\alpha,c_{2}))\right]+c_{3}.$
(12)

We next find values of $\alpha$ and $c_{2}$ that maximize the log-likelihood
$\mathcal{\ell}(\mathcal{D})$. Using a discretized grid search, we find that
the log likelihood is maximized at $c_{2}=0.0135$ and multiple values of
$\alpha$. We choose the largest possible value $\alpha=15$ degrees, or 0.26
radians, with the intention of being conservative.

### Combining the Short and Long-range Transmission Risk

In our simulation, the risk for a susceptible individual is

$\displaystyle\max\left(\mathbb{P}_{\text{droplet}}(\text{transmission}),\mathbb{P}_{\text{aerosol}}(\text{transmission})\right),$

i.e., the larger of the predicted risk due to short and long-range
transmission. This is justified by the fact that we inferred the parameters
for the droplet model assuming all infections in the dataset result from
short-range transmission; as such, the inferred model implicitly accounts for
long-range transmission in the dataset. In practice, the short-range
transmission risk only dominates the long-range transmission risk at short
distances (approximately within three meters), while the risk due to aerosol
is uniform across all locations.

Finally, we recall that the computed risk so far is based on studies on the
original virus strain. Thus, we multiply the calculated risk by 2.4 to account
for the increased transmissibility of the Delta variant, as described in
Appendix A2.

informs2014 references