Within and between classroom transmission patterns of seasonal influenza among primary school students in Matsumoto city, Japan

28 Schools play a central role in the transmission of many respiratory infections. Heterogeneous social 29 contact patterns associated with the social structures of schools (i.e. classes/grades) are likely to 30 influence the within-school transmission dynamics, but data-driven evidence on fine-scale 31 transmission patterns between students has been limited. Using a mathematical model, we analysed a 32 large-scale dataset of seasonal influenza outbreaks in Matsumoto city, Japan to infer social 33 interactions within and between classes/grades from observed transmission patterns. While the 34 relative contribution of within-class and within-grade transmissions to the reproduction number varied 35 with the number of classes per grade, the overall within-school reproduction number, which 36 determines the initial growth of cases and the risk of sustained transmission, was only minimally 37 associated with class sizes and the number of classes per grade. This finding suggests that 38 interventions that change the size and number of classes, e.g. splitting classes and staggered 39 attendance, may have limited effect on the control of school outbreaks. We also found that 40 vaccination and mask-wearing of students were associated with reduced susceptibility (vaccination 41 and mask-wearing) and infectiousness (mask-wearing) and hand washing with increased 42 susceptibility. Our results show how analysis of fine-grained transmission patterns between students 43 can improve understanding of within-school disease dynamics and provide insights into the relative 44 impact of different approaches to outbreak control. 45

46 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint Influenza virus and other directly transmitted pathogens typically spread over social contact 47 networks involving frequent conversational or physical contacts (1–4). There is evidence that schools 48 are important social environments that can facilitate the transmission of influenza via close contacts 49 between students (5–9). Previous studies have collected contact data between students using 50 questionnaires and wearable sensor devices and found strong assortativity of contact rates within 51 classes and grades (10–14), which is likely relevant to the within-school transmission dynamics of 52 respiratory infections and the effectiveness of school-based interventions. However, such insights 53 from contact data also need to be validated with real-world outbreak data because contacts as 54 measured in those studies may not necessarily be fully representative of the types of contacts that lead 55 to transmission (e.g. with regards to proximity and duration). In this light, the differential transmission 56 rates of influenza associated with classes and grades have also been estimated from empirical 57 outbreak data in a few studies (6, 15, 16). However, those studies are limited to the analysis of only 58 one or two schools and included a relatively small number of cases (< 300). Therefore, robust findings 59 across schools with different structures that capture the full range of heterogeneity in within-school 60 transmission dynamics have remained a crucial knowledge gap. 61 Understanding how school population structures (e.g. class and school sizes) shape 62 transmission dynamics is key to making predictions about outbreak dynamics and interventions in 63 these settings. Modelling studies of school outbreaks often require a choice between the ‘density-64 dependent mixing’ and ‘frequency-dependent mixing’ assumptions (17). The density-dependent 65 mixing assumes that the transmission rate between a pair of students is constant regardless of the 66 class/school sizes, while the frequency-dependent mixing assumes an inverse proportionality between 67 them. As a result, the reproduction number is expected to increase with class/school size with the 68 density-dependent mixing assumption and remain stable with the frequency-dependent mixing 69 assumption. Whether the transmission is best characterised by the density-dependent mixing, 70 frequency-dependent mixing or any other alternative assumption may vary between different modes 71 of transmission and exposure settings (18–22). However, choices between the assumptions made by 72 existing studies of school outbreaks vary widely and are not based on a clear empirical concensus (9, 73 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint 23–26). These makes it challenging to interpret simulation studies evaluating school-based 74 interventions (e.g. reduced class sizes) because the estimated effect sizes can heavily rely on the 75 assumed mixing patterns (27–31). 76 To fill this knowledge gap in heterogeneous transmission dynamics at school, we applied a 77 mathematical model of influenza virus transmission to a large-scale dataset from the 2014-15 season 78 in Matsumoto city, Japan, which included diagnosed influenza reports among 10,923 primary school 79 students and their household members. The model accounted for within-school transmissions as well 80 as introductions to and from households and risk from the general community, which constitute key 81 social layers of transmission (32–34). Using this model, we estimated fine-scale heterogeneous 82 transmission patterns among students within and between classes and grades, as well as determinants 83 of transmission rates including school structures and precautionary measures. 84

85 We analysed citywide survey data of 10,923 primary school students (5–12 years old) in 86 Matsumoto city, Japan in 2014/15, which included 2,548 diagnosed influenza episodes among 87 students (Figure 1A). The dataset was obtained from 29 schools with a range of class structures (sizes 88 and the number of classes per grade), allowing for detailed analysis of within and between class 89 transmission patterns (Figure 1B). The attack ratio (i.e. the cumulative proportion diseased) in each 90 school (excluding three distinctively small schools with fewer than 15 students per class) showed 91 weak to null negative correlations with the mean class size and the mean number of classes per grade 92 (Figure 1C). The onset dates of students showed a temporal clustering pattern associated with school 93 structure (Figure 1D). When the students were partitioned into different levels of groupings (i.e. by 94 class, grade, school and overall), the deviation of onset dates from the within-group mean tended to be 95 smaller with finer groupings. 96 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint 97 Figure 1. Transmission dynamics of seasonal influenza in primary schools in Matsumoto city, Japan and estimated effects of 98 interventions for SARS-CoV-2. (A) Epidemic curve of seasonal influenza by illness onset in primary schools in Matsumoto 99 city, 2014/15. Colours represent different schools. Month names denote the 1st day of the month. (B) Scatterplot of the class 100 sizes and the number of classes per grade in the dataset. Each dot represents a class in the dataset. Dots are jittered along the 101 x-axis. Three schools had classes of fewer than 15 students (denoted by dotted horizontal line) and were excluded from the 102 model fitting. (C) The scatterplots of the school attack ratio (%) against the mean class size and the mean number of classes 103 per grade. The correlation indices (r) and the 95% confidence intervals are also shown. (D) Temporal clustering patterns of 104 students’ onset dates with different levels of groupings reproduced from the school transmission model. The distributions of 105 the deviance of each student’s onset from the group mean are displayed at overall, school, grade and class levels. The 106 standard deviation (SD) of each distribution is also shown. 107 108 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint The temporal clustering shown in Figure 1D supports the hypothesis that the transmission is 109 more likely within-class, followed by within-grade and within-school. We explored this further by 110 estimating reproduction numbers within school. Using a mathematical model that accounts for 111 different levels of interaction within and between classrooms and grades as well as introductions from 112 households and community, we estimated the within-school effective reproduction number RS of 113 seasonal influenza in primary schools along with the breakdown of transmission risks associated with 114 class/grade relationships (Figure 2). The relationship between any pair of students in the same school 115 was classified as either “classmates”, “grademates” (in the same grade but not classmates) or 116 “schoolmates” (not in the same grade). The estimated RS was broken down as a sum of the 117 contributions from these students, where the class size (n) and the number of classes per grade (m) 118 were assumed to affect the risk of transmission. The reconstructed overall RS in a 6-year primary 119 school was estimated to be around 0.7–0.9 and was not significantly associated with n or m (Figure 120 2A). Namely, an infected student was suggested to generate a similar number of secondary cases 121 irrespective of the class structure; although our estimates of RS were about 15% smaller for the class 122 size of 40 than 201, the posterior p-value did not suggest a statistical significance (p ~ 0.15 or above). 123 As RS was likely below 1 across class structures, school outbreaks may not have been sustained 124 without continuous introductions from households and community. Transmission to classmates 125 accounted for about two-thirds of RS when each grade has only one class and was partially replaced 126 by transmission to grademates as the number of classes per grade increases, while the sum of within-127 grade transmission (i.e. transmission to either classmates or grademates) remained stable (Figures 2B 128 and 2C). Around 20–30% of overall RS was explained by transmission to schoolmates throughout. We 129 also obtained qualitatively similar results throughout our sensitivity analysis (Figure S3). In a 6-year 130 school with 3 classes of 30 students, the risk of transmission was estimated to be 1.8% (95% credible 131 interval (CrI): 1.4–2.5) from a given infected classmate of the same sex, 1.6% (1.2–2.1) the opposite 132 sex, 0.13% (0.08–0.19) from a given infected grademate and 0.040% (0.029–0.055) from a given 133 1 For example, the estimated relative reduction was 17% (95% credible interval: -16%–40%) for m = 3. . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint infected schoolmate (Table S2). The cumulative risk of infection from the community was estimated 134 to be 2.2% (1.7–2.7) over the season. 135 136 Figure 2. The estimated within-school transmission patterns of seasonal influenza among primary school students in 137 Matsumoto city, Japan. (A) The overall school reproduction number (RS) under different class structures. Whiskers represent 138 the 95% credible intervals (B) The breakdown of RS corresponding to each type of within-school relationships. Whiskers 139 represent the 95% credible intervals. Bottom panels: stacked graph of RS based on the median estimates. 140 141 We incorporated a log-linear regression (35) into the above estimation of RS to account for 142 covariates that may affect the susceptibility or infectiousness of students. The results suggested that 143 vaccines were associated with reduced susceptibility while mask wearing was associated with both 144 reduced susceptibility and infectiousness (Table 1). Conversely, hand washing was associated with 145 increased susceptibility. Reduced chance of transmission during the winter break (27 December 146 2014–7 January 2015) was captured as a 75% estimated decline in the infectiousness of cases whose 147 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint onset dates were during the break. School grade, which serves as a proxy of students’ age, did not 148 show a significant association with either susceptibility (relative value 1.10; CrI: 0.88–1.38) or 149 infectiousness (relative value 0.78; CrI: 0.59–1.04). 150 Table 1. Covariates and effects estimated in the log-linear regression 151 Covariate Frequency in data Relative susceptibility Relative infectiousness School grade (1 year increase) — 1.10 (0.88–1.38) 0.78 (0.59–1.04) Vaccine 47.7% 0.88* (0.81–0.96) 0.97 (0.82–1.18) Mask wearing 51.4% 0.77* (0.71–0.85) 0.65* (0.55–0.78) Hand washing 80.1% 1.54* (1.36–1.76) 1.25 (0.96–1.65) Onset in winter break 5.9% (of cases) — 0.25* (0.15–0.39) Values are median estimates and 95% credible intervals. 152 * Estimates with 95% credible intervals not crossing 1. 153 154 We estimated the breakdown of the source of infection for student cases based on the 155 conditional probability predicted by the model and parameter estimates. The epidemic curve stratified 156 by the estimated source of infection suggested that within-school transmission accounted for the 157 majority of student cases while schools were open and that the within-household transmission was 158 responsible for most of the cases reported during the winter break and shortly after (Figure 3A). The 159 aggregated relative contribution suggested that 54.6% (CrI: 53.5–55.7), 38.7% (CrI: 37.9–39.5) and 160 6.7% (CrI: 5.9–7.5) of the student cases were acquired from school, household and community, 161 respectively (Figure 3B).We estimated the possible relative effects of interventions altering the school 162 population structure on the school reproduction number RS. We assumed that the estimated relative 163 contributions of class/grade relationship to the transmission risk reflect the contact patterns between 164 students which may also be relevant to the dynamics of another influenza outbreak at school (and 165 potentially those of directly-transmitted disease outbreaks in general) and that the responses to 166 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint interventions can be captured by the estimated relationship between RS and the changes in the 167 variables n and m according to each intervention (Table 2). Specifically, in the ‘split class’ scenario, 168 each class was assumed to be split in half and taught simultaneously in separate classrooms, while in 169 the ‘staggered attendance’ scenarios only half of the students attend school at the same time by 170 introducing two different time schedules, e.g. morning and evening classes. The estimated relative 171 effects of school-based interventions on RS in a hypothetical setting of 6-year school with 2 classes 172 per grade (40 students each) showed that splitting classes or staggered attendance alone was unlikely 173 to reduce RS (or may even be counteractive) (Figure 1D), which is consistent with the aforementioned 174 estimates of RS minimally associated with class sizes and the number of classes. By reducing 175 interactions between students from different classes (so-called ‘bubbling’ or ‘cohorting’) by 90%, RS 176 could be reduced by up to around 20%. Combining split classes/staggered attendance with reduced 177 interactions outside classes did not suggest incremental benefit in reducing RS. Given that these 178 interventions typically require additional resources including staff and classrooms, the overall benefit 179 to changing class structures for influenza control may be limited. 180 Table 2. Summary of interventions that changes the size/number of classes 181 Interventions Class size (n) # classes per grade (m) Assumption Baseline (‘no change’) 40 2 Students contact within and between classes and grades proportionally to the estimated transmission patterns in Figure 2. Split class 20 4 Each class is split into two and taught simultaneously in separate classrooms. Students may contact each other between classes. Staggered attendance (within class) 20 2 Each class is split into two and taught separately in two different time slots (e.g. morning and evening). Students in different time slots do not contact each other and thus RS is calculated for students in one slot. Staggered attendance (between class) 40 1 Each class is allocated (as a whole) to either of the two different time slots and taught separately. Students in different time slots do not contact each other and thus RS is calculated for students in one slot. 182 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint 183 Figure 3. Reconstruction of students’ source of infection. (A) Epidemic curve stratified by the reconstructed source of 184 infection. The conditional probability of infection from different sources was computed for each student and aggregated by 185 date of illness onset. (B) Breakdown of the reconstructed source of infection. For each student, the source of infection was 186 sampled based on the conditional probability to provide the proportion of students infected from each source. Bars denote 187 posterior median and whiskers 95% credible intervals. (C) Expected relative changes in the school reproduction number 188 under school-based interventions changing the structure of classes. Dots represent medians and whiskers 95% credible 189 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint intervals. Reduced outside-class transmissions (i.e. from grademates or schoolmates) were also considered (50% reduction: 190 blue; 90% reduction: green). 191 192

193 We used a mathematical model that stratified transmission within and between classes/grades 194 to understand the dynamics of influenza transmission among primary school students. The inferred 195 transmission dynamics of seasonal influenza in Matsumoto city, Japan, 2014-15 season suggested that 196 the within-school reproduction number RS stayed relatively constant regardless of the size or the 197 number of classes (suggesting ‘frequency-dependent mixing’ (36)), in contrast to common modelling 198 assumptions. The estimated RS of 0.8–0.9, more than half of which was attributable to within-class 199 transmissions, is consistent with a previous study in the United States (15). This value is also in line 200 with the reported R0 of 1.2–1.3 for seasonal influenza (37) because our previous study estimated that 201 the students in this dataset had infected 0.3–0.4 household members on average during this 2014-15 202 season (note that R0 corresponds to the overall number of secondary transmissions per student, 203 including at school and household) (18). The value of RS below 1 suggests that an outbreak cannot 204 sustain itself within a school alone and that interactions through importing and exporting infections 205 between households and the general community is likely to play a crucial role in the overall 206 transmission dynamics. We estimated that school, household and community accounted for 55%, 39% 207 and 7% of the source of infection for student cases, respectively. The attributable proportion was 208 lower for schools and higher for households than the previous study (15), which may be explained by 209 different scales of outbreaks in schools and households. In the Matsumoto city dataset, the overall 210 attack ratio at school was lower (19%), students had larger households (average size 5.5) and there 211 were more household cases than student cases (3996 vs 2548), as opposed to 35%, size of 3.4 and 141 212 vs 129 cases in (15). 213 The estimated breakdown of RS revealed a number of notable patterns. As the number of 214 classes per grade increased, the contribution of within-class transmission risk declined and was 215 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint replaced by within-grade transmission. Combined with the almost constant overall RS, this might 216 indicate that contact behaviour between students that contributed to transmission was only minimally 217 affected by the student population density. That is, students may have had a certain number of ‘close 218 friends’ with whom they had more intimate interactions that could facilitate transmission. In a school 219 with more classes per grade, some of such friendship may have come from grademates instead of 220 classmates, but the total number of close friends may have remained similar. This interpretation is in 221 line with published evidence of influenza spreading predominantly in close proximity (38) and is 222 likely to influence the expected effect of interventions not only for influenza but also other respiratory 223 infectious diseases including COVID-19, which share similar routes and range of transmission (39, 224 40). Further disease-specific studies could elucidate the generalisability of these associations in more 225 detail. 226 Our results suggested that interventions such as reducing class sizes or the number of students 227 present (staggered attendance) may not be effective in constrast to what would be expected under the 228 density-dependent mixing assumption (27–31). If interventions altering class structures are not 229 accompanied by additional precaution measures and students try to resume their ‘natural’ behaviours 230 (i.e. the same contact patterns as those in school with the resulting class structures) through so-called 231 social contact ‘rewiring’ (41), the effect of such interventions can diminish or even reverse. For 232 example, if other classes are absent due to staggered attendance, students may increase their 233 interactions with classmates instead of their previous close friends in other classes. Our results are 234 also consistent with a recent study of interventions against COVID-19 in US schools that did not find 235 a significant risk reduction associated with reducing class sizes (42). Given the additional logistical 236 resources required to implement these interventions, we propose that reducing the class sizes or the 237 number of attending students should be considered only if they enable effective implementation of 238 precaution measures such as physical distancing, environmental cleaning or forming social bubbles. 239 Using a log-linear regression analysis combined with a transmission model, we identified 240 several precautionary measures associated with the susceptibility or infectiousness of students. 241 Vaccines were associated with reduced susceptibility and masks with a reduction in both 242 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint susceptibility and infectiousness. Influenza vaccine effectiveness in the 2014-15 season was suggested 243 to be particularly low in Japan due to vaccine mismatch and estimated to be 26% (95% CrI: 7–41%) 244 for primary-school-age children (6–12 years old) (43). Our estimate of a relative susceptibility of 0.88 245 (CrI: 0.81–0.96) in vaccinated students, which translates into a vaccine effectiveness of 12% (CrI: 9–246 19%), is broadly consistent with this prior estimate. While existing evidence for the effectiveness of 247 mask policies for the control of respiratory infections is still limited (44, 45), our estimates of small 248 protective effects acting on the relative susceptibility (0.77; CrI: 0.71–0.85) and infectiousness (0.65; 249 CrI: 0.55–0.78) lie within a plausible range based on evidence available to date (44, 46–48). Increased 250 susceptibility associated with hand washing in our analysis, however, does not align with existing 251 findings (49, 50). Although the underlying cause for this is unclear, the original report on the 252 Matsumoto city dataset also reported a higher odds ratio (1.4; CrI: 1.27–1.64; unadjusted for 253 differential exposure) and attributed it to the possible congregation of students washing hands in 254 communal settings at school (51). 255 Several limitations of this study should be noted. First, the transmission patterns within 256 schools were estimated from a single dataset of seasonal influenza in primary schools (aged 5-12 257 years) in Matsumoto city, Japan, and it is unclear to what extent the results can be extrapolated to 258 other settings, e.g. secondary schools or schools in other countries. Some features of our results may 259 still be relevant to transmission dynamics in different types of schools if they reflect general social 260 contact behaviours of schoolchildren; however, the relative contribution of within-class/within-grade 261 interactions may become smaller for older students (52). The data points used in the inference mostly 262 consisted of classes of size 20-40 (those with a size smaller than 10 were excluded as they might be 263 operated differently) and most schools had no more than 5 classes per grade. The scope of the 264 estimated effect of the school-based interventions was also limited to within this range for internal 265 consistency and thus may not necessarily be applicable to class structures outside this range (e.g. 266 splitting a class of 20 students into two). Extrapolating the estimated transmission patterns to other 267 respiratory infectious diseases also warrants caution because their epidemiological characteristics may 268 not be identical, although we believe that such an approach may still be useful for diseases sharing 269 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint similar modes of transmission. Modelling studies using social contact data often assume 270 proportionality between contacts and the transmission of directly-transmitted diseases (e.g. measles, 271 influenza and COVID-19) and have many successful applications (7, 33, 53–57). Using the estimated 272 transmission patterns of influenza as a proxy for other diseases essentially rests on the same 273 assumption, which nonetheless has limitations and should eventually be validated by disease-specific 274 studies. Second, some aspect of the outbreaks may have been missing from the dataset. Since the 275 illness data of teachers were not available, they were not considered throughout the analysis. 276 However, their role in seasonal influenza transmission may have been minor given a large number of 277 student cases and the smaller risk in adults (58, 59). Although our student incidence data likely had 278 good case ascertainment given encouraged medical attendance and confirmation by rapid diagnostic 279 kits (18), a certain proportion of infections (e.g. asymptomatic or very mild) may have been missing. 280 We believe that students feeling unwell due to influenza mostly attended medical institutions and 281 received a test as it was encouraged by schools. Nonetheless, it should be noted that this could have 282 been a source of bias in the estimated transmission patterns. Students with very mild symptoms (e.g. 283 only slight sore throat) may visit a medical institution only if they know of other classmates also 284 diagnosed with influenza. If such cases were common, the contribution of within-class transmissions 285 in our results might have been an overestimate. Third, since the dataset was obtained from an 286 observational study, the identified determinants of transmission may not be causal and should not be 287 viewed as conclusive evidence. The results of our log-linear regression were mostly in line with 288 existing findings, however, our dataset may still be biased due to unmeasured confounders such as 289 health awareness. Our estimates of the relative effect of school-based interventions were based on the 290 assumption that students’ behaviours follow the fixed patterns according to the school structure even 291 under interventions. That is, when the class size or the number of classes were changed by an 292 intervention, students were assumed to change their behaviour according to the new school structure 293 (as if it were the original structure) by e.g. rewiring close contacts in a timely manner. This is a 294 hypothetical expectation that may not exactly be observed in actual interventional settings; for 295 example, it may take time for students to resume close contacts after the class is split, which can bring 296 RS lower than our prediction at least temporarily. We have also neglected the possible effect of the 297 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint interventions on the transmission outside the school. The actual effects of these interventions should 298 ideally be validated by empirical data, as in (42). 299 Our analysis disentangled the transmission dynamics of seasonal influenza among primary 300 school students and highlighted the relative importance of within-class and within-grade transmission. 301 Since class and school sizes were minimally associated with the within-school reproduction number, 302 school-based interventions that change classroom structures, e.g. reduced class sizes and staggered 303 attendance, may have limited effectiveness. Empirical evidence on fine-grained heterogeneous 304 transmission patterns at school as was obtained from this study would inform public health planning 305 for future outbreaks of influenza and, potentially, other directly transmitted infectious diseases that 306 thrive in schools. 307

308 Data 309 We analysed a citywide school-based influenza survey data from the 2014/15 season. The 310 survey was conducted in Matsumoto city (population size: 242,000 (60)), Japan, enrolling 13,217 311 students from all 29 public primary schools in the city. During the survey period (from October 2014 312 to February 2015), the participants were asked to fill out a questionnaire when they were back from 313 the suspension of attendance due to diagnosed influenza (prospective survey). In March, the 314 participants were asked to respond to another survey on their experience during the study period, 315 regardless of whether they had contracted influenza (retrospective survey). A total of 2,548 diagnosed 316 influenza episodes were reported in the prospective survey, which accounted for 96% of the cases 317 officially recognised by the schools during the study period. Primary schools in Japan often requested 318 students suspected of influenza to seek diagnosis at a medical institution. All students reporting an 319 influenza episode in the prospective survey answered that they had received a diagnosis and at least 320 95% of them were noticed of type A influenza (indicating that they were lab-confirmed). In the 321 retrospective survey, 11,390 (86%) participants responded, among which 8,375 reported that they did 322 not have influenza during the study period. 323 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint We combined those who responded to the prospective survey (“case group”) and those who 324 reported no influenza experience in the retrospective survey (“control group”) and obtained a dataset 325 of 10,923 students. Of those, 71 students from 3 schools with less than 15 students per grade were 326 excluded because they may have different schooling patterns from other schools (e.g. some students 327 in different grades shared classrooms). We used individual profiles (sex, school, grade, class, 328 household composition), onset dates, influenza episodes of household members and precaution 329 measures students engaged in (vaccine, mask, hand washing) in the subsequent analysis. Further 330 details of the dataset can be found in the original studies (51, 61). 331 The secondary data analysis conducted in the present study was approved by the ethics 332 committee at the London School of Hygiene & Tropical Medicine (reference number: 14599). 333 Inference model 334 We modelled within-school transmission considering class structures as follows. We defined 335 the “school proximity” d between a pair of students i and j attending the same school as 336 𝑑 = # 1 (different grades, same school) 2 (different classes, same grade) 3 (different sex, same class) 4 (same sex, same class) (1) To investigate the potential effect of reduced class sizes and the number of attending students, we 337 modelled the transmission between students as a function of two variables: the class size n and the 338 number of classes per grade m (i.e. the number of students per grade is nm). Namely, we assumed that 339 in the absence of any individual covariate effects, the cumulative transmission rate between student i 340 and j in proximity d over the infectious period is represented as 341 𝛽!" = 𝛽# =𝑛!,# ? %&! =𝑚!,#? %'! , (2) where 𝛽#, 𝛾#, 𝛿# are parameters to be estimated. When i and j are in the same grade (i.e. d = 2, 3, 4), 342 the average class size and the number of classes in that grade were used as 𝑛!,# and 𝑚!,#. When d = 1, 343 the school average was used as 𝑛!,# and 𝑚!,#. The exponent parameters within the same class were 344 assumed to be equal: 𝛾( = 𝛾) and 𝛿( = 𝛿). 345 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint We modelled the daily hazard of incidence for student i as a renewal process. Let ℎ* be the 346 onset-based transmission hazard as a function of serial interval s (normalised such that ∑ ℎ+ , +-. = 1; 347 ℎ+ = 0 for s ≤ 0). We used a gamma distribution of a mean of 1.7 and a standard deviation of 1.0 for 348 influenza, which resulted in a mean serial interval of 2.2 days (62). The daily hazard of disease onset 349 attributed to school transmission is given as 350 𝜆! /(𝑇) = 𝑣! I 𝑤"𝛽!" ℎ0%0" " , (3) where vi and wi represent the relative susceptibility and infectiousness, respectively, which are 351 specified for each individual by a log-linear regression model to account for covariates (see 352 Supplementary materials for detailed methods). 353 In addition to the above within-school transmission, we also considered within-household 354 transmission and general community transmission. Within-household transmission was incorporated 355 as the Longini-Koopman model (63) with parameters from a previous study on the same cohort of 356 students (18). General community transmission was modelled as a logistic curve fitted to the total 357 incidence in the dataset to reflect the overall trend of the epidemic. See Supplementary materials for 358 further details of the model. 359 We constructed the likelihood function and estimated the parameters by the Markov-chain 360 Monte Carlo (adaptive mixture Metropolis) method. We obtained 1,000 thinned samples from 361 100,000 iterations after 100,000 iterations of burn-in, which yielded the effective sample size of at 362 least 300 for each parameter. Using the posterior samples, we computed the proximity-specific 363 reproduction number Rd in a hypothetical 6-year school with given n and m (assumed to be constant 364 schoolwide) as 365 𝑅# = ⎩ ⎨ ⎧ 5𝑛𝑚 ⋅ 𝛽.𝑛%&# 𝑚%'# (𝑑 = 1) 𝑛(𝑚 − 1) ⋅ 𝛽1𝑛%&$ 𝑚%'$ (𝑑 = 2) 𝑛 ⋅ 𝛽( + 𝛽) 2 𝑛%&% 𝑚%'% (𝑑 = 3, 4) (4) and defined the within-school reproduction number RS as a sum of them. 366 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint We predicted the relative reduction in RS under intervention measures changing the number of 367 attending students and class structures by using posterior samples. Interventions were assumed to 368 change n and m as shown in Table 1, and the predictive distribution of the relative change in RS was 369 computed for each intervention. The estimated RS represents the value in a hypothetical condition 370 where an infectious student spends the whole infectious period at school; the effect of absence due to 371 symptoms or the staggered attendance was not included in this reduction. 372 All analysis was performed in Julia 1.5.2 and R 4.1.0. Replication code is available on 373 GitHub (https://github.com/akira-endo/schooldynamics_FluMatsumoto14-15). 374

375 This research was partially funded by Lnest Grant Taisho Pharmaceutical Award. AE was financially 376 supported by The Nakajima Foundation and The Alan Turing Institute. Yang Liu is supported by Bill 377 & Melinda Gates Foundation [INV-003174], National Institute for Health Research [16/137/109], 378 European Commission [101003688] and UK Medical Research Council [MC_PC_19065]. KEA is 379 supported by European Research Council Starting Grant [757688]. AJK [206250] and SF [210758] 380 are supported by the Wellcome Trust. 381 Conflict of interest 382 AE received a research grant from Taisho Pharmaceutical Co., Ltd. 383 Data and code availability 384 Due to potentially sensitive information included, the original dataset is not made public and is 385 available from the corresponding author upon reasonable request. A processed dataset with an 386 increased level of anonymity, which can still qualitatively reproduce the main study finding (i.e. 387 breakdown of the school reproduction number breakdown by the class/grade relationship without 388 . CC-BY 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.08.21259917doi: medRxiv preprint adjustment for covariates) is publicly available along with the accompanying code on a GitHub 389 repository (https://github.com/akira-endo/schooldynamics_FluMatsumoto14-15). 390 Prior publication 391 Earlier version of this manuscript is available at Research Square [https://doi.org/10.21203/rs.3.rs-392 322366/v1]. 393

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