We model COVID-19 progression in an individual as having the states susceptible (S), exposed (E), presymptomatic (P), symptomatic (Sym), and recovered (R). Individuals start in the susceptible state and then transition to the exposed state when they are infected. Exposed individuals are not able to infect others. Individuals transition from the exposed state to the presymptomatic state at which point they are able to infect others, but have no symptoms. Individuals may either transition from presymptomatic to symptomatic states (showing symptoms while remaining infectious) or directly to the recovered state without ever showing symptoms. Symptomatic people eventually enter the recovered state, where they are no longer infectious. Presymptomatic or symptomatic individuals infect susceptibles at constant rate β when they are together, where β may depend on the individuals involved, their exact state, their proximity and the environment.

Individuals stay in the exposed state for the duration of the latent period, after which they become presymptomatic. Latent periods are modeled as gamma-distributed with mean μ and standard deviation σ. For each presymptomatic individual a gamma-distributed presymptomatic infectious period (PIP) and a gamma-distributed infectious periods are generated (with means and standard deviations (μP, σP) and (μi, σi) respectively). The individual is infectious for the duration of the infectious period, starting from when they enter the presymptomatic state, and ending when they recover. They enter the symptomatic state after the PIP, if they have not already recovered by that time. See the Supplemental Material for information about the parametrization and distribution of the latent period, the PIP, and the infectious period. With probability α they never show symptoms (and are asymptomatic); otherwise symptoms appear after the PIP.

Rather than assuming that all students in the class are equally likely to transmit the infection to each other, we model individual and contact group effects. We assume that the nclass students are broken into smaller groups of ngroup students. We start with a base rate β of transmission which represents the rate of transmission of from one infectious individual to another in the same group. Our default value for β is 0.003 transmissions per contact per hour (0.006 in environments with increased transmission). Another source of variability is in the infectiousness of individuals. As discussed in the introduction, some evidence indicates that certain individuals are superspreaders and so have atypically large β compared to others. The most important instance of this is when the index case has high β. In order to capture this, we model the index case as having a separate transmission rate β0 = findex β where findex = 1 or 3, depending on whether the index case has the same infectiousness or a higher infectiousness than others. We also model reduced infectiousness of asymptomatic individuals. Their transmission rate is fasymp β where we choose fasymp = 0.8. We explore the impact of these assumptions in the supplemental material. The final effect modifying transmission rate is to decrease it when the infectious person and the susceptible person are in different contact groups. The effect is to multiply β by faero = 0.25. We model the effect of these different heterogeneities multiplicatively, so that if, for example, the index case is asymptomatic, the rate of transmission to a susceptible in another group is findex fasymp faero β. We note that our maximum value of β (when both the environment and the infectiousness of the index case are most conducive to transmission) is 0.018 transmissions per contact per hour. This is considerably smaller than the estimates of index β for widely-reported outbreaks in adults [38], by up to a factor of 30 for some events.

Table 1 lists the parameter values used in our simulations and provides supporting citations. We run our simulations twice: once with the index case is symptomatic, and once when the index case is asymptomatic, because this turns out to be a crucial factor in determining cluster size.

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