COVID-19 Test Allocation Strategy to Mitigate SARS-CoV-2 Infections across School Districts

In response to COVID-19, schools across the United States closed in early 2020; many did not fully reopen until late 2021. Although regular testing of asymptomatic students, teachers, and staff can reduce transmission risks, few school systems consistently used proactive testing to safeguard return to classrooms. Socioeconomically diverse public school districts might vary testing levels across campuses to ensure fair, effective use of limited resources. We describe a test allocation approach to reduce overall infections and disparities across school districts. Using a model of SARS-CoV-2 transmission in schools fit to data from a large metropolitan school district in Texas, we reduced incidence between the highest and lowest risk schools from a 5.6-fold difference under proportional test allocation to 1.8-fold difference under our optimized test allocation. This approach provides a roadmap to help school districts deploy proactive testing and mitigate risks of future SARS-CoV-2 variants and other pathogen threats.


Classroom
Each classroom is composed of 20 students and a single teacher. Students and teachers are assigned to a classroom at the beginning of the simulation and remain in the same room throughout the simulation. Therefore, each student interacts with 19 other students, as well as the same teacher, while in the classroom. Students spend 6 hours a day in the classroom.

Bus
All students commute to and from school by taking a bus. Each bus transports the same 20 students throughout the simulation and is driven by the same driver, and students are randomly assigned to a bus, independently of the classroom to which they belong. Therefore, each student interacts with 19 other students as well as the same driver while on the bus.
Students spend 1 hour on the bus each day, half an hour at the beginning and at the end of the day.

Break
During the noon break students interact with other students throughout the school in cliques of 10 students; in addition, each student interacts with two adults randomly selected from the teachers and staff. The cliques of students, as well the two adults each student interacts with, are randomly determined independently of classrooms and buses, and remain constant throughout the simulation. Therefore, each student interacts with 9 other students as well as two adults while on break. Students spend 1 hour on break each day (Appendix Table 1).
Outside of school, students interact with the adults in their households. The number of adults in each student's household is determined according to data collected by the US Census Bureau regarding living arrangements of children under age 18 (2). For simplicity, we do not explicitly model siblings within households who attend the same school. Household transmission between such siblings can amplify school-based outbreaks if siblings who are infected at home return to school while infectious. However, the household attack rate of 16.6% (3), and the limited number of sibling pairs in different classrooms (4), would make such occurrences relatively small. In addition, the baseline quarantine strategy of quarantining an entire individual's household upon a positive test would prevent some of those sibling infections from spreading further in school.
Rather than explicitly modeling contacts within households, we use a published estimate of the household attack rate of COVID-19 and assumed infected individuals, whether adults or children, symptomatic or not, transmit the disease to 16.6% of their susceptible household (3).
The exact time of infection is determined randomly based on the relative infectiousness of an individual through time (details below).
All interactions with the broader community are abstracted and included through a single daily community incidence value that is kept constant throughout the simulation. We determined the number of new infections due to community interactions through a binomial process. For instance, if we denote the daily community incidence of COVID-19 by p, e.g., p = 70 new daily cases per 100,000 population, then the number of newly infected individuals in a group with N individuals is sampled from a binomial distribution as Binomial(N,p).
To account for time spent in school, the daily incidence of new cases among members of the school environment on weekdays is half that of the community, but incidence is the same on weekends.

COVID-19 Natural History
The natural history of COVID-19 is modeled according to the diagram shown in Appendix Figure 1. Infected individuals move to the exposed compartment (E) before progressing to either a presymptomatic (PY) or pre-asymptomatic (PA) compartment, from which they then move to the symptomatic infectious (IY) and asymptomatic infectious (IA) compartments, respectively. From there all infected individuals recover (R) and become immune to the disease. The transition times from one compartment to another are determined at the time of infection and follow the probability distributions listed in Appendix Table 2.
The infectiousness of all individuals varies through time, and given the average sojourn times in each compartment, the infectiousness profile follows a gamma distribution with shape and scale parameters of 2.0 and 1.55, shifted left by 2.3 days. The resulting profile is shown in blue in Appendix Figure 2, and based on the time of its maximum value (mode) this results in peak infectiousness of ≈0.7 day before symptom onset, with 45% of total infectiousness for an individual occurring in the pre-(a)symptomatic compartment, following the results from He et al. (5). The curve is normalized so the average infectiousness is 1.0.
As the time spent in each compartment by different individuals is randomly sampled from the probability distributions listed in Appendix Table 2, the infectiousness profile in blue in Appendix Figure 2 is adjusted for the specific time spent by an individual in the pre-(a)symptomatic and (a)symptomatic compartments using the standard times of symptom onset (or transition to infectious asymptomatic compartment), start of infectiousness (transition of pre-(a)symptomatic compartment), and recovery as the control points. For instance, individual 1 shown in green in Appendix Figure 2 spends less time in the pre-symptomatic compartment than average so the green curve before time 0 has the same shape as the blue one, but is compacted from 2.3 days to 1.8 days. Then individual 1 spends more time in the symptomatic compartment than average, so the green curve after time 0 has the same shape as the blue curve, but it is stretched to 10 days.
In addition, individuals who remain asymptomatic are assumed to be two-thirds as infectious as symptomatic persons at any given point (6), with 20% of children and 57% of adults becoming symptomatic (7,8).

Infection Events Modeling
When an infected individual indexed by j interacts with a susceptible individual indexed by k at time t in school for one time step of half an hour, the probability that j infects k is given by: where ωj = 1.0 if individual j is symptomatic or pre-symptomatic and ωj = 2/3 if individual j is asymptomatic or pre-asymptomatic; here, ij(t) represents the relative infectiousness of individual j at time t as shown in Appendix Figure 2, and β is an input parameter that is selected so that the unmitigated reproduction number R0 of children in school is equal to a chosen input. It represents the probability that a symptomatic individual with relative infectiousness of 1.0 at time t infects a susceptible contact in a half-hour interval.
We use the following to determine β: dI = dP + dS the total infectious period in days, which is the sum of the pre-(a)symptomatic and (a)symptomatic periods τ proportion of symptomatic students hx, Cx the number of hours spent in school context x (of the 3 contexts detailed above) and the number of contacts in that context ωɑ = 2/3, ωs = 1 scaling factor of the relative infectiousness of asymptomatic and symptomatic individuals through their infection Δt = 1/2-hour single time step duration in the simulations We can calculate the desired R0 in school as the sum of R0 in the three different school The basic reproduction number in a given place is simply the probability of infecting a given contact multiplied by the number of contacts: R0(x) = Cx • p(x), where p(x) represents the probability of infecting a single individual contact in context x over the individual's entire infectious period. Then: where we make the simplification that the probability of infecting an individual is directly proportional to the total time spent in contact with them.
We now only need to calculate the probability of infecting an individual who is a classroom contact, p(class). This probability is averaged over symptomatic and asymptomatic individuals. To simplify calculations, we calculated the probability that a symptomatic individual would infect another person in their classroom, p(class), with subscript s to denote symptomatic, which is given by: We can then write the relationship between the probability of a symptomatic infectious individual infecting a contact at some point in time, with the probability of infection at each time step in the simulation via: By definition, for a symptomatic individual we have qt = β • ij(t), which depends on t through the individual's relative infectiousness. We simplify this by qt = β using the fact that ij(t) is constructed to have an average value of 1. As a result, we have the following: where N represents the average number of time steps that an infected individual has with other contacts in their classroom while infectious: and we use 5/7 to represent the fact that school days occur on weekdays only.
Finally, we obtain .

Testing and Isolation
In all scenarios, we assumed that 90% of symptomatic individuals will seek testing once symptoms occur. The exact time after symptom onset at which individuals seek testing is random and is sampled from a triangular distribution with an average of 1 day, a lower bound of 0.5 day, and an upper bound of 1.5 day. We assume that the tests that symptomatic individuals use are perfect, and separate from the surveillance testing budget. In addition, 2 hours after getting tested, the symptomatic individual starts their isolation, and if applicable their contacts quarantine themselves at the same time. For surveillance testing the results are assumed to be instantaneous. The infected individual then starts their isolation immediately, and their contacts quarantine themselves immediately as well. In the baseline scenarios all isolations and quarantines last 14 days, according to the Austin Independent School District (AISD) policy during the 2020-2021 school year (9).
In our simulation, surveillance tests are all done on Mondays at 8 AM, right after students take the bus to school and before the first class. If students are isolated or quarantined for 2 weeks, they come back to school on the second Monday after testing positive.
In addition, surveillance tests are allocated across classrooms every week. For instance, if 50% of a school's students are tested every week, then 50% of the students in each classroom are tested each week, according to a defined schedule so that every student will be tested every other week. This method of test allocation within a school performed better in our experiments than randomly selecting the students for testing, or testing entire classrooms some weeks while no student is tested in other classrooms.
We assumed that individuals strictly quarantine themselves, so that they cannot be infected with contacts from the broader community. However, in this case, students could still be infected by one of their household members.
In our base-case scenario, we assumed that surveillance tests were perfect, but we ran some sensitivity analysis relaxing this assumption. We ran some scenarios where the tests had a sensitivity of 95% for symptomatic individuals, 80% for pre-(a)symptomatic and asymptomatic individuals, and a 99% specificity, which corresponds to pre-Delta estimates published for the Abbott BinaxNOW tests (10,11). Parameters are provided in Appendix Table 2.

Optimization Model
We optimize the allocation of tests across a set of schools to minimize a chosen risk metric. The results shown in the main text seek to minimize the maximum risk across schools and the associated model is presented in the following section. We have explored other objective functions, and we show the corresponding model formulations in this section as well as the resulting allocations as a sensitivity analysis further below.
As part of our optimization model formulation, we use the preprocessed results from the simulation model above as inputs, so the optimization part of the framework is run independently and subsequently to the disease transmission model.

Notation
Set and indices: s ∈ S set of schools in the system. Parameters: Ns number of students in school s; B testing budget, expressed as the proportion of students in the entire system that can be tested weekly.
Variables: ts proportion of students tested each week in school s; Is(ts) proportion of students in school s infected on-campus over the horizon under testing regime ts.
The decision variable ts represents the testing regimen in a school and is expressed as the proportion of the school's students tested each week. For example, ts = 50% means that every week 50% of the school's students are screened so that, on average, students get tested every other week, while with ts = 33% students are tested every 3 weeks on average. Evaluating Is(ts) requires running the simulation model of disease transmission that we sketch above for 300 simulations, given a specific value of ts.

Model
The objective of the optimization model is to minimize the maximum risk experienced by any school in the system subject to two constraints.
First, we cannot allocate more tests than are available in the budget: Second, we limit testing to a weekly frequency in each school: The risk metric we aim to minimize is the average of (i) the on-campus expected infection rate across simulations Ε{Is(ts)}, and (ii) the conditional value-at-risk (CVaR) of the infection rate at a level β = 90%, CVɑR90{Is(ts)}. Therefore, the risk for school s given a testing regimen ts is . CVaR, also called expected shortfall, is a widely used risk measure in stochastic optimization, thanks to its coherence properties, ease of interpretation, and computational tractability (13,14). Consider a random variable, X, that we would like to keep "small," such as the proportion of a population that is infected with COVID-19. CVaR is the conditional expectation given that X exceeds its β-level quantile. Thus, in our case, with β = 0.90, CVɑR90{Is(ts)} computes the conditional expectation of the worst 10% of the outcomes, or the average proportion of infected students at a school, when restricting that average to the 30 simulated scenarios out of 300 with the largest proportion of infections. Hence, including this term helps determine a test-allocation strategy that controls tail risk. Formally, for a random variable X with cumulative distribution function F(x), value-at-risk (VaR) and CVaR are defined as follows: .
Then the optimization problem can be expressed as the following: Put together and reformulated, this gives us the following optimization model, where the decision variables are the various amounts of testing ts in each school: Due to the nonlinear, and effectively black-box, nature of the functions Is(ts) with respect to ts and the fact that we summarize the risk of each school using the infection rate's expected value and CVaR, this is a nonlinear black-box continuous optimization problem that can be solved with standard solvers. We used the COBYLA method implemented in SciPy/Python to solve the problem (15,16).
Our optimization model is nonlinear and nonconvex but has two key properties that allow us to readily check whether a solution is globally optimal. First, the risk function associated with each school is a decreasing function of the allocation ts. Second, we are solving a continuous minimax problem in which we are focused on the school with the worst-case risk. Thus, we can first allocate resources to the worst-case school to decrease its risk to that of the school with the second-highest risk, and repeat this scheme until the budget is exhausted, dealing with obvious edge cases. This allows us to verify that the solution found via COBYLA is indeed a globally optimal solution.
Testing cannot be more frequent than weekly, i.e., ts<1. When solving problem (P), we may observe that some schools, indexed by say, s ∈ S′, are allocated enough tests for weekly testing. To help reduce the dimension and aid the solver in its search, we can fix the corresponding decision variables of the schools with weekly testing ts, s ∈ S′, to 1, and rerun COBYLA to optimize the allocation for the remaining schools, excluding those schools from the first constraint in (P). The problem then becomes the following: After sketching two alternative formulations, we describe another means by which we help the optimization algorithm in terms of preprocessing output from the simulation model.

Alternative Objective Functions
Below we show the optimization model when minimizing different objective functions.
In Appendix Figure 15, we show the resulting optimal allocations from solving those problems for the hypothetical school system.

Varying CVaR Level and Relative Weight
Two straightforward modifications to the original optimization problem are to change the level ꞵ of CVaR as well as the respective weights of CVaR and of the expectation in the risk level of each school. The problem is easily modified, using some value, w ∈ [0,1] for the weight of CVaR, we have the following: By setting w to 0 or 1 we can focus either on the expected infection rate or on the CVaR of infection rates respectively, thus ignoring tail risk entirely or focusing solely on it.

Total Expectation
A different approach is to try to minimize total infections occurring across the entire system rather than trying to minimize the risk for the highest-risk school, as in the main text.
Then the size of different schools, through the number of students Ns, directly impacts the objective function. The problem is formulated as follows:

Inputs Preprocessing
To run the optimization model, we need to preprocess the results from the transmission model to smooth out some of the stochasticity of the simulations. We do this in two steps.
First, we take the results from the 300 simulations for a single parameter set, which corresponds to a specific school with a certain testing frequency, and we fit a gamma distribution to the proportion of students infected on-campus (Appendix Figure 3, panel A). This ensures that the risk metrics we calculate are not biased by single simulations and that the CVaR metric stays continuous with respect to the exact level β chosen.
Second, using those fitted gamma distributions, we calculate the risk for all schools for each of the 21 testing amounts simulated, from no students tested to all students tested weekly, in 5% increments. This produces the blue dots in Appendix Figure 3, panel B. We then fit a nonincreasing curve to these dots, specifically we fit a cubic curve up to the point where 75% of students are tested weekly and then a linear curve. This piecewise definition of the curve helps us keep the number of parameters necessary for fitting low enough to have a parsimonious model while having a good fit.
This process ensures that the risk level of a school decreases as the testing frequency increases, which might not always be the case in our simulations due to stochasticity, especially when the number of infections is already low. Additionally, explicitly pre-calculating the risk level of a school as a function of the testing frequency ensures rapid execution of the optimization routine because the COBYLA algorithm might require a large number of iterations when the risk level of each school must be evaluated.

AISD Details
The list of high schools from the Austin Independent School District (AISD) included in our analysis is given in Appendix Table 3 along with the number of students modeled in our simulations (17). The number of students in each school is based on enrollment reports from the Texas Education Agency (TEA) multiplied by the reported in-person attendance of 6% in AISD high schools at the beginning of January 2021 (18,19). We rounded the numbers of students to get classes of 20 students and calculated the number of adults working in the school by scaling the numbers given in Appendix Table 2 to the numbers of students in each high school. To assign R0 to each school, we used the reported COVID-19 cases in each AISD school from the district's public dashboard on March 8, 2021 (20), as well as the total enrollment and staff data per school used by TEA for allocation of tests in the context of the K-12 COVID-19 testing project (21).
We then regressed the number of reported cases on the total school population and obtained a best fit line of the form: y = y0 + α • x = 5.82 + 0.0095 • x with R 2 = 0.44, as shown in Appendix Figure 4.
We then assigned an R0 in each school proportional to the square root of the ratio of reported cases yi to predicted cases: with a base value of R0 = 1. For a school that reported yi cases with a total population xi we assigned the following: .
The values assigned to each school are listed in Appendix Table 3.

COVID-19 Incidence in School Catchment Areas
To calculate the relative community incidence of COVID-19 cases for each school we combine catchment area information from AISD and COVID-19 burden data in the Austin metropolitan statistical area (MSA).
We first obtain the cumulative COVID-19 hospitalization rate, hj, per 100,000 persons for each postal code j in the Austin MSA from March 2020-January 11, 2021 (22); then, we calculate the average hospitalization rate in the metropolitan area, hMSA. From there we computed the relative burden experienced by each postal code j as follows: . Next, we estimated the proportion of students in each school i coming from postal code j (pij) by using the proportion of school i's catchment area located in postal code j as a proxy. To do so, we use GIS data from AISD and Austin MSA (23,24).
We denote by Ai the area of the catchment area of school i, the area covered by postal (i.e., ZIP) code j (Zj), and Ai ∩ Zj the area of the intersection of the two. We then have, Combining the two quantities we then estimate the relative community incidence si of each school as the weighted average of the postal codes relative burden via the following:

Additional Results
Appendix Table 4 gives the cumulative on-campus attack rate in each school for three allocation strategies: no testing, prorated allocation of 14-day total testing capacity, and optimal allocation of 14-day prorated total testing capacity. The results are given for the scenarios in which all schools have different transmission rates (first three substantive columns), and when they all have the same transmission rate (R0 of 1.0, last three columns). The prorated and optimal allocations when schools have different transmission rates correspond to the results shown in Figure 4 panels A, B.
The additional results below analyzing the 11 high schools in AISD assume rapid tests with a sensitivity of 95% for symptomatic individuals, 80% for asymptomatic people, and a 99% specificity, based on reported estimates of the Abbott BinaxNOW antigen tests (10,11). Appendix Figure 5 shows results similar to those shown in Figure 4 Figure 6 shows results similar to those shown in Figure 5 (main text), except that here the schools have the same transmission risk of R0 = 1.0. Appendix Figure 7 shows the risk level of each school under an optimal allocation when the on-campus transmission rate used as input to the optimization problem is different from the actual transmission rate of each school. Specifically, we derived an optimal allocation assuming all schools have the same on-campus transmission rate R0 of 1.0 when schools actually have different on-campus transmission rates. The gray dots correspond to the resulting risk levels under that allocation, while the blue dots correspond to the risk level achieved when using the correct transmission rate to derive the optimal allocation, and the orange dots correspond to a pro rata allocation. The risk of most schools ends up being similar under the two optimal allocations, with the allocation derived using the wrong transmission rates typically resulting with in a risk level that is closer to the optimized risk level than that of the pro rata allocation, with the exception of schools H and I for which the pro rata allocation happens to be very close to optimal in that scenario.

Toy System Results
Appendix Table 5 below contains the details of the numbers shown in Figure 1, panel A in the main text. It gives the expected proportion of students infected on-campus for different quarantine strategies, testing frequencies, and in-school R0. Appendix Table 6 contains the details of the numbers shown in Figure 2, panel A in the main text. It gives the proportion of school days students either miss school, infected or not, or are at school while infected.

Toy System Optimization
We show the results of the optimal test allocation on a toy system of hypothetical schools (Appendix Figures 8, 9, 14). There are six schools of 500 students each in this system, three with a low daily community incidence of new cases (35 per 100,000) and three with a high daily community incidence (70 per 100,000). In each of the two groups the schools have different inschool transmission rates, low (unmitigated R0 = 1.0), moderate (unmitigated R0 = 1.5), and high (unmitigated R0 = 2.0).
Appendix Figure 8 shows the number of averted infections through surveillance testing with a 2-week testing frequency budget, compared to no surveillance testing, for both a prorated allocation and an optimal allocation. Using the distributions of on-campus infections for a school under the scenario without testing, and under a scenario with testing, we can calculate the number of averted infections through testing using inversion sampling. We denote by Fno testing and Ftesting the cumulative distribution functions (CDF) for the number of on-campus infections of the scenarios without testing and with testing, respectively. We then generate 300 random numbers Uj, j ∈ [1, 300], uniformly distributed between 0 and 1, and we calculate the number of averted infections as follows: . As we move from no testing to a prorated allocation, we decrease the risk of all schools, and when we move from the prorated allocation to an optimal allocation, the performance of the three lowest-risk schools worsens slightly, i.e., the low-low, low-moderate, and high-low schools (incidence-transmission pairs). Meanwhile the number of cases averted at the remaining three schools grows significantly; hence, the total system risk decreases in an optimal allocation. Appendix Figure 9 shows the optimal allocation of tests to all schools under different testing budgets, with the 14 days column corresponding to the allocation that yields the distributions of averted cases shown in Appendix Figure 8. No matter the testing frequency, under an optimal allocation the testing capacity is diverted from the same three low-risk schools to the higher-risk schools. The low-low school receives the fewest tests under all budgets, while the high-high school receives the most. For instance, for a budget that would allow us to test all students in the system every 10 days on average, it would be optimal to test all students in the high-high school every week, while it would only be necessary to test students in the low-low school every 4 weeks to achieve the same overall level of risk. As a result, the risk profiles of all schools are very similar under the optimal allocation below that shows the distribution of oncampus infections for all schools under prorated and optimal allocations (Appendix Figure 14).

Quarantine Strategy and Imperfect Surveillance Tests
It is now widely accepted that antigen tests for COVID-19 are effective at detecting infectious people, but there have been debates regarding their true performance (25). As such, we ran some sensitivity analyses using imperfect surveillance tests. Specifically, we used a test sensitivity of 95% for symptomatic individuals, 80% for asymptomatic (including presymptomatic) people, and we used a specificity of 99%, corresponding to published estimates of the Abbott BinaxNOW tests (10,11). Appendix Figure 10 shows results similar to those shown in Figure 1A and 1B (main text), the proportion of students infected in schools under different testing frequencies, for the same two quarantine strategies considered in Figure 1 (main text), as well as a scenario using imperfect tests and classroom quarantines.

School Days Missed
In Figure 2, panel A (main text) we detail the proportion of school days missed for a school with low community incidence and moderate transmission risk, for two quarantine strategies. In Appendix Figure 11, we show analogous results for the other hypothetical schools considered in our system, as well as the case in which imperfect tests are used.
The results are qualitatively similar across schools. We notice that holding other factors constant, a higher community incidence leads to more school days missed as more introductions of the disease in the school lead to more students being quarantined.
The bottom panels in each figure consider the case in which entire classrooms are quarantined while using imperfect tests with 99% specificity and with 80% and 95% sensitivity for asymptomatic and symptomatic individuals, respectively. While these tests reduce cases to a similar extent as perfect tests (see Appendix Figure 10) the false positives (students testing positive despite not being infected) lead to students missing more school time while healthy, with school days missed increasing with the testing frequency.
We can also see that when the transmission risk is high, increasing the testing frequency does not lead to more school days missed when quarantining entire classrooms. Indeed, when the transmission risk is low enough some of the infected individuals found through surveillance testing would not infect anyone else, but entire classrooms are still quarantined, so that more frequent testing leads to more missed days. On the other hand, when the transmission rate is high more frequent testing allows us to find infected individuals earlier, before more infections occur, thus breaking transmission chains. This effect is also seen by looking at the proportion of days in school while infected when only households are quarantined: when the transmission risk is low that proportion decreases slowly as testing increases while it decreases dramatically as testing frequency increases when transmission risk is high.

Quarantine Duration and Contacts Quarantined
While many school districts have used a policy of quarantining contacts of a person testing positive for COVID-19 for 14 days, the Centers for Disease Control and Prevention (CDC) has continually been updating its interim guidance throughout the pandemic, saying in February 2021, that 10 days of isolation were sufficient in most cases, and in January 2022, it further shortened the isolation period to 5 days (26). We evaluated the impact of shorter isolation periods, either 7 days or 10 days, and compared it with our baseline assumption of a 14-day isolation period. Appendix Figure 12 shows the tradeoff between reducing infections and minimizing missed school days for different quarantine strategies at various testing frequencies for schools with a moderate transmission risk.
The different quarantine strategies are represented by different marker types and darker colors indicate more frequent surveillance testing. While quarantining entire classrooms for longer periods reduces infections more than shorter quarantines, it also leads to more school days missed for students. When testing is frequent enough, nearing weekly testing, longer quarantines are only marginally more effective at preventing infections (points closer to the x-axis) at the expense of more school days missed (points further from the y-axis).

Total Testing Budget
One way to visualize the allocation of tests across a set of schools is to graph the risk level of all schools as a function of the testing frequency in the same graph. Appendix Figure 13 shows the risk level of each of the six hypothetical schools in the system we consider, as we move from no surveillance testing to weekly testing. The risk of each school is again defined as the average of the expected on-campus infection rate and CVaR.
As the testing frequency increases the risk of each school decreases. The objective of our optimization is to find the lowest horizontal line on the graph such that the testing budget is respected. A horizontal line corresponds to all schools having the same risk, unless some school has reached the maximum testing frequency of 7 days, or a school does not need any testing to reach the same risk level as the other schools.
The left y-axis represents each school's risk, while the right y-axis represents the necessary budget required to achieve the corresponding maximum risk across schools. We can see that the lower the risk across schools becomes, the more expensive it becomes to further decrease it; to achieve a similar decrease requires much more frequent testing, so many more tests every week.
Given a certain budget we can draw a horizontal line at the corresponding point on the right axis and the intersection of this line with each of the schools' curves gives us the optimal allocation that minimizes the maximum risk across schools.
We can then verify that the full risk profiles of the six schools are very similar under the optimal allocation for the two budgets shown by the horizontal lines above. Appendix Figure 14 shows the distribution of on-campus infections in all the schools for three testing strategies. The no testing strategy shows the risk of each school without testing, the pro rata allocation shows the risk of all schools under the suboptimal allocation in which all schools receive the same number of tests, while under the optimal allocation all schools have a similar risk profile, and the total system risk is lowest.

Objective Function Optimized
In Appendix Figure 15 we show the sensitivity of the optimal allocation of tests in each school as we parametrically vary the objective function. We consider two testing budgets, first a budget in which all students can be tested every 4 weeks on average, and then every 2 weeks.
In the left panels we fix the weight of the CVaR term to 50%, and thus we have equal weight for the expected value of the infection rate when defining the risk of each school, R(s), and we vary the CVaR level β. That is we solve the problem (P50%β) where β goes from 5% to 95%. In the middle panel we fix β = 90% and we solve (Pw,90%) for w increasing from 0% to 100%, i.e., increasing the weight on the CVaR term. In the right panels we solve the problem (Ptotal), which aims to minimize the total system-wide expected infections.
When we only have a budget to test students every 4 weeks the impact of w and β on the solution of (Pwβ) is limited, with only the school with the pair of low community incidence and moderate R0 being allocated marginally more tests as w and β increase.
However, when we have a budget to test students every 2 weeks, we see much more variation in the allocation. This comes from the fact that with that budget the school with the pair of high community incidence and high R0 is being allocated nearly enough tests to test weekly.
As can be seen in the earlier Appendix Figure 13, the risk of that school does not change much as testing changes from every 9 days to every 7 days, so that small changes in w or β can impact the school risk enough to then cause large changes in the allocations to the other schools. While the changes in allocation for the school with high community incidence and high R0 can seem large for small changes in the school risk parameters, the allocations to the other schools do not change drastically.
Lastly, in the right panels we see that when minimizing the total infections across all schools the same schools receive more tests than under a pro rata allocation, but there tends to be smaller differences in the number of tests allocated across schools as all schools receive a share closer to their pro rata allocation.