A new threshold reveals the uncertainty about the effect of school opening on diffusion of Covid-19.

Studies on the effects of school openings or closures during the Covid-19 pandemic seem to reach contrasting conclusions even in similar contexts. We aim at clarifying this controversy. A mathematical analysis of compartmental models with subpopulations has been conducted, starting from the SIR model, and progressively adding features modeling outbreaks or upsurge of variants, lockdowns, and vaccinations. We find that in all cases, the in-school transmission rates only affect the overall course of the pandemic above a certain context dependent threshold. We provide rigorous proofs and computations of the thresdhold through linearization. We then confirm our theoretical findings through simulations and the review of data-driven studies that exhibit an often unnoticed phase transition. Specific implications are: awareness about the threshold could inform choice of data collection, analysis and release, such as in-school transmission rates, and clarify the reason for divergent conclusions in similar studies; schools may remain open at any stage of the Covid-19 pandemic, including variants upsurge, given suitable containment rules; these rules would be extremely strict and hardly sustainable if only adults are vaccinated, making a compelling argument for vaccinating children whenever possible.

Introduction

The question of keeping schools open or closing them has turned out to be one of the most debated issues of the Covid-19 pandemic. Schools have been closed at the early stages of the pandemic in almost every country, with classes held online for most of last year[1,2]; but the, sometimes hurriedly arranged, remote teaching has created great difficulties for more than one billion students, their teachers and communities[3,4].

Many studies have been carried out trying to clarify the potential effects of school opening on the course of the Covid-19 outbreaks: they appear to be reaching different, and sometimes conflicting, conclusions. Several works[5-8] find that closing schools has little impact on the number of cases, while others conclude that it is very effective[9-11]; a number of other studies determine that the influence of school opening on the course of the pandemic depends crucially on some implementation details, such as level of blending, use of masks etc.[12-18]. Even studies of the same situation reach opposite conclusions[19,20]. When trying to analyze available Covid-19 data using simulations of compartmental models, we observed a remarkable instability: the observed effect of school opening depended in a crucial way on small changes in the models parameters, and we kept oscillating between making the schools the culprits of the pandemic, or completely absolving them; see Fig. 3a–c below.

Figure 3

Active cases in two different scenarios.

This situation raises the issue of understanding the mechanisms behind and the nature of a possible transition in the effect of school opening policies. Ideally, one would like to identify one or more parameters, measurable at least in some theoretical sense, summarizing the effects of school opening, with an explicitly, for as much as possible, known effect on the overall epidemiological indicators. A clearer understanding in this sense could inform data collection, analysis and release of information, and help providing guidelines for policy makers in concrete situations.

Our aim is to report of the identification of a mechanism which could explain the observed instability of effects and diversity of conclusions. To highlight the fundamental mechanisms at work, we started from the simplest compartmental models[21,22] with two subpopulations, studying how changes in the transmission rate in one population affect the overall course of the infection. We first performed a theoretical study of the mathematical models, validating then the results using simulations with incrementally added more realistic features, and a comparative analysis of various data based studies.

The outcome of our analysis has been the existence of a perhaps surprising threshold, below which further reduction of in-school transmission, for instance by school closure, has a minimal effect, but above which school opening becomes the leading factor driving infections. We observe the transition in all scenarios, albeit with different vales of the threshold. Of particular interest is the case of a vaccination campaign of adults, in which the threshold for in-school transmission rates turns out to be extremely small, giving a strong indication of the need of a children vaccination campaign. The presence of a threshold for the in-school transmission rate can explain the divergent conclusions of several statistical studies, as they might have been observing the two opposite conditions. In addition, if in the situation under investigation school transmission is close to the threshold, epidemiological models, which necessarily rely on, often hard to estimate, parameters, might end up forecasting one or the other scenario depending on tiny changes in the calibration. Awareness of a threshold is crucial for modeling, data collection and analysis, and policies determination[23].

Let us clarify, though, that, to focus on our main objective, we disregard many other relevant epidemiological issues concerning school opening, such as the possible role and availability of teachers[24]; the sustainability of school opening[25]; as well as psychological, cultural, educational aspects; that need then to be added when planning concrete policies.

In addition, variants, especially those which might be vaccine resistant, constitute a very challenging situation: from an abstract point of view, the situation is similar to that of an initial outbreak, as there are little awareness and immunization capabilities, so, broadly speaking, our first scenario below applies. On the other hand, the population has developed new individual and social responses, and therefore new assessment of various parameters of our investigation should be suitably modified. It is too early now to be able to make such adjustments, but the situation should be reevaluated should a resistant variant become prevalent.

Results

There is a phase transition in the effect of school transmission rates on the overall epidemic course during an outbreak (or a variant upsurge)

The effect of the in-school transmission rate on the course of the epidemic undergoes a phase transition with threshold

The total number of active cases is almost constant for all , and has a sharp increase for . An effective containment of the effect of a change of is achieved ifwhile there is a substantial effect ifwhere is generally chosen to be a small integer (smaller than to make sense of (2)), analogous to the number of deviations away from a mean, used to describe a transition as plotted in Fig. 4. We take in “Parameter calibration” section. Since in concrete cases, see “Parameter calibration” in "Methods" section, the right hand sides of (2) and (3) are close to .

Figure 4

Derivative of the largest eigenvalue of with respect to , with and .

Calculations are done in a linear approximation of the SIR model, which applies to the Covid-19 pandemic as the numbers of active cases are kept relatively low by containment measures in the early stages of the outbreaks[26]. In the linear approximation it is possible to formally compute the total number of cases up to a certain time , which corresponds to when a lock down is imposed. If the target is to contain the increase in the number of total cases up to to a given percentage , an explicit formula allows to compute the maximal allowed value of .

In a realistic example with total population and recovery rate normalized to 1, setting , see “Parameter calibration” in "Methods" section, the critical point is . Assuming an initial fraction of of active cases in Subpopulation 2 and none in Subpopulation 1, a rescaled time frame of (corresponding to approximately 50 days), and , a suitable value of gives .

The first part of Fig. 1, for , shows a simulation of the active cases with the above values. One can see that school opening has a moderate effect for small values of , and then the effect becomes dramatic as the values increase past the critical point.

Figure 1

Daily active cases for various scenarios: outbreak or new strain upsurge, lockdown, and vaccination. In each case, there is critical value for the in-school transmission rate . Cyan curve is with closed schools, green for safe opening, orange for critical values, red for values above criticality.

It follows that closing schools, i.e. setting , is of limited impact if the reproduction rate in school is somewhat lower than , the external reproduction rate, and of substantial impact otherwise. This provides harmless school opening options, assuming that one has access to the reproduction rates in the subpopulations.

The phase transition is preserved under lock-down, albeit with a different critical point

An analogous effect takes place when a lockdown is imposed. If at some time transmission rates are reduced to values , corresponding to a subcritical reproduction number, then the effect of on the total number of active cases undergoes the same phase transition as during the outbreak, but with critical pointMore precisely, letindicate the attack rate of the epidemic, i.e. the fraction of the initially susceptible population that is eventually infected by the disease in the course of the epidemic from to complete eradication. It turns out that a sufficient condition to ensure that does not exceeds iswhere F (see (26) below) is a function that depends on the proportions of active cases and susceptible individuals at time .

In a realistic example continuing the one for the outbreak, with we get . In addition, we take , see “Linear approximation during the initial phase of an outbreak or new strain upsurge” in "Methods" section; in order to contain the increase in attack rate to no more than 30% one needs now to have

Although in a different scenario, this is smaller than the value 6.344 found in the outbreak, as there the aim was just to avoid producing an even more extended diffusion of the infection.

The second part of Fig. 1, for , illustrates active cases in the lockdown scenario, with the above values of the model parameters.

When considering a complete outbreak-lockdown cycle, the attack rate undergoes a similar transition, depending on the values of the two transmission rates and . If the pair is sufficiently closed to (0, 0), then there is little change in , while there is a drastic change for larger values of the two transmission rates (see Fig. 7).

Figure 7

Total number of cases at resolution of outbreak for varying values of and .

Success of widespread vaccination of non-schooling individuals requires internal reproduction number in schools to be subcritical

If a vaccination campaign for not-in-school individuals is carried out, the total number of cases from a restart of the epidemic to the complete disappearance due to vaccination undergoes an analogous phase transition, with threshold

The attack rate is only moderately changed for below the threshold, while the outcome of the vaccination process is substantially disrupted for larger values of .

Notice that if then the in-school reproduction number is i.e. is subcritical.

In a realistic case, continuing with the data from the example above, we now suppose a vaccination program is introduced targeted to a 60% coverage in about 3 months (Israel kept this pace at the time we are writing, with schools almost completely closed). In Fig. 1, this corresponds to , and now is the susceptible at the start of this simulation, which is equal to the susceptible population calculated for the end of the previous simulation for lockdown (corresponding to t = 18 in Fig. 1). Then , and the critical value for is 5.02836. To achieve a sensible containment that limits the number of extra infections to no more than 30% one needs to take . See “Simulations of the phase transition during vaccination” in "Methods" section for further details on this simulation.

From the point of view of containing the epidemic, schools can be kept open at all times, with strict control measures

Taken together, the results obtained from simple SIR models with subpopulations show that, although the values of the critical points are different, opening of schools would not seriously affect the course of the pandemic at all times, provided the internal transmission rate is kept low enough. On the other hand, if the control is released, then the effect of school opening becomes dramatic.

Figure 1 summarizes the numbers of active cases in the three scenarios we have analyzed: the cyan curve corresponds to closed schools, while the green one is a subcritical pattern; the red curves, instead, show the risk that the pandemic spirals out of control because of insufficiently controlled school opening.

Notice that the explicit values that we give in Formulas (1), (4) and (6) are relevant from the theoretical point of view, as they indicate that the critical thresholds are different. Their specific values, however, could be hard to estimate from these formulas. The initial fraction of infected in (1), for instance, is almost impossible to estimate, due to the initial absence of awareness and testing. Other methods and more details models would be needed for a careful estimation of the threshold in concrete cases.

An analogous behavior takes place in more elaborate and realistic models, involving presymptomatic, asymptomatic, testing, isolation etc. Critical values appear for the in-school transmission rate, below which the effect of school opening on the epidemic trajectory is extremely contained. We provide simulations in “A SPIAR model” in section (see Fig. 10), and evidence from case studies here below.

Figure 10

Daily cases for the three scenarios of outbreak, lockdown, and vaccination in an SPIAR model. In each case, there is critical values for the in-school transmission rate , as for SIR model. Cyan curve is with closed schools, green for safe opening, orange for critical values, red for values above criticality.

Discussion

Evidence of phase transition appears in several data driven case studies

The effect of a phase transition seems to appear in all data driven studies (see “Other case studies” in section). Most studies reach a definite conclusion: in some cases, the data or the model after calibration correspond to a subcritical regime, so that the study ends up asserting the almost irrelevance of school opening on the pattern of the epidemic for all the analyzed cases; in other cases, the study determines a supercritical setting, and then comes to the opposite conclusion.

Some works include one or more parameters that can be modulated to envision the effect of school reopening. In these cases, one can see the effect of a sharp transition from a subcritical, acceptable reopening, to an excessively impactful one. In España et al., Figs. 4 and 5, for instance, one can see that up to 50% capacity the effect of opening schools is almost negligible, while it becomes substantial above 75%; this is a likely indication of a critical point between these values[17]. For convenience, their Fig. 4 is reproduced here in Fig. 2.

Figure 5

The largest eigenvalue of as function of , with and .

Figure 2

The impact of school reopening strategies in time as simulated in[17] from data from Bogotà, Colombia, for various values of the capacity, i.e. the percentage of students allowed back at school. Each column shows a different capacity level. Top panel shows the median daily incidence of deaths for each reopening strategy based on grades. Bottom panel shows the estimated attack rate for each of the reopening scenarios. Vertical black line shows the timing of school reopening (January 25, 2021). All scenarios were simulated up to August 31, 2021.

A very detailed study of school opening in The Netherlands is conducted in Rozhnova et al. , and their conclusions are a clear description of the phase transition[12]. Using a data driven, elaborate model, Rozhnova et al. claim that their “analyses suggest that the impact of measures reducing school-based contacts depends on the remaining opportunities to reduce non-school-based contacts[12]. If opportunities to reduce the effective reproduction number () with non-school-based measures are exhausted or undesired and is still close to 1, the additional benefit of school-based measures may be considerable, particularly among older school children.” The first scenario of Rozhnova et al.  corresponds to a subcritical in-school transmission rate, so that the effect of closing schools would be very moderate. The second scenario seems to correspond to an in-school transmission rate around the critical value, so that both containment, in- or out-of-school, are effective[12].

Yuan et al. uses a detailed compartmental model and data from the second semester 2020 in Toronto, an outbreak context, to estimate the likely impact of school opening; with parameters estimated and collected from literature, the paper finds that opening school has little effect on the overall course of the pandemic: in all scenarios presented in their Figs. 2 and 4 the difference between school opening and closure is extremely contained[16]. The findings of the research is then consistent with our phase transition scenario. In particular Yuan et al., find that “school reopening was not the key driver in virus resurgence, but rather it was community spread that determined the outbreak trajectory”; in other words, the parameters of the models, although not explicitly given in the paper, are such that the external transmission is preponderant[16]. As an additional finding, it is observed in Yuan et al. that, according to their model, “brief school closures did reduce infections when transmission risk within the home was low”[16]: in this case, a reduced transmission rate outside makes the one in school likely supercritical.

The role of phase transitions

Phase transitions, like the one occurring at or the one we detect for in-schoool transmission rates, are fundamental in epidemiology[23]. They consist of the fact that changes in one parameter, the in-school transmission rate in our case, produce almost no effects except when the threshold is crossed; at that point, however, a small change in the parameter determines completely different behaviors.

The presence of a threshold for the in-school transmission rate can explain the divergent conclusions of data driven studies, as they might have been observing two different phases. The presence of a phase transition can also dramatically disrupt forecasts based on calibrated compartmental models: one calibration might lead to a subcritical phase, in which the transmission in school is irrelevant, and another, based on possibly similar data, might lead to a supercritical phase, in which in-school transmission is the driving factor of the pandemic. This phenomenon is known to affect epidemic forecasts[27], and we think it might be the reason beyond the mentioned conflicting conclusions of several studies.

To make things worse, even retrospective studies trying to evaluate the role of school openings or closures on the evolution of the pandemic run the risk of being completely untrustworthy. Covid-19 data are affected by enormous errors, due to the presence of asymptomatic, lack and partial reliability of testing, difficulty in assessing close contacts etc. It follows that estimation of parameters for both statistical and model based studies are affected by large errors. In the presence of a threshold, even small errors can lead to incorrect attribution of the situation under observation to one phase, or to conflicting attributions to two opposite phases by different studies. In such scenario, a retrospective study could misclassify the effect of school opening or closure; and different studies even based on almost the same data might end up reaching opposite conclusions. We explicitly illustrate this phenomenon with a simulation in the next section.

Finally, awareness about the presence of a phase transition suggests the type of measurements that could be carried out, analyzed and finally released to the public. In our case, for instance, one could consider adapting our model to specific local situations, and then measuring in-school transmission rates; these can then be used as basis for local policies about school opening, and also as a possible public indicator of the potential risk of interventions on schools. The information that the in-school transmission rate is approaching a critical level would certainly stimulate and justify the reinforcement of containment measures. The findings about vaccination strongly support vaccinating children as well.

Confounding effects on retrospective studies

In the noisy, synthetic data in Fig. 3a, the number of daily infected in a population have been generated with the same parameters, except that

In school transmission rates are supercritical in the first case, and subcritical in the second. But the different number of initial cases, a value that is subject to errors of various order of magnitudes and is quite arbitrarily assigned in the various studies, makes the two trajectory basically indistinguishable.

In a retrospective study one is forced to assign an initial value to the number of infected, and then estimate other parameters from the observations. Both scenarios are then plausible, depending on the chosen initial values. As Fig. 3b confirms, the research would conclude in the first scenario, that closing schools would have been basically useless. In the second scenario, however, the opposite conclusion would be drawn, as illustrated by Fig. 3c.

Conclusions

We have identified the presence of different phases for the effect of the in-school transmission rates on the course of a Covid-19 like epidemic.

Such results provide evidence in favor of keeping the schools open when specific epidemiological conditions are met and preventive measures are respected: the key condition is that the transmission rate in schools must be kept below a certain threshold that depends on the situation. As the threshold might not be easily determinable nor achievable, however, there can be contingent motives for school closure if policy makers, as it happened in most locations during the Covid-19 pandemic, are not aware or able to exploit the critical threshold. Awareness about the critical threshold can in any case suggest directions for the analysis of locally adapted models, data collection and exploration, and public release, and can give policy makers sound instruments for containing school closures.

In addition, the presence of a threshold is the likely cause of the opposing views that many studies have presented, some asserting almost irrelevance of school opening, and others pointing to its significant effects. Minimal changes in the overall conditions, or in values of the estimated parameters may determine one phase of the other; this may result in different attributions of responsibility to school opening, and creates the possibility of an arbitrary identification of the phase due to parameter estimation in the presence of very noisy data.

Finally, we have seen that with a vaccination campaign being carried out largely for out-of-school individuals only, there is a threshold below which schools can still be opened; it corresponds, however, to an internal reproduction number that would eradicate the virus if schools were completely isolated. As measure to achieve such a low reproduction number are highly demanding, this level of containment seems to be sustainable for brief periods only, after which vaccination for children becomes the only viable possibility to return to normality.

Limitations, related and future works

Although the presence of a phase transition in the effect of the school transmission rate in the overall course of the epidemic seems to have been unnoticed so far in the literature, there are many works related to ours.

For a different perspective, focused on the sustainability of opening from the point of containing the number of cases of a single school, one can see[25].

Compartmental models with two subpopulations are discussed in many works in general terms[28]; and then applied to the school opening issue in data driven analyses[7,11,16,17]: we discuss the relation of some of these results with our work in “Other case studies” in "Methods" section.

Finally, other papers[6-8] make a purely statistical evaluation of the effect of school opening (see “Other case studies” in "Methods" section).

Our work has several limitations. Our results are based on abstract, simplified models, and, although they seem to be stable when more detailed features are included, we cannot ensure that they always take place in more complex models.

Even when a critical value can be estimated, ensuring that the transmission rates are below their relative thresholds is clearly a matter of distancing, testing, and other measures[29]. We do not elaborate here on how to develop a set of possible interventions, and on how to measure their success in containing the transmission rates in schools.

There are several directions for future work and research.

From the practical point of view, our analysis needs to be adapted to local and contingent situations adding specific details to the model, and collecting and analyzing appropriate data before becoming a viable tool for policy makers.

From the mathematical point of view, it would be interesting to ascertain the presence and behavior of the phase transition in the non linear models. Also, it would be relevant to explore the analogous phenomenon when the cross terms are not small with respect to those in the main diagonal, a situation which could explain the dynamics of vaccinated vs. unvaccinated population. Finally, one could evaluate the presence of similar phenomena with more than two subpopulations, in order to detect which combinations drive the pandemic, and which internal transmissions can be disregarded up to a certain threshold.

Table 1

Recap of the model parameters and their selected values for each scenario in SPIAR model.

Parameter	Outbreak	Lockdown	Vaccination
β_22	2	0.5	1.5
β_12=β_21	0.25	0.1	0.25
s	0.9	0.9	0.9
ξ	0.3	0.3	0.3
k	1	1	1
ε_1	0.75	0.75	0.75
ε_2	0.9	0.9	0.9
v	0	0	0.05