  More than one billion students are out of school because of Covid-19, forced
to a remote learning that has several drawbacks and has been hurriedly
arranged; in addition, most countries are currently uncertain on how to plan
school activities for the 2020-2021 school year; all of this makes learning and
education some of the biggest world issues of the current pandemic.
Unfortunately, due to the length of the incubation period of Covid-19, full
opening of schools seems to be impractical till a vaccine is available. In
order to support the possibility of some in-person learning, we study a
mathematical model of the diffusion of the epidemic due to school opening, and
evaluate plans aimed at containing the extra Covid-19 cases due to school
activities while ensuring an adequate number of in-class learning periods. We
consider a SEAIR model with an external source of infection and a suitable loss
function; after a realistic parameter selection, we numerically determine
optimal school opening strategies by simulated annealing. It turns out that
blended models, with almost periodic alternations of in-class and remote
teaching days or weeks, are generally optimal. Besides containing Covid-19
diffusion, these solutions could be pedagogically acceptable, and could also
become a driving model for the society at large. In a prototypical example, the
optimal strategy results in the school opening 90 days out of 200 with the
number of Covid-19 cases among the individuals related to the school increasing
by about 66%, instead of the about 250% increase that would have been a
consequence of full opening.
