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Evaluation of reopening strategies for educational institutions during COVID-19 through agent based simulation

Abstract

Many educational institutions have partially or fully closed all operations to cope with the challenges of the ongoing COVID-19 pandemic. In this paper, we explore strategies that such institutions can adopt to conduct safe reopening and resume operations during the pandemic. The research is motivated by the University of Illinois at Urbana-Champaign’s (UIUC’s) SHIELD program, which is a set of policies and strategies, including rapid saliva-based COVID-19 screening, for ensuring safety of students, faculty and staff to conduct in-person operations, at least partially. Specifically, we study how rapid bulk testing, contact tracing and preventative measures such as mask wearing, sanitization, and enforcement of social distancing can allow institutions to manage the epidemic spread. This work combines the power of analytical epidemic modeling, data analysis and agent-based simulations to derive policy insights. We develop an analytical model that takes into account the asymptomatic transmission of COVID-19, the effect of isolation via testing (both in bulk and through contact tracing) and the rate of contacts among people within and outside the institution. Next, we use data from the UIUC SHIELD program and 85 other universities to estimate parameters that describe the analytical model. Using the estimated parameters, we finally conduct agent-based simulations with various model parameters to evaluate testing and reopening strategies. The parameter estimates from UIUC and other universities show similar trends. For example, infection rates at various institutions grow rapidly in certain months and this growth correlates positively with infection rates in counties where the universities are located. Infection rates are also shown to be negatively correlated with testing rates at the institutions. Through agent-based simulations, we demonstrate that the key to designing an effective reopening strategy is a combination of rapid bulk testing and effective preventative measures such as mask wearing and social distancing. Multiple other factors help to reduce infection load, such as efficient contact tracing, reduced delay between testing and result revelation, tests with less false negatives and targeted testing of high-risk class among others. This paper contributes to the nascent literature on combating the COVID-19 pandemic and is especially relevant for educational institutions and similarly large organizations. We contribute by providing an analytical model that can be used to estimate key parameters from data, which in turn can be used to simulate the effect of different strategies for reopening. We quantify the relative effect of different strategies such as bulk testing, contact tracing, reduced infectivity and contact rates in the context of educational institutions. Specifically, we show that for the estimated average base infectivity of 0.025 ($$R_0 = 1.82$$), a daily number of tests to population ratio T/N of 0.2, i.e., once a week testing for all individuals, is a good indicative threshold. However, this test to population ratio is sensitive to external infectivities, internal and external mobilities, delay in getting results after testing, and measures related to mask wearing and sanitization, which affect the base infection rate.

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 LAMP 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.

Data availability

The data and codes for this program are available in the following websites: 1. Summary description of the project is available at https://apps.heart-analytics.com/covid19/, 2. UIUC data is available at https://covid19.illinois.edu/on-campus-covid-19-testing-data-dashboard/, 3. Data for all other 85 universities are available at www.covidedutrends.com, 4. The simulation codes are available at https://github.com/heart-group/institutions-model, 5. A preprint version of the paper is available at https://www.medrxiv.org/content/10.1101/2020.09.04.20188680v4.

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Acknowledgements

We acknowledge the support of C3.ai Digital Transformation Institute in funding this research. We also acknowledge the support of the UIUC SHIELD program and program director Ronald Watkins for providing inputs and encouragement for this work.

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Correspondence to Ujjal K. Mukherjee.

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Appendices

Additional notations used for the derivation

Define the following:

• $${\mathscr {S}}_t$$ as the set of individuals who are susceptible;

• $${\mathscr {U}}_t$$ as the set of individuals who are infected but undetected;

• $${\mathscr {P}}_t$$ as the set of individuals who are infected and detected;

• $${\mathscr {R}}_t$$ as the set of individuals who are recovered;

• $${\mathscr {N}}_t$$ is the set of mobile individuals who are not isolated, i.e., $${\mathscr {N}}_t = {\mathscr {S}}_t \cup {\mathscr {U}}_t \cup {\mathscr {R}}_t$$;

• $${\mathscr {C}}_t^i$$ is the set of contacts of individual i at time t;

• $${\mathscr {C}}_t$$ is the set of all contacts in the contact list for day t to be tested in day $$t+1$$;

• $$\Delta {\mathscr {P}}_t$$ is the set of new detected positive individuals on day t;

• $$x\rightarrow y$$ as x meets (comes in contact with) y;

• $$\#$$ as the cardinality of a set such that $$\#{\mathscr {U}}_t = u_t$$;

• $${\mathscr {A}}=\{x:x\in {\mathscr {X}}\}$$, an arbitrary a set $${\mathscr {A}}$$ of elements of type x belonging to a super-set $${\mathscr {X}}$$;

• $${\mathscr {X}}\cup {\mathscr {Y}}$$, is the union of set $${\mathscr {X}}$$ and set $${\mathscr {Y}}$$;

• $${\mathscr {X}}\cap {\mathscr {Y}}$$, is the intersection of set $${\mathscr {X}}$$ and set $${\mathscr {Y}}$$;

• ijk are individual members of an institution.

Derivation of the number of new infections: $$\beta _t^0 m_t^Is_tu_t/n_t + \beta _t^0 m_t^E s_t\rho _t^E$$

At time t, the number of infected undetected individuals are $$u_t$$, the number of susceptible individuals are $$s_t$$, and the number of recovered individuals are $$r_t$$. The infected and detected individuals, $$p_t$$, are isolated and do not participate in the infection dynamics. The base infection rate, defined as the probability that a suseptible individual gets infected when she comes in contact with an infected individual, is $$\beta _t^0$$. A susceptible individual at time t get infected at time $$t+1$$ if the susceptibe individual comes in contact with one or more infected individual. The probability that a susceptible individual i at time t becomes infected at time $$t+1$$, denoted by $${\mathbb {P}}\{i\in {\mathscr {U}}_{t+1} | i \in {\mathscr {S}}_t\}$$ is obtained by considering first, the probability that a susceptible individual meets k undetected individuals, where k can take values in $$\{1,2,\ldots ,m_t^I\}$$ (recall $$m_t^I$$ is the internal contact rate); and then, multiplying this probability by the probability of infection and finally, summing over k. The number of ways in which a susceptible individual meets k infected individuals is given by, $$\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}n_t - u_t\\ m_t^I - k\end{array}}\right)$$, where $$n_t = s_t+u_t+r_t$$. Similarly, the number of ways in which the susceptible individual can meet $$m_t^I$$ individuals equals $$\left( {\begin{array}{c}n_t\\ m_t^I\end{array}}\right)$$. Therefore, the probability that susceptible individual meets k infected individual in period t is given by

\begin{aligned} {\mathbb {P}}\{i\rightarrow {\mathscr {C}}_t^i = \{j:j\in {\mathscr {N}}_t, i\ne j\}: \#{\mathscr {C}}_t^i = m_t^I; \#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\}=k\} = \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}n_t - u_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}n_t\\ m_t^I\end{array}}\right) } \end{aligned}

for the individual by i. The probability that the susceptible individual gets infected given that the the individual meets k infected individuals is given as

\begin{aligned} {\mathbb {P}}\{i\in {\mathscr {U}}_{t+1}|i\in {\mathscr {S}}_t; \#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\}=k\} = 1-(1-\beta _t^0)^k \approx k\beta _t^0. \end{aligned}

The approximation $$1-(1-\beta _t^0)^k \approx k\beta _t^0$$ is acceptable since $$\beta _0$$ is usually much smaller than 1, approximately in the range of 0.01–0.05. Therefore, the probability that a susceptible individual meets k infected individual and gets infected is given by

\begin{aligned} \begin{aligned} {\mathbb {P}}\{i\in {\mathscr {U}}_{t+1}; \#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\}=k|i\in {\mathscr {S}}_t\}&= {\mathbb {P}}\{i\in {\mathscr {U}}_{t+1}|i\in {\mathscr {S}}_t; \#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\}=k\} \times {\mathbb {P}}\{ \#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\}=k\}\\&= k\beta _t^0 \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}n_t - u_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}n_t\\ m_t^I\end{array}}\right) }. \end{aligned} \end{aligned}
(4)

Therefore, the probability a susceptible individual becomes infected at time t is

\begin{aligned} \begin{aligned} {\mathbb {P}}\{i\in {\mathscr {U}}_{t+1}|i\in {\mathscr {S}}_t\}&= \beta _t^0 \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}n_t - u_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}n_t\\ m_t^I\end{array}}\right) } =\beta _t^0 m_t^I \frac{u_t}{n_t}. \end{aligned} \end{aligned}
(5)

The last expression is obtained by replacing the summation with the expectation of the hypergeometric distribution. The probability of external infection is also similarly obtained as $$\beta _t^0m_t^E\rho _t^E$$, where $$\rho _t^E$$ plays the role of $$u_t/n_t$$. Therefore, the total number of new infections in period t is given by the product of the number of susceptibles at time t with the probability of each susceptible getting infected. Therefore, expected number of new infections in time t is given by

\begin{aligned} \beta _t^0 m_t^I \frac{u_t}{n_t} + \beta _t^0m_t^E\rho _t^E. \end{aligned}

Derivation of the probability of infection of an individual among the contact list, $$\phi (\beta _t^0, m_t^I, \Delta p_{t})$$

We first derive the probability of infection through within-institutional-transmission (internal) of an individual i who is among the contact list and who was not infected in period $$t-1$$, i.e., $$\phi (\beta _t^0, m_t^I, \Delta p_{t})={\mathbb {P}}\{i\in {\mathscr {U}}_t, i \in {\mathscr {C}}_{t} | i\notin {\mathscr {U}}_{t-1}\}$$. The probability of infection of individuals in the contact list given that the individual was not infected in the previous period is derived by considering that individual i can meet $$k\in \{1,\ldots ,m_t^I\}$$ infected individuals, of which at least one individual needs to be among the detected positive cases $$\Delta p_{t}$$ that is the individual i is in the contact list of the detected individuals. Recall that the probability of infection for an individual in the contact list is obtained by adding the probability that the individual was already infected in the previous period, and the probability that the individual is newly infected after the previous period, i.e., $$\phi (\beta _t^0, m_t^I, \Delta p_{t})$$. Also, for simplicity, in the following expressions we ignore the notation that the individual i was not infected in the previous period. Therefore, we have the following expression:

\begin{aligned} \begin{aligned} {\mathbb {P}}\{i\in {\mathscr {U}}_t, i \in {\mathscr {C}}_{t}|i\notin {\mathscr {U}}_{t-1}\} = \sum _{k=1}^{m_t^I}&{\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k; {\mathscr {C}}_t^i\cap \Delta {\mathscr {P}}_t \ne \emptyset \} \times {\mathbb {P}}\{i\in {\mathscr {U}}_{t} | i\notin {\mathscr {U}}_{t-1}\}. \end{aligned} \end{aligned}
(6)

The term $${\mathbb {P}}\{i\in {\mathscr {U}}_{t} | i\notin {\mathscr {U}}_{t-1}\}$$ is derived as above and equals $$1-(1-\beta _t^0)^k$$. The probability of meeting k individuals of which at least one is in $$\Delta p_{t}$$ is given by

\begin{aligned} \begin{aligned} {\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k; {\mathscr {C}}_t^i\cap \Delta {\mathscr {P}}_t \ne \emptyset \}&= {\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k \} - {\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k; {\mathscr {C}}_t^i\cap \Delta {\mathscr {P}}_t = \emptyset \}. \end{aligned} \end{aligned}
(7)

The probability of meeting k individuals in a small population is modeled using the Hypergeometric probability. The mobile population $$n_t = s_t+u_t+r_t$$ at time t is divided into two groups, (i) undetected infected group, $$u_t$$, who can transmit infection, and (ii) the susceptible and recovered group, $$s_t+r_t$$, who cannot transmit infection further. Therefore, the first probability, $${\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k\}$$, is given by the ratio of the product of the number of ways of choosing k from $$u_t$$ possibilities and choosing the rest $$m_t^I - k$$ from $$s_t+r_t$$ group, to the number of ways of choosing the total $$m_t^I$$ contacts from $$n_t$$ individuals. Similarly, the second probability, $${\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k; {\mathscr {C}}_t^i\cap \Delta {\mathscr {P}}_t = \emptyset \}$$, is obtained by the ratio of the product of the number of ways of choosing k infected individuals from $$u_t - \Delta p_t$$ individuals (since, we are interested in the probability of detecting contacts with infected individuals who are not in the detected group, $$\Delta p_t$$), the number of ways of choosing 0 individuals from $$\Delta p_t$$ newly detected infected individuals, and choosing the rest of the $$m_t^I-k$$ contacts from $$s_t+r_t$$ individuals, to the number of ways of choosing the total $$m_t^I$$ individuals from $$n_t$$ individuals. Thus, we have

\begin{aligned} &{\mathbb {P}}\{\#\{{\mathscr {C}}_t^i\cap {\mathscr {U}}_t\} = k; {\mathscr {C}}_t^i\cap \Delta {\mathscr {P}}_t \ne \emptyset \} =\frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t-\Delta p_t\\ m_t^I\end{array}}\right) }. \end{aligned}
(8)

Therefore, combining the above, the function $$\phi (\beta _t^0, m_t^I, \Delta p_{t})$$ is

\begin{aligned} \begin{aligned} \phi (\beta _t^0, m_t^I, \Delta p_{t})&=\sum _{k=1}^{m_t^I} \left[ \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}\Delta p_t\\ 0\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) }\right] \times \left[ 1-(1-\beta _t^0)^k\right] . \end{aligned} \end{aligned}
(9)

However, the above expression considers only infections through internal transmission. However, individuals in the contact list can also get infected by external sources. Therefore, to extend the function $$\phi (\beta _t^0, m_t^I, \Delta p_{t})$$ to include infections from outside the organizations, the expression becomes (after considering that $$\left( {\begin{array}{c}\Delta p_t\\ 0\end{array}}\right) = 1$$)

\begin{aligned} \begin{aligned} \phi (\beta _t^0, m_t^I, \Delta p_{t})&=\sum _{k=1}^{m_t^I} \left[ \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) }\right] \times \left[ 1-(1-\beta _t^0)^k\right] + \beta _t^0 m_t^E \rho _t^E. \end{aligned} \end{aligned}
(10)

We consider the regime where $$\beta _t^0 \ll 1$$ and $$\rho _t^E\ll 1$$. This is acceptable since the infectivities are usually of the order of 5%, and the maximum environmental positivty is below 10%. Therefore, this assumption holds for the case of COVID-19. This assumption leads to the approximation $$1-(1-\beta _t^0)^k \approx k\beta _t^0$$, that gives

\begin{aligned} \begin{aligned} \phi (\beta _t^0, m_t^I, \Delta p_{t}) \approx \beta _t^0 \sum _{k=1}^{m_t^I} \left[ \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) }\right] k + \beta _t^0 m_t^E \rho _t^E = \kappa _t \beta _t^0, \end{aligned} \end{aligned}
(11)

where, we have

\begin{aligned} \begin{aligned} \kappa _t :=\sum _{k=1}^{m_t^I} \left[ \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) }\right] k + m_t^E \rho _t^E. \end{aligned} \end{aligned}
(12)

Approximating $$\kappa _t$$

From the above derivation, we get

\begin{aligned} \begin{aligned} \kappa _t&= \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } + m_t^E \rho _t^E \\&= \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } - \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}s_t+r_t\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t-\Delta p_t\\ m_t^I\end{array}}\right) } \times \frac{\left( {\begin{array}{c}s_t+u_t+r_t-\Delta p_t\\ m_t^I\end{array}}\right) }{\left( {\begin{array}{c}s_t+u_t+r_t\\ m_t^I\end{array}}\right) } + m_t^E \rho _t^E. \end{aligned} \end{aligned}
(13)

As mentioned earlier, for analytical simplicity, we consider the regime where $$\Delta p_t \ll s_t+u_t+r_t = n_t$$, where recall that $$n_t$$ is the size of the mobile part of the population. This is acceptable because in the case of UIUC data, we observe that the maximum detected $$\Delta p_t < 500$$ in a population of 50K individuals.

\begin{aligned} \begin{aligned} \kappa _t&\overset{(a)}{=} \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t\\ k\end{array}}\right) \left( {\begin{array}{c}n_t-u_t\\ m_t^I - k\end{array}}\right) }{\left( {\begin{array}{c}n_t\\ m_t^I\end{array}}\right) } - \sum _{k=1}^{m_t^I} k \frac{\left( {\begin{array}{c}u_t-\Delta p_t\\ k\end{array}}\right) \left( {\begin{array}{c}(n_t -\Delta p_t)-(u_t-\Delta p_t)\\ m_t^I-k\end{array}}\right) }{\left( {\begin{array}{c}n_t -\Delta p_t\\ m_t^I\end{array}}\right) } + m_t^E \rho _t^E\\&\overset{(b)}{=} m_t^I \frac{u_t}{n_t} - m_t^I \frac{u_t - \Delta p_t}{n_t-\Delta p_t} + m_t^E \rho _t^E\\&=m_t^I \frac{\Delta p_t (n_t-u_t)}{n_t (n_t - \Delta p_t)} +m_t^E \rho _t^E. \end{aligned} \end{aligned}
(14)

The step in (b) follows from noting that the first summation in (a) is the expectation of a hyper-geometric distribution with parameters $$(n_t, u_t, m_t^I)$$, while the second one is the same for a hyper-geometric distribution with parameters $$(n_t - \Delta p_t, u_t - \Delta p_t, m_t^I)$$.

Derivation of the contact traced positive individuals

As indicated in the earlier section, the probability of an individual in the contact list is equal to the prior probability of infection, which is given by $$\frac{u_{t-1}}{n_{t-1}}$$, and the probability of getting infected in day t, given that the individual was not infected earlier. The probability that an individual in the contact list is infected on day t given that the individual was not infected earlier is derived in the previous section as $$\kappa _t\beta _t^0$$, where, the value of $$\kappa _t$$, as we have derived earlier in  (14). The probability that an individual was not infected earlier is given by $$1-\frac{u_{t-1}}{n_{t-1}}$$. Therefore, the total probability of infection of a contact traced individual is given by

\begin{aligned} {\mathbb {P}}\{i\in {\mathscr {U}}_t, i\in {\mathscr {C}}_t\} = \frac{u_{t-1}}{n_{t-1}} + \left( 1-\frac{u_{t-1}}{n_{t-1}}\right) \kappa _t \beta _t^0. \end{aligned}

Appendix 2: List of universities in the empirical dataset with location information

Name City State Zip-Code
Eastern Illinois University Charleston IL 61920
Emory University Atlanta GA 30322
Florida State University Tallahassee FL 32306-1037
George Washington University Washington DC 20052
Illinois State University Bloomington-Normal IL 61761
Kansas State University Manhattan KS 66506
Louisiana State University and Agricultural & Mechanical College Baton Rouge LA 70803-2750
Loyola University Chicago Chicago IL 60660
Northern Illinois University Dekalb IL 60115-2828
Northwestern University Evanston IL 60208
Rochester University Rochester Hills MI 48307
Rutgers University-Camden Camden NJ 8102
Tufts University Medford MA 02155-5555
Tulane University of Louisiana New Orleans LA 70118-5698
University at Buffalo Buffalo NY 14260-1660
University of Arkansas at Little Rock Fayetteville AR 72701
University of California-Berkeley Berkeley CA 94720
University of California-Los Angeles Los Angeles CA 90095-1405
University of California-San Diego La Jolla CA 92093
University of Colorado Boulder Boulder CO 80309-0017
University of Colorado Denver Denver CO 80204-2026
University of Connecticut Storrs CT 6269
University of Florida Gainesville FL 32611
University of Georgia Athens GA 30602
University of Illinois at Chicago Chicago IL 60607
University of Illinois at Urbana-Champaign Champaign IL 61820-5711
University of Kentucky Lexington KY 40506-0032
University of Maryland Baltimore Baltimore MD 21201-1627
University of Massachusetts-Amherst Amherst MA 1003
University of Miami Coral Gables FL 33146
University of Michigan-Ann Arbor Ann Arbor MI 48109
University of Minnesota-Twin Cities Minneapolis MN 55455-0213
University of Nebraska-Lincoln Lincoln NE 68588
University of North Carolina at Chapel Hill Chapel Hill NC 27599
University of North Dakota Grand Forks ND 58202-8193
University of Notre Dame Notre Dame IN 46556
University of Oklahoma-Norman Campus Norman OK 73019-3072
University of Pennsylvania Philadelphia PA 19104-6303
University of Pittsburgh-Pittsburgh Campus Pittsburgh PA 15260
University of South Carolina-Columbia Columbia SC 29208
University of Vermont Burlington VT 05405-0160
Vanderbilt University Nashville TN 37240
Virginia Polytechnic Institute and State University Blacksburg VA 24061-0131
West Virginia University Institute of Technology Beckley WV 25801
Wichita State University Wichita KS 67260-0124
William & Mary Williamsburg VA 23187-8795
Ball State University Muncie IN 47306
Binghamton University Vestal NY 13850-6000
Boston College Chestnut Hill MA 2467
Boston University Boston MA 2215
Brown University Providence RI 2912
Case Western Reserve University Cleveland OH 44106
Clark University Worcester MA 01610-1477
Clemson University Clemson SC 29634
Columbia University in the City of New York New York NY 10027
Dartmouth College Hanover NH 03755-3529
Duke University Durham NC 27708
Harvard University Cambridge MA 2138
Haverford College Haverford PA 19041-1392
Johns Hopkins University Baltimore MD 21218-2688
Lehigh University Bethlehem PA 18015
Massachusetts Institute of Technology Cambridge MA 02139-4307
Mississippi State University Mississippi State MS 39762
New York University New York NY 10012-1091
North Carolina State University at Raleigh Raleigh NC 27695-7001
North Carolina State University at Raleigh Raleigh NC 27695-7001
Northeastern University Boston MA 02115-5005
Northern Michigan University Marquette MI 49855-5301
Ohio Northern University Ada OH 45810-1599
Ohio State University-Main Campus Columbus OH 43210
Oklahoma State University-Main Campus Stillwater OK 74078-1015
Old Dominion University Norfolk VA 23529
Pennsylvania State University-Main Campus University Park PA 16802-1503
Purdue University-Main Campus West Lafayette IN 47907-2040
Radford University Radford VA 24142
Rensselaer Polytechnic Institute Troy NY 12180-3590
Rice University Houston TX 77005-1827
Stanford University Stanford CA 94305
Stony Brook University Stony Brook NY 11794
SUNY at Albany Albany NY 12222
Syracuse University Syracuse NY 13244
Temple University Philadelphia PA 19122-6096
Texas A & M University-College Station College Station TX 77843-1248
The University of Texas at Austin Austin TX 78705
University of Wisconsin-Madison Madison WI 53706-1380
Virginia Commonwealth University Richmond VA 23284-2512

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Mukherjee, U.K., Bose, S., Ivanov, A. et al. Evaluation of reopening strategies for educational institutions during COVID-19 through agent based simulation. Sci Rep 11, 6264 (2021). https://doi.org/10.1038/s41598-021-84192-y

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