Planning of school teaching during Covid-19.

Learning and education are two of the biggest world issues of the current pandemic. Unfortunately, it is seen in this work that, due to the length of the incubation period of Covid-19, full opening of schools in the Fall of 2020 seems to be impractical unless the spread of the virus is completely under control in the surrounding region (e.g. with fewer than 5 active cases every million people). In order to support the possibility of some in-person learning, we model the diffusion of the epidemic within each single school by an SEAIR model with an external source of infection and a suitable loss function, and then evaluate sustainable opening plans. It turns out that blended models, with almost periodic alternations of in-class and remote teaching days or weeks, are generally (close to) optimal. In a prototypical example, the optimal strategy prescribes a school opening of 90 days out of 200 with the number of Covid-19 cases among the individuals related to the school increasing by about 67% with respect to no opening, instead of the about 200% increase that would have been a consequence of full opening. As clinical fraction is low in children, these solutions could lead to very few or no symptomatic cases within the school during the whole school year. Using the prevalence of active cases as a proxy for the number of pre- and asymptomatic, we get a preliminary indication for each country of whether either full opening, or blended opening with frequent testing, or no school opening at all, is advisable.