Introduction

The COVID-19 pandemic caused by the SARS-CoV-2 virus has caused significant disease and economic burdens since its first outbreak in December 2019. Because of the absence of an effective vaccine, as of June 2020 at the time of this study and since March 2020, the main intervention for the prevention of COVID-19 transmissions had been to reduce contacts between people through lockdowns of non-essential organizations and services [1]. However, lockdowns are a threat to the economic stability of a nation as seen by the unprecedented rise in unemployment rates [2, 3]. Therefore, while lockdowns are a good short-term strategy, for a long-term strategy, or until a vaccine becomes widely available, it has become necessary to identify alternate strategies and lifestyles that control the disease burden while minimizing the economic burden. Interventions that are effective include the use of face masks, physical distancing between persons at a recommended 6ft, and contact tracing and testing or mass testing to enable early diagnosis in the asymptomatic stage of infection [4]. However, removal of lockdowns should be strictly accompanied by a reopening plan that rapidly and efficiently enables the adoption of the above interventions to avoid an epidemic rebound. In addition to public health agencies, all members of a community, in both public and private sectors, play a key role in the development and implementation of a reopening plan that is most suited for their organization [5]. Among these sectors, universities and colleges bear a special burden to develop a reopening plan that include changes to a range of activities related to teaching, research, dining, housing, and extra-curricular activities.

We developed a compartmental differential equations model to simulate epidemic projections of a potential COVID-19 outbreak in a population of 38,000 individuals, which is representative of undergraduate and graduate students, faculty, and staff in a typical residential university in the United States. We simulated epidemic projections of potential outbreaks under varying combinations of contact tracing and testing, and mass testing, to identify combinations that would reduce the effective reproduction numberR_e to a value below the epidemic threshold of 1. R_e is directly proportional to the duration of infectiousness, transmission rate (the probability of transmission per contact per day, representing the infectiousness of the virus), and contact rate (the number of contacts per person per day) [6]. Asymptomatic testing through trace and test or mass tests lead to diagnosis in the asymptomatic phase of the infection, and thus, if persons diagnosed with infection are successfully quarantined, it reduces the duration of exposure [7–9] and thus reduce R_e. Physical distancing by the recommended 3 or 6ft and use of face masks can reduce transmission rate, and thus reduce R_e [10, 11]. Reducing contact rate such as through transitioning to remote work to reduce population density on campus directly reduces R_e. Thus, different types of interventions help reduce each of these components of R_e. Here, we evaluated different combinations of test rates, transmission rates, and contact rates that help reduce R_e to below 1 to identify minimum levels of testing, physical distancing and face mask use, and population density necessary for effective control of an outbreak.

While it is generally known that increasing contact tracing and testing is necessary, studies evaluating testing at an organizational level, such as university, were only recently emerging at the time of this study in June 2020. One study that analyzed contact tracing in the general populations estimated that reducing R₀ of 1.5 to an R_e of 1 requires more than 20% of contacts traced, reducing R₀ of 2.5 to an R_e of 1 requires more than 80% of contacts traced, and reducing R₀ of 3.5 to an R_e of 1 requires more than 100% of contacts traced [12]. A modeling study applied to the Boston area [13] estimated that the best way-out scenario is a Lift and Enhanced Testing (LET) with 50% detection and 40% of contacts traced. According to this, the number of individuals that need to be traced per 1000 persons is below 0.1 under partial reopening and below 0.15 under total reopening. Models for a university were only recently emerging at the time of this study in June 2020 [14–17], but typically, most studies combine transmission rate and contact rate as one metric in the evaluation of testing.

In this study, instead of using a product of transmission rate and contact rate as one metric as typically done, we evaluated these separately, due to the following reasons. First, it helps systematically evaluate different interventions considering that different types of interventions help reduce each of the three components of R_e, testing reduces duration of exposed infectious stage, transitioning to remote classes reduce contact rate, vaccinations reduce the number of contacts who are potential disease carriers, and face mask use and 6ft distancing reduces transmission rate. Second, while adoption of each of these decisions are made at an organizational level, adherence and feasibility of face mask and 6ft distancing are highly influenced by individual behaviors and thus have a larger range of uncertainty. Third, while physical distancing and use of face masks can reduce transmission rate, the baseline transmission rate and expected reductions could vary based on multiple factors such as indoor vs. outdoor settings and ventilation, proper use and type of face mask [10, 11, 18, 19], mode of transmission [20–23], and viral load in the index person [8, 24]. Fourth, though we specifically focus this study on COVID-19 caused by the original SARS-CoV-2 virus, studying varying levels of transmission rates could help extrapolate findings to new variants or future outbreaks of viral respiratory infections with similar disease progressions [24], especially in the early stages when specific data is lacking but when the same non-pharmaceutical interventions, such as face masks, physical distancing, remote instructions, and testing, are suitable options.

To systematically inform these analyses, we first evaluated different combinations of trace and test rate, mass test rate, and transmission rate for a range of contact rates, to identify the threshold contact rates that maintain infection cases below certain set levels of tolerance. We then used the contact rate thresholds to identify the population size thresholds, i.e., the maximum population size on campus, which could help inform decisions related to campus activities such as the fraction of classes to transition to remote. We also used the contact rate thresholds to identify the vaccine coverage thresholds for a post-vaccine era, i.e., the vaccine coverage necessary for a campus to return to a normal population size. We also identify, under each intervention combination, the number of trace and tests and quarantines. These metrics could collectively help inform development of a preparedness plan for reopening a university during the COVID-19 pandemic or to set protocols in the event of future outbreaks.

Methodology

Simulation methodology

We developed a compartmental model for simulating epidemic projections over time. The epidemiological flow diagram for the compartmental model is depicted in Fig 1A. Each box is an epidemiological state, and each arrow represents a transition from one state to another. Note, each compartment is further split by age and gender, but for clarity of notations, we do not include it in the equations below.

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Fig 1. Overview of the extended SEIR compartmental model.

(A) Compartmental model flow diagram. (B) Natural disease progression of SARS-COV-2 virus in infected patients. S = susceptible, L = exposed and not infectious (Latent stage) (asymptomatic), E = asymptomatic and infectious, I = symptomatic and infectious, Q_L = exposed and not infectious (Latent) and Quarantined (diagnosed), Q_E = asymptomatic and infectious and Quarantined (diagnosed), Q_I = Infectious and Quarantined (diagnosed), H = Hospitalized, R = Recovered, and D = Deaths.

https://doi.org/10.1371/journal.pone.0255864.g001

Let π_t = [S, L, E, I, Q_L, Q_E, Q_I, H, R, D] be a vector, with each element representing the number of people in a compartment at time t, specifically,

S = the number of susceptible individuals at time t,

L = the number of exposed, but asymptomatic and not infectious individuals (latent stage; also, the non-infectious phase of incubation stage) at time t,

E = the number of asymptomatic or pre-symptomatically infectious individuals (infectious phase of the incubation stage) at time t,

I = the number of infectious individuals (symptomatic and infectious stage) at time t,

Q_L = the number of exposed, asymptomatic and not infectious (latent) and quarantined individuals (diagnosed) at time t,

Q_E = the number of asymptomatic or pre-symptomatically infectious and quarantined individuals (diagnosed) at time t,

Q_I = the number of infectious and quarantined individuals (diagnosed) at time t,

H = the number of hospitalized individuals at time t,

R = the number of recovered individuals at time t, and

D = the number of deaths at time t.

Epidemic states L, E, and I were formulated such that each state represented a distinct phase along the natural disease progression (see Fig 1B), and they collectively included all phases. Over time, persons from S can transition to L, E, and I, and upon diagnoses, transition to Q_L, Q_E, or Q_I, and further to H, R, or D, (transitions represented by arrows in Fig 1A) as discussed below.

Let,

p = transmission rate (probability of transmission per contact per day),

c = contact rate (number of contacts per person per day),

N = total population who are alive,

a_B = symptom-based testing rate,

a_C,t = rate of testing through contact tracing at time t,

a_U,t = rate of testing through mass testing at time t,

ρ = test sensitivity for asymptomatic testing (through mass tests or trace and test),

days_L = duration in latent period,

days_incub = duration in incubation period,

days_IR = time from onset of symptoms to recovery,

= time from diagnosis to recovery,

= time from diagnosis to hospitalization,

days_HR = time from hospitalization to recovery,

days_HD = time from hospitalization to death,

prop_hosp = proportion hospitalized, and

prop_severe = proportion of cases that are severe.

Then, we can write the equations for transition rates (arrows in Fig 1A) as follows:

, which assumes that only infected persons in E and I can transmit, persons in Q_E and Q_I self-quarantine, and persons in L and Q_L are not infectious.

, which assumes that only a proportion of cases that are severe (prop_severe) get diagnosed immediately because of exhibition of symptoms, we use the proportion hospitalized as a proxy for severe cases; the denominator is based on the assumption that the duration in state E is equal to the difference between the duration of the incubation period and the latent period.

, which follows from above.

, which assumes that under symptom-based testing, only persons who show moderate to severe symptoms get diagnosed and those who show mild symptoms do not.

, for prop_hosp we use the proportion of persons hospitalized among those diagnosed through symptom-based testing.

, which assumes that under the implementation of both mass testing and contact tracing and testing, persons diagnosed through mass test will not be tested again on the same day through contact tracing (as our time unit is daily).

, which is similar to above.

, which assumes that persons with mild cases that did not get diagnosed through symptom-based testing have a chance of getting tested through additional testing options, and self-quarantine upon diagnosis. Note that we did not separately model asymptomatic cases but incorporated that into the symptom-based testing rate (a_B) by considering that 35% of cases are mild to no symptoms and thus do not have a chance of being diagnosed through symptom-based testing.

, theoretically, should be the same as r_E,I, however, as the rate of transitioning from Q_I to H is fixed to fit to the proportion hospitalized under symptom-based tests, if extensive testing is conducted, the number of persons in Q_I would increase, thus, incorrectly inflating the number of persons who are hospitalized; To avoid this, we modified the equation to consider that the number of persons flowing into Q_I would be equal to the proportion flowing from I to Q_I under symptom-based testing.

, which follows from the above equation.

Note: r_s,s is the testing rate (either through mass test or trace and test). We assumed that susceptible persons go back to the susceptible state after testing, i.e., we did not explicitly track false positives.

The values and ranges for the above epidemic parameters used in the compartmental simulation model are presented in Table A in S1 Text.

We simulate the epidemic over time using the following system of differential equations where, Q_t = a matrix of transition rates between states (arrows in Fig 1A), and dt = time-step. We use a time-unit of per day for the transition rates in Q_t and set , and thus, the model simulates every 10^th of a day.

The expansion of the system of differential equations are as follows:

We can further expand by substitution of the rate terms with their equations as follows:

Results

When vaccines are unavailable (V = 0%), there is no single intervention that can effectively control an outbreak. However, there are multiple feasible combinations of testing, contact rate for non-essential workers (), and population size on campus () that can be implemented to effectively control an outbreak to keep cases below the relaxed to medium tolerance levels, though none to keep cases below the tight tolerance level (Table 1). Examples of feasible combinations under the relaxed tolerance level include: mass tests only at 25% per day, contact rate for non-essential workers at 2 to 6 per day, and campus population size at 26% to 42%; or trace and test only at 33%, contact rate for non-essential workers at 4 to 8 per day, and campus population size at 31% to 47% (see full list in Table 1). Under the medium tolerance level, only scenarios with combination tests were feasible, examples include: 5% mass test, 50% trace and test, contact rate for non-essential workers at 2 to 5 per day, and campus population size at 26% to 36%; or 33% mass test, 50% trace and test, contact rate for non-essential workers at 8 to 14 per day, and campus population size at 47% to 73% (see full list in Table 1). Note: the range in population size results correspond to mid-points of the range for contact rate (C) of 16 and 24 in Table 1.

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Table 1. Feasible combinations ^⁋ of testing, contact rate, and population size on campus for effective control of a disease outbreak in the absence of a vaccine.

https://doi.org/10.1371/journal.pone.0255864.t001

The corresponding peak numbers of trace and tests per day (per 10,000 persons) in the above feasible scenarios were at a reasonably manageable level. The relaxed tolerance level had a higher value (14 to 55 per day) than the medium tolerance level (3 to 11 per day) considering the population size on campus were lower in the latter case because of the tighter tolerance (Table 1). The peak number of quarantines per day (per 10,000 persons) for the above feasible scenarios also seem manageable. As with above, the relaxed tolerance level had a higher value (6 to 25 per day) than the medium tolerance level (5 to 6 per day). Combinations of testing, contact rate, and population size for the full range of transmission rates evaluated are presented in S1 Table.

When vaccines become partially or fully available, to keep the population size on campus at 100% (), the level of testing necessary to effectively control an outbreak would depend on the vaccine coverage in the population (Table 2). To keep infection cases within the relaxed tolerance level, implementing symptomatic-testing-only will be sufficient if at least 95% () of the population are vaccinated (Table 2). With the addition of mass tests only, 5%, 10%, 20%, 25%, and 33% mass tests per day would be sufficient if at least 89% to 95%, 84% to 89%, 74% to 84%, 63% to 79%, and 53% to 68% () of the population are vaccinated (Table 2), respectively, the range corresponding to transmission rate of 5% to 8%, i.e., the unvaccinated continue to use face masks and maintain physical distancing at current compliance levels.

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Table 2. Combinations of testing, vaccine coverage, and population size for effective control of a disease outbreak.

https://doi.org/10.1371/journal.pone.0255864.t002

If vaccine coverage (V) is below 53% (the threshold noted above), it would be necessary to also reduce the population size (Table 2). If vaccine coverage (V) is between 38% and 53%, 23% and 38%, or 8% and 23%, in addition to mass tests at 33% per day, it would be necessary to maintain a population size threshold () of at most 90%, 75%, or 60% on average, respectively, (Table 2) and the unvaccinated continue to use face masks and maintain physical distancing at current compliance levels. Note: the population size threshold noted here is the average of the values reported for contact rate (C) of 16 and 24 in Table 2.

Instead of adding mass tests only, addition of trace and tests only to symptom-based testing at the lowest rate of 10% (or highest rate of 50%) will also be sufficient to keep the population size on campus at 100% () if at least 84% to 95% are vaccinated (or 42% to 63% are vaccinated) (Table 2). If vaccine coverage (V) is below 42% it would be necessary to also reduce the population size, keeping it to at most 89% on average if vaccine coverage is between 27% and 42%, and to at most 75% on average if vaccine coverage is between 12% and 27%. Considering that 50% trace and test, equivalent to 2 days from infection to isolation is a very tight timeline, which may be more feasible only with digital tracing, we also evaluated at a maximum of 20% trace and test, equivalent to 5 days from infection to isolation. This level of 20% trace and test only will be sufficient to keep the population size on campus at 100% if vaccine coverage is at least 74% to 84%. If vaccine coverage is below that, it will also require a reduction in population size, e.g., to 75% on average if only 44% to 54% of the population are vaccinated (Table 2). All the above scenarios for trace and tests also correspond to the continued use of face masks and physical distancing at least at current compliance levels (transmission rate of 5% to 8%). The combinations of testing and vaccination coverage under the full range of transmission rates are presented in S2 Table.

The total cases of infections and deaths over a 90-day semester if a fully unvaccinated population is on campus (contact rates of 16 to 24 per person per day as reported for regular face-to-face instructions [42, 43]) suggest an exponential growth in infections in most testing scenarios, even if face mask and physical distancing are used at levels reported during the pandemic (transmission rates of 5% to 8%) (S3 Table). With contact rate of 6 to 8 per person per day (corresponding to reported numbers when several universities moved to partial or full remote instructions [35, 36]) and use of facemask and physical distancing at levels reported during the pandemic, an exponential growth in infections was prevented with the following testing scenarios: 33% per day mass test only, at least 33% trace and test only, any of the combination tests, and any of the delayed combination tests (Table 3). In these suitable scenarios, the peak number of trace and tests, per 10,000 persons, varied from 2 to 64 per day, and the peak number of quarantines, per 10,000 persons, varied from 3 to 26 per day (Table 3).

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Table 3. Suitable testing options ^⁋ for effective control of a disease outbreak keeping contact rates at reported levels ^{† ‡}.

https://doi.org/10.1371/journal.pone.0255864.t003

Discussions

This work estimates, under varying combinations of mass test, trace and test, and transmission rate, the contact rate thresholds that would help efficiently control an infectious disease outbreak on residential university campuses in the United States. The metric typically used in the COVID-19 literature for evaluating testing strategies is the reproduction number R₀, which combines the contact rate and transmission rate. As interventions that influence transmission rates are different than those that influence contact rates, separating these parameters help systematically evaluate metrics to inform epidemic control protocols on university campuses. In this study, we extracted four main metrics. First, the contact rate threshold among non-essential workers after accounting for the higher contact rate among essential workers, which could help inform the size of social circles at the individual-level and schedule group activities such as in labs and offices. Second, population size threshold, i.e., the maximum proportion of the actual population size, which could help university-level activity decisions such as the fraction of classes that should be moved to remote instruction. Third, the threshold values for vaccination coverage for the campus to return to normal, i.e., the minimum vaccination coverage for having 100% of the population back on campus, which would help plan for the period post introduction of vaccines. Fourth, the threshold values for population size if vaccine coverage is below required thresholds, which could help decisions in the event that vaccines are not widely available that coverage (proportion vaccinated) is not at levels sufficient to fully resume normal activities. The fourth metric could especially be useful in the transitionary phase to normality (until vaccines become fully available) and where the results suggest lowering the population size by just a small number, which could be achieved by moving only a few classes online, such that, the overall population density on campus on any given day is lower but most students have most (if not all) of their classes as face-to-face.

While the implementation of the decisions related to the above metrics are driven at the university-level, adherence and feasibility to use of interventions such as face mask and physical distancing could vary by individual behaviors [37, 39, 40]. By separately modeling contact rates and transmission rates in this study, we extracted results corresponding to transmission rates (of 5% to 8%) that match reported behaviors for face mask use and physical distancing [11, 30], and thus evaluated the university-level decisions under these adherence or feasibility ranges.

Our analyses suggest that implementing only testing, only face mask use and physical distancing, or only population size reductions will not be sufficient, but require combinations of these interventions to successfully control an outbreak on university campuses. Further, in the absence of vaccinations, at reported levels of face mask and physical distancing, testing alone without reducing population size would also not be sufficient to control an outbreak. This suggests that university campuses have high population densities that, for effective control of highly virulent infections such as SARS-CoV-2, it would require reducing the population size such as through remote learning.

Although individual interventions are not sufficient, there are multiple choices for combinations of interventions to choose from if vaccines are unavailable. If, along with continuing face mask and physical distancing at current levels, the population size is kept to at most 34% (or 44%) of the actual population size, mass tests only of 25% (or 33%) per day would help control an outbreak (Table 1). The choice between mass tests of 25% per day vs. 33% per day should consider the costs of a greater proportion remote learning (quantitative and qualitative costs) vs. costs of both testing more often and testing a larger population.

An alternative to mass tests only would be trace and test only, along with continuing face mask and physical distancing at current levels and reducing population size. Trace and test only would also be sufficient at rates of 33% (or 50%) if population size is kept to at most 39% (or 52%) (Table 1). These population size range are close to the 34% (or 44%) reported above for 25% (or 33%) per day mass tests only. Trace and test of 33% and 50% correspond to 3 days and 2 days, respectively, from the time an infected person makes contact with an individual to effective isolation of that individual. Feasibility of this short turnaround times would determine the choice between use of mass test vs. trace and test. Turnaround times are expected to be shorter with digital contact tracing, such as smart phone apps, compared to manual tracing, and feasibility and adoption of apps could be higher among university students than general population. However, studies related to its feasibility and adoption followed by adherence to isolation, among other issues such as privacy and alternative digital technologies are only recently emerging [44–47]. Our results also suggest that, if these turnaround times are not achievable and further if there are any delays in trace and test initiation, then trace and test alone is not favorable (none of the delayed trace and test were feasible (Table 2)) and should instead adopt either mass tests only or mass test with trace and test. Use of mass test with trace and test could improve trace and test due to potential early diagnosis of index persons. Our results suggest that, if mass tests can increase trace and test to 50% (within two days from contact to isolation), there is more flexibility in trade-offs between mass test rates and contact rate thresholds, and thus more flexibility in population size (Table 2).

In the event that vaccines are available, the full population can be back on campus and resume normal activities provided at least 95% of the population is vaccinated (Table 2). If vaccine coverages are lower than 95%, resuming normal activities with the full population size on campus would require additional asymptomatic testing, with the level of testing depending on vaccine coverage. Mass tests of at least 25% per day would be sufficient if vaccine coverage is at least 70%, or mass tests of at least 33% per day would be sufficient if vaccine coverage is at least 59%. If vaccine coverage is below 59%, to control an outbreak, in addition to mass tests at 33% per day, it would also require lowering the population size to 90%, 75%, and 60%, if vaccine coverage is at 46%, 31%, and 16%, respectively (Table 2).

Corresponding to the reported compliance to face mask and physical distancing and reported contact rates of 6 to 8 per person per day (a population size of 31% to 42% as per our estimations), from surveys [35, 37, 39] conducted over the year 2020 when universities transitioned a large proportion of classes to remote instructions and vaccines were unavailable, our results suggest the need for at least 33% mass test only or 33% trace and test only (Table 3). Scenarios that did not meet these criteria led to exponential growths in infections. These results generally match observed cases over the Fall 2020 semester, where several campuses saw cases into the thousands within the first two weeks of opening and were able to quickly control the spread within two to three weeks by temporarily transitioning to remote instructions [48]. While the universities were able to effectively control the outbreak quickly, it was also observed by this study [48] that the infections rapidly spread into the neighboring community, which were less successful in controlling the spread. Therefore, we believe, results obtained from our study, which set tight tolerance levels on infection cases, would be beneficial for developing epidemic response plans that consider the interests of the broader community. Our results also suggest that, with asymptomatic testing only, it would be necessary to have a vaccination coverage threshold of >95% for a university to fully return back to normal. This threshold is much higher than the typical 70% to 80% range used for herd-immunity in the literature for the general population [49], to a small extent because of setting a tighter tolerance but to a large extent because of the higher population density characteristic of university campuses. The latter can also be observed in R₀ values estimated for universities, which in some instances went above 10 even with online instructions [48], while the herd-immunity in the general population is approximately calculated as 1−1/R₀ using a R₀ of 3.5.

Our work is subject to limitations. Our model is deterministic. We used an average contact rate for all persons in order to estimate threshold values that could help inform university-level decisions. We did not model contact rates to be representative of actual expected networks between individuals. We did not explicitly model other interventions that could reduce transmission rate such as controlled ventilation, filtering air and controlling air flow, which are likely to impact transmissions [50]. The transmission rates also have a large range of uncertainty due to varying individual behaviors, the data used for streamlining the analyses in this study are based on limited data availabilities, however, the extrapolations over the wide range of transmission rates could be utilized. We did not model false positives for any of the testing scenarios and thus susceptible persons immediately return back to susceptible compartment after testing. We did not model other flu like illnesses and thus we did not assess the additional healthcare resource needs such as testing and quarantining because of similarity in symptoms with COVID-19. In estimation of vaccination thresholds, we did not consider the natural immunity developed among persons who may have been infected previously. The estimation of vaccination thresholds assume that the virus is still prevalent in the larger community and thus there is a chance of the infection entering the population, such as through local or global travel. We assume that the population density is similar across university campuses with contact rates between 16 and 24, and thus, this assumption should be considered when generalizing to campuses.

In conclusion, the results from this study could be used to collectively inform decisions related to testing, population size reductions through remote instructions, size of social circles, personnel scheduling in labs and offices, under scenarios of both unavailability or partial availability of vaccines, and within the observed levels of compliance to face mask use and physical distancing. The analyses conducted here specifically streamlined the results to the COVID-19 disease caused by the SARS-CoV-2 virus. However, given the wide range of transmission rates evaluated here, which were based on results from a meta-analysis study that evaluated SARS-CoV-2 and other viruses of similarly high virulence [11], broader observations from this study could be extrapolated for use in early stages of new outbreaks of similar viral respiratory infections with similar incubation periods [24], where non-pharmaceutical intervention options such as face masks, physical distancing, remote instructions, and testing are the suitable options. As was the case at the time of conducting this study, in the early stages of an outbreak, there is uncertainty in the baseline transmission rate, efficacy of face mask use and physical distancing [11]. Thus, the results over the range of transmission rates might only serve as a preliminary guide, until more information becomes available for more streamlined analyses.

Supporting information

Acknowledgments

We would like to acknowledge Sonza Singh, Shifali Bansal, Seyedeh Nazanin Khatami, and Arman Mohseni Kabir for their assistance in data collection in initial stages of the study, and Dr. Laura Balzer, Dr. Michael Ash, and Dr. Hari Balasubramanian for their comments and inputs.