Mitigation of infectious disease at school: targeted class closure vs school closure

Background School environments are thought to play an important role in the community spread of infectious diseases such as influenza because of the high mixing rates of school children. The closure of schools has therefore been proposed as an efficient mitigation strategy. Such measures come however with high associated social and economic costs, making alternative, less disruptive interventions highly desirable. The recent availability of high-resolution contact network data from school environments provides an opportunity to design models of micro-interventions and compare the outcomes of alternative mitigation measures. Methods and results We model mitigation measures that involve the targeted closure of school classes or grades based on readily available information such as the number of symptomatic infectious children in a class. We focus on the specific case of a primary school for which we have high-resolution data on the close-range interactions of children and teachers. We simulate the spread of an influenza-like illness in this population by using an SEIR model with asymptomatics, and compare the outcomes of different mitigation strategies. We find that targeted class closure affords strong mitigation effects: closing a class for a fixed period of time – equal to the sum of the average infectious and latent durations – whenever two infectious individuals are detected in that class decreases the attack rate by almost 70% and significantly decreases the probability of a severe outbreak. The closure of all classes of the same grade mitigates the spread almost as much as closing the whole school. Conclusions Our model of targeted class closure strategies based on readily available information on symptomatic subjects and on limited information on mixing patterns, such as the grade structure of the school, shows that these strategies might be almost as effective as whole-school closure, at a much lower cost. This may inform public health policies for the management and mitigation of influenza-like outbreaks in the community. Electronic supplementary material The online version of this article (doi:10.1186/s12879-014-0695-9) contains supplementary material, which is available to authorized users.

We implement the same mitigation strategies as in the main text, and, in order to compare their relative efficiencies, we focus on the fraction of stochastic realizations that yield a global attack rate higher than 10%, on the final average number of cases and on the temporal evolution of the number of infectious individuals. We compare the results obtained by implementing the various mitigation measures with the baseline case, represented by the situation in which the only mitigation strategy consists in the isolation of the symptomatic individuals once they are detected.

Results
In Tables I, II and III we show the fraction of stochastic realizations leading to an attack rate (AR) higher than 10%, for various mitigation measures, compared with the same result obtained when no closure was implemented. Compared with the results presented in the main text, faster spread lead to a decrease in the probability of large outbreaks even in the baseline case, while larger fractions of asymptomatic individuals make large outbreaks more probable.
Overall however, the same phenomenology is observed for all parameter values: the reduction of the probability of a severe outbreak occurs for all strategies, and increases for smaller triggering thresholds and longer closure durations. The closure of the whole school always leads to the most important effect, but the grade closure has as well a strong impact on the probability of occurrence of large outbreaks.
Tables IV, V and VI give the final number of cases for the parameter sets (i), (ii) and (iii), computed for the realisations leading to an attack rate larger than 10% for the various mitigation strategies. Faster spread lead to smaller number of cases, while increasing p A has the opposite effect. Overall, all strategies lead to a reduction in the final number of cases. Interestingly, and similarly to the case shown in the main text, this reduction is very similar for the various closures (class, grade or whole school) if the closure triggering threshold is small enough: in cases of large outbreaks, closing only one class can be as effective as closing the whole school. As in the main text, we also note that large confidence intervals are observed, limiting the predictability of the final number of cases. Finally, as the duration of the closures is increased, the impact on the spread saturates at a value close to the sum of the latent and infectious periods, as in the main text.  I. Percentage of realizations leading to an attack rate higher than 10%, for the various mitigation strategies with various thresholds and closure durations. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". β = 6.9 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, pA = 1/3. TABLE II. Percentage of realizations leading to an attack rate higher than 10%, for the various mitigation strategies with various thresholds and closure durations. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". Here β = 1.4 · 10 −3 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 0.5 day, 1/γ = 1 day, pA = 1/3.  IV. Average final number of cases, computed for the realisations leading to an attack rate larger than 10% for the various mitigation strategies; the brackets provide the 5 th and 95 th percentiles. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". β = 6.9 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, pA = 1/3. TABLE V. Average final number of cases, computed for the realisations leading to an attack rate larger than 10%, for the various mitigation strategies; the brackets provide the 5 th and 95 th percentiles. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". Here β = 1.4 · 10 −3 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 0.5 day, 1/γ = 1 day, pA = 1/3. TABLE VI. Average final number of cases, computed for the realisations leading to an attack rate larger than 10%, for the various mitigation strategies; the brackets provide the 5 th and 95 th percentiles. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". Here β = 3.5 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 2 days, 1/γ = 4 days, pA = 1/2.

Figures 1 and 2 complement the results by
showing the temporal evolution of the median number of infectious individuals, for realisations leading to an attack rate larger than 10%. Figure 1 shows the effect of various closure durations, for a closure triggering threshold of three symptomatic individuals and for the targeted class and grade closure strategies. For larger β, µ and γ, the epidemic curves unfold on shorter time-scales than in the case shown in the main text, as expected. As a result, the various strategies do not change much the epidemic peak timing, and have a smaller influence on the global duration of the spread, especially for the fastest spread. Moreover, in both cases, very similar epidemic curves are obtained when the closure duration becomes larger than the sum of 1/µ and 1/γ, while shorter durations yield a smaller effect. The optimal closure duration is thus close to the sum of the latent and infectious periods. Figure 2 finally compares the epidemic curves for the targeted class closure, targeted grade closure, and whole school closure strategies, at fixed closure duration and closure-triggering threshold.

WEIBULL-DISTRIBUTED DURATIONS OF THE LATENT AND INFECTIOUS PERIODS
In the main text and in the results shown above, the durations of the latent and infectious periods are taken at random from Gaussian distributions of averages 1/µ and 1/γ. These distributions have a standard deviation equal to one tenth of their average, meaning that the effective durations are rather close to the average values.
In order to check that our results are not sensitive to the specific distribution of latent and infectious periods distributions, we present here results obtained with other distributions, namely Weibull distributions of average 1/µ for the latent period and 1/γ for the infectious period, respectively. We consider distributions of parameter shape k = 3 and k = 4, corresponding respectively to standard deviations equal approximately to 0.28 (for k = 4) and 0.36 (for k = 3) times the average (e.g., for k = 3 and an average of 2 days, the standard deviation is of 17.4 hours). These distributions are shown alongside the Gaussian distribution used in the main text in Figure 3  Tables VII-X and Figures 4-5 show that the percentages of runs leading to large outbreaks and the outbreak sizes depend slightly on the choice of the distribution of latent and infectious periods, but that the results remain qualitatively similar to the ones described in the main text. to Weibull distributions of shape parameter 4): β = 3.5 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 2 days, 1/γ = 4 days, pA = 1/3. TABLE X. Average final number of cases, computed for the realisations leading to an attack rate larger than 10%, for the various mitigation strategies; the brackets provide the 5 th and 95 th percentiles. The baseline case given by the simple isolation of symptomatic children is indicated as "No closure". Parameter values (durations distributed according to Weibull distributions of shape parameter 4): β = 3.5 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 2 days, 1/γ = 4 days, pA = 1/3. Only runs with an attack rate (AR) higher than 10% are taken into account. Parameter values: β = 3.5 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 2 days, 1/γ = 4 days, pA = 1/3. Latent and infectious period durations are distributed according to Weibull distributions of shape parameters 3 (top) and 4 (bottom).  (6 days). The no-closure scenario is provided for reference. Only realizations with an attack rate (AR) higher than 10% are taken into account. Parameter values: β = 3.5 · 10 −4 s −1 , βcom = 1.4 · 10 −9 s −1 , 1/µ = 2 days, 1/γ = 4 days, pA = 1/3. Latent and infectious period durations are distributed according to Weibull distributions of shape parameters 3 (left) and 4 (right).

COMPARISON OF TARGETED AND RANDOM CLASS AND GRADE CLOSURES
We consider here strategies based on random closure of classes: whenever the number of symptomatic infectious individuals detected in any class reaches a certain threshold, (iv) one random class, different from the one in which symptomatic individuals were detected, is closed ("random class closure" strategy) (v) this class and a randomly chosen one in a different grade are closed ("mixed class closure" strategy).
We use here β = 6.9 · 10 −4 s −1 , β com = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, p A = 1/3. Tables XI and XII compare the effect of the targeted class and grade closure with the partially random closure of one or two classes. Closing one class chosen at random (different than the one in which the infectious individuals are detected) leads only to a marginal decrease in the probability to obtain an attack rate higher than 10% and in the number of individuals affected by large spreads.
In the mixed strategy on the other hand, the class in which the infectious individuals have been detected is closed, and a second one, chosen at random in a different grade, is closed as well. This leads to an effect almost as strong as the targeted grade closure strategy. Figures 6 and 7 report the temporal evolution of the median number of infectious individuals in the targeted and random strategies, leading to the same conclusion: the targeted closure of a class leads to a much smaller peak than the closure of a random class; the mixed strategy leads on the other hand to an epidemic curve that is very close to the case of a targeted grade closure. Percentage of runs leading to an attack rate larger than 10% for the targeted and random closure strategies. β = 6.9 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, pA = 1/3. . β = 6.9 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, pA = 1/3. Only realizations with attack rate (AR) larger than 10% are taken into account. In the mixed closure strategy, the class in which the infectious individuals are detected is closed, as well as a second class from a different grade. β = 6.9 · 10 −4 s −1 , βcom = 2.8 · 10 −9 s −1 , 1/µ = 1 day, 1/γ = 2 days, pA = 1/3. Only realizations with attack rate (AR) larger than 10% are taken into account.