  We aim at clarifying the controversy about the effects of school openings or
closures on the course of the Covid-19 pandemic. The mathematical analysis of
compartmental models with subpopulations shows that the in-school contact rates
affects the overall course of the pandemic only above a certain threshold that
separates an influence phase from a non-influence one. The threshold, that we
calculate via linear approximation in several cases, seems to appear in all
contexts, including outbreaks or new strains upsurge, lockdowns, and
vaccination campaigns excluding children, albeit with different values. Our
theoretical findings are then confirmed by several data-driven studies that
have previously identified the phase transition in specific cases. Specific
outcomes of this study are: opposite conclusions reached by studies of the same
or similar situations might depend on, possibly small, differences in modeling
or in parameter estimation from the very noisy Covid-19 data, that result in
identifying different phases; it is possible to keep schools open at any stage
of the Covid-19 pandemic, but suitably strict rules must be applied at all
times or else this becomes highly detrimental to virus containment efforts; as
the threshold during vaccination turns out to correspond to the internal
transmission rate that would lead to virus extinction if the school population
was isolated, the needed strict control can be sustained only for very brief
periods; as a result, either schools will have to face a prolonged closure or
children need to be vaccinated as well.
