## Introduction

The majority of the educational institutions in the United States, ranging from primary schools to universities, have temporarily ceased in-person classes and other activities due to the ongoing COVID-19 pandemic. While the importance of reopening is widely recognized1, there is lack of consensus on the strategies necessary to safely reopen these institutions2. The Center for Disease Control (CDC) has issued reopening guidelines that include extensive hand hygiene, cloth face coverings, repetitive disinfection, physical barriers and spacing of individuals inside enclosed surroundings, frequent testing, etc3,4. Sharp increase in COVID-positive cases from reopening with in-person interactions prompted eventual re-closures5. For example, Cherokee County School District in the state of Georgia, USA, quarantined 250 staff members and students after reopening in August, 20206. Similarly, the University of North Carolina, Chapel Hill, USA, canceled in-person classes after finding > 130 confirmed infected cases in the very first week after reopening7,8. Motivated by these observations, we explore the question of whether educational institutions and other organizations can safely commence in-person operations amidst the COVID-19 pandemic. In particular, we identify measures that are necessary to ensure the safety of the members of an institution and the public at large. To do so, we employ a combination of analytical modeling, data analysis and agent-based simulation. We first develop a mathematical model that captures the dynamics of the infection process with bulk testing and contact tracing. Then, we estimate some of the analytical model parameters from real data from a number of universities in the United States. Finally, we use the parameter estimates to conduct an agent-based simulation experiment to evaluate strategies for safe reopening.

SARS-CoV-2 is a novel strain of coronavirus that currently does not have an approved cure9,10,11. For mitigation, a variety of strategies have been implemented across the globe, ranging from complete lock-down of large geographical areas12 to partial restrictions on mobility and mask enforcement in public places13. A particular challenge associated with this virus is its asymptomatic transmission in which many infected individuals remain asymptomatic from a few days to several weeks and yet transmit the disease to susceptible people14,15. We mention the results from Hao et al.16 to highlight the seriousness of asymptomatic transmission17. As Ceylan18 reveals, Italy’s infected population may have ranged between 2.2 and 3.5 million in number as of May 4, 2020, while detected infections numbered a mere 200K. The potency of asymptomatic transmission is no different within an educational institution. Thus, we posit that a reopening strategy is difficult to design without the ability to conduct rapid bulk testing (testing everyone once every few days) so that one can detect and arrest the spread of infections through systematic isolation and quarantining of those who test positive for infection. Our work is motivated and guided by the SHIELD program of the University of Illinois at Urbana-Champaign (UIUC). In this program, the university is currently testing > 10K students and staff every day (that amounts to 0.2 tests per individual per week) through saliva-based tests.

#### Efficiency of contact tracing

Efficiency of contact tracing is understood as the probability with which a contact of an infected positive individual is identified and tested. We report our empirical findings for contact tracing efficiencies of 90% and 80% in Table  3. The results indicate that contact tracing efficiency has much more impact on the epidemic dynamics when bulk testing capabilities are small. This impact almost disappears when bulk testing capabilities increase. For example, with bulk testing 1K individuals daily, contact tracing efficiency drop from 90 to 80% leads to a drop of mean $$f_S$$ from 0.753 to 0.712 (5.4% reduction). The same numbers with 15K daily tests are 0.891 and 0.890, respectively. While contact tracing helps, our results yield that bulk testing has a much larger impact. With around 10K daily tests with parameters for UIUC, we typically found the number of contacts of positive individuals $$c_t \approx 650$$ on an average, and with a probability of infection slightly higher (factor of $$\kappa$$) than that of random selection approximately 20 positive cases are detected. As a result, the total number of infections detected via contact tracing is much smaller as compared to about 200+ COVID-positive individuals detected via bulk testing. Judging based on our experiments, we find it unlikely for contact tracing alone to define a viable infection containment strategy, given the large proportion of asymptomatic carriers of COVID-19.

#### Base infectivity and preventative measures

Universities have adopted several measures that directly impact the base infectivity levels, such as mask wearing and frequent sanitization of its premises. Some institutions have even pursued punitive measures for violation of mask wearing measures such as financial penalty, sanctions, and restrictions on accessing institution facilities. For example, at UIUC, several students were placed under probation for violation of regulations related to COVID-19 measures after the initial surge of infections immediately following reopening in August. At UIUC, our estimation puts $$\beta ^0$$ in the range 0.01–0.11, with a mean of 0.025. We simulate the effect of adopting less stringent preventative measures and report the results of agent-based simulations with $$\beta ^0 \in \{ 0.025, 0.040, 0.055, 0.070\}$$ for multiple levels of testing T. We plot the outcomes in Fig. 6. Interestingly, Fig. 6a reveals that with 1K daily tests, the entire population will get infected within 50 days for $$\beta ^0 \ge 0.04$$. Similar catastrophic results ensue even with higher testing capacities (see Fig. 6b–d) at high values of $$\beta ^0$$’s. The impact of $$\beta ^0$$ on the infection dynamics is rather pronounced, underscoring the importance of preventative measures. This sensitivity to $$\beta ^0$$ is not surprising, given that $$\beta ^0$$ directly changes the potency of each meeting between a susceptible and an infected individual. The consequence of each new infection then accumulates fast, given the nature of the epidemic dynamics. Besides bulk testing, it is thus imperative for institutions to enforce mask wearing, place hand sanitizers at various locations, periodically clean classrooms and laboratories, etc. This same sentiment is resonated in existing literature37.

#### Contact rates

Contacts create opportunities for infection transmission. With the parameters for UIUC (where average $$m^I$$ is 5 with a range 1–15), we evaluate the effect of varying $$m^I$$ from 2 to 11 in steps of 3 in Fig. 7. Increasing internal contact rate severely impacts the transmission of infection with testing capacities of 1K and 5K per day. The impact, however, becomes minimal with higher daily testing capacities of 10K and 15K. Strategies to reduce internal contacts include spacing out classroom sitting arrangements, staggering class and meeting times, using larger capacity rooms for classes and meetings, and adopting a hybrid of online and in-person operations as feasible. Our experiments demonstrate that increased bulk testing decreases the need for severely restricting internal contacts, revealing that contact restrictions and testing play a complimentary role in infection mitigation.

The effect of the number of external contacts $$m^E$$ is similar and the results are omitted for brevity. While an institution may not possess the means to directly control $$m^E$$, targeted information and awareness campaigns can indirectly reduce $$m^E$$ by educating the members of the consequences of infection transmission.

#### Varying testing frequencies among sub-populations

The agent-based simulation results presented so far assume that the institution has a population with homogeneous mobilities that we estimate from data. In practice, student groups and faculty/staff typically have different mobilities and hence, belong to different risk categories in terms of their potencies to transmit the disease. Personal communication with the UIUC SHIELD program indicates that they expect the contact rates among the student population to be at least double that of faculty and staff. Based on these expectations, the program has delineated different guidelines for these population groups. Specifically, students were asked to test at least twice a week and the faculty and staff to test once a week over initially, which moved to thrice a week testing for students and twice a week testing for staff and faculty on November 2, 2020 due to increased positivity. Here, we study the impact of risk-based modulation of bulk-testing frequencies through agent-based simulations. To that end, we divide the population of 50K agents in the simulation into two groups—40K students and 10K faculty/staff. We assume that students have an internal contact rate of $$m^I=5.5$$, compared to that of $$m^I=3$$ for faculty/staff. The numbers are chosen such that the average $$m^I$$ becomes 5, that approximately equals the rate we estimated from data. Students are then tested at double the rate compared to the faculty/staff. Table 4 presents the simulation outcomes.

Compared to the uniform testing frequency, the targeted risk-based testing indeed reduces the overall infection load. The gain from modulation of the testing frequency among the population is higher when the testing capacity is especially limited. For example, the increase in the mean value of $$f_S$$ is $$4.24\%$$ (from 0.753 for uniform testing to 0.784 for risk-based testing) with a daily testing level of 1K. The corresponding increase with 10K daily tests reduces to $$0.79\%$$ (from 0.883 for uniform testing to 0.890 for risk-based testing). Our experiments affirm that targeted testing among the group with a higher mobility (and hence, higher chances of infection) will lead to faster identification and isolation of more COVID-positive individuals, leading to higher values of $$f_S$$. Such a strategy is especially useful during the initial stages of the infection when testing infrastructure is likely to be limited. While we have only studied two risk classes, a more nuanced risk-stratification of the population can lead to further reductions in infection loads.

#### Efficiency in isolating COVID-positive patients

While we have so far assumed that isolation is 100%, in reality isolation efficiency tends to vary significantly. For example in China, it was found that 75–80% of all clustered infections occurred within family. Therefore, in many countries such as in China, South Korea and Singapore COVID-19 patients were isolated in separate facilities rather than at home38,39,40. In the context of an institution such as UIUC, creation of separate isolation facilities provides high isolation efficiency41, however, isolation efficiency may vary depending on adherence behavior of infected and non-infected individuals. Also, testing is an effective strategy to mitigate infection transmission only if positive detection is followed by proper isolation measures. Here, we study the impact of varying degrees of isolation efficiency $$\psi$$ through our agent-based simulations. This efficiency captures the probability that an individual who tests positive in fact isolates. Table  5 shows the average daily fraction of the susceptible population over 120 days for $$\psi =$$ 100%, 90%, 70% and 50%. The efficacy of testing drops sharply with isolation efficiency and the impact is more pronounced when the number of daily tests is low (see the case with $$T=$$ 1K). Increased volumes of bulk testing can offset the inefficiencies of isolation in part, but that comes at higher costs of building the testing infrastructure.

#### Delay in obtaining test results

Delay in receiving test results, either due to the nature of testing or due to limited testing capacity as compared to the demand for testing, can have adverse effect on the infections within an institution. In Table 6, we record $$f_S$$ from our experiments with delays $$\delta$$ varied from zero to 4 days in steps of 2 days. The case with $$\delta =0$$ days corresponds to the setting we considered so far, which is in line with rapid saliva testing at UIUC, where the test results are often made available within 12 h of testing. As our experiments demonstrate, delay in revelation of test results has a significant impact on the efficacy of testing, even when number of daily tests are high. This is not surprising, given that delay in isolation of infected individuals renders the test somewhat ineffective if these individuals continue to interact with people, awaiting test results.

#### Test sensitivities

Our final study seeks to understand the impact of the sensitivity of tests on the infection mitigation strategy. Early reports22,36 claim saliva-based RT-qPCR tests to have an average sensitivity of 92%, i.e., they are able to correctly detect 92% of the cases that are COVID-positive. In contrast, some other reports19,20 show that under certain conditions, particularly with different duration of infections, the test sensitivity can vary widely, and nasal swab based RT-qPCR tests tend to demonstrate much superior accuracy than the saliva based tests. While we consider bulk testing within institutions, where each individual gets tested relatively frequently (once to twice per week), and the duration of infections may not have a as high a variation as in the case of the general population, yet, we check for sensitivity of bulk testing and isolation policies to varying test sensitivities. In Table 7, we present the outcomes of agent-based simulations with test sensitivities in $$\{90\%, 80\%, 70\%, 60\%\}$$ with varying degrees of time delays between testing and reporting of test results. All experiments for this study utilized $$T=$$ 10K daily tests. While both the rate of false negatives of the tests and said time delay have adverse effects, the latter appears to be the dominant factor. Higher sensitivity of tests is desirable, no doubt. Even if that efficiency drops, rapid bulk testing appears crucial to effectively control the infection growth within the institution.

## Conclusions

The reopening of institutions during the COVID-19 pandemic is challenging. The initial experience of reopening in August and September 2020 demonstrate that reopening requires careful planning and measures to mitigate rapid infection spread within an institution. Per a recent media report, several universities have clocked more than 500 cases, such as the University of Alabama at Birmingham (972 cases), the University of North Carolina at Chapel Hill (835 cases), University of Central Florida (727 cases), Auburn University in Alabama (557 cases), Texas A&M University (500 cases), University of Notre Dame (473 cases), and the University of Illinois at Urbana-Champaign (448 cases) within days or weeks of reopening. Our work is motivated to answer if there is any possible policy path that allows institutions to manage the disease, if not fully stop it.

To study epidemic mitigation strategies, we first formulated a dynamical system model to describe the spread of COVID-19 within an institution. The key features of this model include the asymptomatic transmission of the disease, the effect of two channels of testing (contact tracing and bulk testing) and subsequent isolation of those who test positive. The analytical model is parameterized. We used COVID-19 data from 86 universities in the US (including that from the UIUC SHIELD program) to estimate some of these parameters via non-linear regression. The range of parameters were utilized as inputs to an agent-based simulations setup. The outcomes of this simulation are sample paths of the epidemic within the institution. The mean and the range of the outcomes helped us to derive important insights into the efficacy of various parameters and reopening strategies. Having grounded our study to the context of the UIUC SHIELD program data and cross-validated with data from 85 other universities, we believe that our observations are fairly robust and suitable to guide policies at educational institutions.

Our study yields three key observations. First, preventative measures such as mask wearing, social distancing and reduction of contact rates among individuals are indispensable to even consider reopening. Such measures are vital to reduce the potency of asymptomatic transmission. Second, contact tracing is not enough to contain the infection spread. Even though testing infrastructure is expensive, bulk testing capabilities are crucial to contain the disease. The key design parameter is the ratio of the total number of daily tests to the institution population. Additional measures can help combat the disease propagation such as increasing testing frequencies for subgroups with higher mobilities and increasing the efficiency of isolation of patients who test positive. Third, the testing technology should be able to provide test results quickly. The rapidity of the testing cycle appears even more important than test sensitivity (within reasonable limits). Therefore, institutions considering reopening must invest in COVID-testing for its members that is cost-effective, easy to administer in high volumes, and has a quick turnaround time to results.