Effectiveness of Isolation Policies in Schools: Evidence from a Mathematical Model of Influenza and COVID-19

Background: Non-pharmaceutical interventions such as social distancing, school closures and travel restrictions are often implemented to control outbreaks of infectious diseases. For influenza in schools, the Center of Disease Control (CDC) recommends that febrile students remain isolated at home until they have been fever-free for at least one day and a related policy is recommended for SARS-CoV2 (COVID-19). Other authors proposed using a school week of four or fewer days of in-person instruction for all students to reduce transmission. However, there is limited evidence supporting the effectiveness of these interventions. Methods: We introduced a mathematical model of school outbreaks that considers both intervention methods. Our model accounts for the school structure and schedule, as well as the time-progression of fever symptoms and viral shedding. The model was validated on outbreaks of seasonal and pandemic influenza and COVID-19 in schools. It was then used to estimate the outbreak curves and the proportion of the population infected (attack rate) under the proposed interventions. Results: For influenza, the CDC-recommended one day of post-fever isolation can reduce the attack rate by a median (interquartile range) of 29 (13 – 59)%. With two days of post-fever isolation the attack rate could be reduced by 70 (55 – 85)%. Alternatively, shortening the school week to four and three days reduces the attack rate by 73 (64 – 88)% and 93 (91 – 97)%, respectively. For COVID-19, application of post-fever isolation policy was found to be less effective and reduced the attack rate by 10 (5 – 17)% for a two-day isolation policy and by 14 (5 – 26)% for 14 days. A four-day school week would reduce the median attack rate in a COVID-19 outbreak by 57 (52 – 64)%, while a three-day school week would reduce it by 81 (79 – 83)%. In both infections, shortening the school week significantly reduced the duration of outbreaks. Conclusions: Shortening the school week could be an important tool for controlling influenza and COVID-19 in schools and similar settings. Additionally, the CDC-recommended post-fever isolation policy for influenza could be enhanced by requiring two days of isolation instead of one.

The model generalizes the classical SEIR model in which there is a sharp division between the exposed but not infectious population, and the infectious population (the E and the I in SEIR). In our model, each stage of the infection has a certain rate of infectiousness and symptoms (which can be zero, for the classical E compartment).   Table S2.
Interpretation Naïve persons in cohort i (for immunity, , see below) ) Infected persons in cohort i on day d of infection who are not isolated Similarly, but who are isolated (e.g. at home)

Recovered in cohort i
We model vaccination as a reduction in the initially susceptible population for each cohort, which depends on the vaccination rate and the vaccine efficacy: . This transformation could be used to account for ) − v e any resistance to infection, whether induced, or arising from genetics or prior exposure to similar pathogens.
Typical outbreak curves are shown in Figure S1 . Figure S1 . The forecasted epidemic curve of an influenza outbreak in a median school size based on US Department of Education data. Simulation is for the case with no formal isolation policy (where isolation occurs due to voluntary choice alone). Transmission is reduced during weekends, resulting in visible ripples in the outbreak curves every 7 days. ILI, influenza-like illness.

Contact Rates
We assumed that persons have the most physical contact with others in their cohort. The rate of contact with other cohorts is controlled by a parameter which was varied in our sensitivity analysis. To adjust for generally higher contact rates during the winter months, we use a seasonal term cf. 3,4 which multiplies the baseline contact rate, , b 0 by a factor that peaks on January 1st. : Here, D is the day of the year counting from January 1st. The value is multiplied by if the day falls on a b b h weekend or by if the school is in closure or vacation (see Table S2). b c

Sensitivity Analysis
In sensitivity analysis, we assigned the parameters to truncated normal distributions with the mean, standard deviation (SD) and range obtained from previous research or calibration. For influenza, we then adjusted the transmission parameter of the model to give a 25% attack rate (thus matching the typical rate 5 , after accounting for symptomatic rate 6 ). For COVID-19, we adjusted the transmission parameter to give an 11.3% attack rate observed in a representative school outbreak (see Calibration section below). Experiments with alternative attack rates give qualitatively similar results (results not shown). To estimate the rate of symptomatic COVID-19 cases, our priority was to use only studies that had unbiased sampling of all at-risk persons, since surveillance-based methods tend to capture individuals with higher expression of symptoms 6 . Therefore, we adopted the estimate of Poletti et al. 8 which tested all household members of known COVID-19 cases and is unique in reporting the symptom rate for ages 0 to 19, instead of all age ranges. It used a symptom definition of upper or lower respiratory tract symptoms, or fever ≥37.5 °C and a sample size of N=692.  Table S3 and Table S4 below   Shedding at day d  Table S3 and Their estimate is consistent with a large random survey from Spain 14 that reported a rate of 34.53% but that included adults that have a higher rate of symptomatic infections 8,15 .

Modeling of isolation behaviors and the effect of isolation policy
We model the complex interaction between the factors that affect the population. As quantified below, the rate of returning to school is decreased at the more severe stages of the disease, when the disease has a higher symptomatic rate, or when the patients are more aware of their symptoms. As described below, the symptom-based isolation policy is modeled as a decrease in the rate at which students return to school.
To be exact, let the values express the fraction of persons returning from home on day after they become r d d infected when no policy is in place. It is modeled as a function of four factors: the symptoms in the previous days , the symptom propensity (i.e. rate of symptomatic infections) , the attention to symptoms ) (f s 0, ] l ∈ [ 1 0, ] y ∈ [ 1 , and the compliance to policy (if any) . To be precise, in the simplest case (no isolation policy), , or would decrease the return rate: . The introduction of the isolation policy f d−1 l y yf r d = 1 − l d−1 decreases the return to school rate at day by making the persons more attentive to the recent days of symptoms. d Namely, under a one-day isolation policy, the rate is modified to: This model ensures that the rate of symptoms sets an upper bound on the effectiveness of the f l d−1 symptom-isolation policy. In the case of 100% compliance and 100% symptom attention, the return rate is 1 − lf d−1 and not lower. Under a two-day isolation policy, the rate on day is given by replacing in Eqn A.3 with d f d−1 , and in general, for longer isolation policies (see Table S3 and Table S4).

Viral Shedding and Symptom Burden
Influenza viral shedding appears to vary by subtype. However, the patterns of shedding were similar in both children and adults 16,17 , and between the seasonal and p(H1N1) outbreaks [17][18][19] . We allowed for a proportion of the l infections to be asymptomatic 16,20 , which in our model increases the rates of return to the community and transmission, while also reducing the effectiveness of control policies. Meta-analysis of influenza studies 21 was used to determine shedding and symptom rates by disease state (see Table S3, Table S4). Table S3. Daily symptom and shedding rates for influenza from 21 . Viral shedding rate is based on log10 of titers, and is multiplied by the transmissibility parameter in the model. Symptom scores are relative to their peak. New influenza-like illness (ILI) rates is the probability of newly reporting ILI on a given day. Predicted return rates when symptom propensity is and symptom attention is . Return rates: A = no policy, B = policy of one .84 l = 0 y .5 = 0 day of isolation with 0.5 compliance, C = policy of one day of isolation with 100% compliance. As the rate of compliance rises, the return rate generally decreases since students self-isolate at home. Rows are days from the time of infection. Total Score is an overall indicator of illness reported in 21 Figure S2 . Estimated SARS-CoV-2 infectiousness and symptom propensity in symptomatic cases. Infectiousness is based on 22 with linear interpolation added before day 5 and after day 28 [23][24][25][26][27] . Fever scores for patients with fever are based on collections of case reports 28 . Table S4. Estimated symptoms, shedding rates, and return rates for symptomatic persons infected with SARS-CoV-2. Asymptomatic infections are accounted for through a separate parameter. Return rates calculated for when attention is focused on fever, symptom propensity rate is , and symptom attention is . .1809 l = 0 y .5 = 0 Return rates: A = no policy, B = policy of one day of isolation with 0.5 compliance, C = policy of one day of isolation with 100% compliance. As the rate of compliance rises, the return rate generally decreases since students self-isolate at home. Rows are days from the time of infection.

Model Validation and Calibration
The model has been validated by comparing it to empirical data on three influenza outbreaks and one outbreak of COVID-19. The available data generally indicates the daily number of new ILI cases or the daily number of new absences, and both were also calculated from the model. In all calibrations, we found the model to give a tight fit to the data matching the attack rate, the peak date, and the overall shape, as illustrated in Figure S3, Figure S4, and Figure S5 . For COVID-19, the fit is reported in Table S5. .
The symptom data indicates that ILI symptoms first arise 24 hours after influenza infection and peak around 2.5 days after infection. Accordingly, we assume that, of the population reporting ILI symptoms, ¼, ½, and ¼ first reports on day 1, 2, and 3, respectively.
We calculated the new absentees per day by considering the number of infected and the return rate. The number of isolated persons is given by: . Furthermore, the number of newly-absent (i.e. isolated) students is given (1 )I (t) ∑ s − r s s by . We will use this formula in calibrating the model to absenteeism data below. Unknown parameter values and ranges were estimated using a genetic algorithm 29 . The algorithm adjusted the values of the unknown parameters to match the well-characterized A(H1N1)v influenza outbreak in a boarding school 30 . Parameter ranges were estimated from the distribution of parameters in the top 10% of the solutions. A separate set of simulations with seasonal influenza calibrated to an outbreak of seasonal influenza 31