One of the biggest challenges raised by the Covid-19 outbreaks involves school openings and in-class vs. remote learning, affecting more than one billion students worldwide. One of the main issues herein concerns the potential effect of in person school activities on the spread of the epidemic: many studies have been conducted, often reaching divergent conclusions. In this paper, we first identify one apparently unobserved feature of the interrelation between virus transmissions in two subpopulations : as the contact rate of one subpopulation is varied, there is a phase transition between almost irrelevance to heavy influence of the transmission rate in the subpopulation on the overall outbreak; such transition is seen to occur during outbreaks or new strains upsurge, lockdowns, and vaccination campaigns , albeit at different transition points. Applied to schools, our results suggest that suitably, and possibly very tightly, controlled openings have no bearing on the overall spread of the pandemic. This is the case even during a widespread vaccination for which school age children arenot eligible. To curb the mathematical complexity of the results, we restrict here to SIR models, but we trust, and illustrate by simulations, that our conclusions apply in more general setting; our results provide a clarification to the diversity of opinions on school opening effects, and help envisioning plans for upcoming school activities. Case studies in the literature illustrate the occurrence of phase transitions .
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 .
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One of the biggest challenges raised by the
We aim at clarifying the controversy about the effects of school openings or closures on the course of the
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outbreaks involves school openings and in-class vs. remote learning, affecting more than one billion students worldwide. One of the main issues herein concerns the potential effect of in person school activities on the spread of the epidemic: many studies have been conducted, often reaching divergent conclusions. In this paper, we first identify one apparently unobserved feature of the interrelation between virus transmissions in two subpopulations : as the contact rate of one subpopulation is varied, there is a phase transition between almost irrelevance to heavy influence of the transmission rate in the subpopulation on the overall outbreak; such transition is seen to occur during
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
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, albeit at different transition points. Applied to schools, our results suggest that suitably, and possibly very tightly, controlled openings have no bearing on the overall spread of the pandemic. This is the case even during a widespread vaccination for which school age children arenot eligible. To curb the mathematical complexity of the results, we restrict here to SIR models, but we trust, and illustrate by simulations, that our conclusions apply in more general setting; our results provide a clarification to the diversity of opinions on school opening effects, and help envisioning plans for upcoming school activities. Case studies in the literature illustrate the occurrence of phase transitions
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