key: cord-0709405-mohnt9zc
authors: Best, A.; Singh, P.; Ward, C.; Vitale, C.; Oliver, M.; Idris, L.; Poulston, A.
title: The impact of varying class sizes on epidemic spread in a university population
date: 2021-03-01
journal: nan
DOI: 10.1101/2021.02.26.21252501
sha: 23f2592308e7eb9ddcaf3305b594e93684d48a40
doc_id: 709405
cord_uid: mohnt9zc

A common non-pharmaceutical intervention (NPI) during the Covid-19 pandemic has been group size limits. Further, educational settings of schools and universities have either fully closed or reduced their class sizes. As countries begin to reopen classrooms, a key question will be how large classes can be while still preventing local outbreaks of disease. Here we develop and analyse a simple, stochastic epidemiological model where individuals (considered as students) live in fixed households and are assigned to a fixed class for daily lessons. We compare key measures of the epidemic - the peak infected, the total infected by day 180 and the calculated $R_0$ - as the size of class is varied. We find that class sizes of 10 could largely restrict outbreaks and often had overlapping inter-quartile ranges with our most cautious case of classes of 5. However, class sizes of 30 or more often result in large epdiemics. Reducing the class size from 40 to 10 can reduce $R_0$ by as much as 30%, as well as signficantly reducing the numbers infected. Intermediate class sizes show considerable variation, with the total infected varying as much as from 20% to 80% for the same class size. We show that additional in-class NPIs can limit the epidemic still further, but that reducing class sizes appears to have a larger effect on the epidemic. We do not specifically tailor our model for Covid-19, but our results stress the importance of small class sizes for preventing large outbreaks of infectious disease.

https://github.com/abestshef/classsizeSEIR). The underlying epidemiological 66 model is an SEIR (Susceptible-Exposed-Infected-Recovered) framework where, 67 within a setting ('home' or 'class'), the dynamics would be given by the 68 following ordinary differential equations,

where β is the transmission coefficient (with βI the 'force of infection'), ω is 75 the rate of progression from exposed to fully infected and γ is the recovery 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 (which was not certified by peer review)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 March 1, 2021. ; https://doi.org/10.1101/2021.02.26.21252501 doi: medRxiv preprint explore the impact on the epidemic. We also compare results where 96 transmission is high (β = 0.5γ) and low (β = 0.2γ). While these values would 97 appear to produce very high values of the basic reproductive ratio, R 0 , in the 98 mean-field model (β = 0.5γ, N = 1000 =⇒ R 0 = 500), it is well known that 99 the actual R 0 is considerably lower in invididual-based models, especially when CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 1, 2021. ; https://doi.org/10.1101/2021.02.26.21252501 doi: medRxiv preprint Figure 2 : Measures of epidemic severity for average houseshold size of 10 for different average class sizes from 100 simulation runs at each class size. Boxplots of the peak (a,d,g) and total infected by day 180 (b,e,h) and the calculated R 0 (c,f,i), where infection is low, β = 0.2γ (a,b,c), high, β = 0.5γ (d,e,f) and high at home but low in class (g,h,i). The orange lines denote the median, the boxes the 25th and 75th centiles, the whiskers to 1.5 the interquartile range and circles any outliers. The dashed lines mark the inter-quartile range (IQR) for the class of 5, and colouring of the boxes whether the IQRs of each class size do (blue) or do not (yellow) overlap the class of 5's IQR.

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. reduced from 33% to 10%, the median total from 99% to 64% and the median 186 R 0 from 3.66 to 2.62. The class size of 10 has an overlapping IQR with the 187 class of 5 for the peak infected, but no classes overlap for total infected. or smaller the median R 0 is less than 1, and even for class sizes of 50 the peak 194 is lower than 10% in every simulation. All class sizes' peak IQRs overlap with 195 9 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 1, 2021. ; https://doi.org/10.1101/2021.02.26.21252501 doi: medRxiv preprint Figure 4 : Measures of epidemic severity for average houseshold size of 5 for different average class sizes from 100 simulation runs at each class size. Boxplots of (left) the peak and (middle) total infected by day 180 and (right) the calculated R 0 , where infection is (top) low, β = 0.2γ, (middle) high, β = 0.5γ and (bottom) high at home but low in class. The orange lines denote the median, the boxes the 25th and 75th centiles, the whiskers to 1.5 the interquartile range and circles any outliers. The dashed lines mark the inter-quartile range (IQR) for the class of 5, and colouring whether the IQRs of each class size do (blue) or do not (yellow) overlap the class of 5's IQR.

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. of 10 with NPIs, we see the median peak reduced from 25% to 4%, the median 217 total infected from 96% to 14% and median R 0 from 2.42 to 1.56. Compared 218 to the class of 5, class sizes of 25 and below have overlapping IQRs for the 219 peak, but only class sizes of 10 and 15 for the total infected. We also see 

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

We additionally consider what happens when we alter the time in class to be considerably, in some cases reducing the median total infected from 96% to 304 just 14% in our model. to infection or simply imposed -will reduce the degree of mixing and effective 325 population size, such an approach would clearly be expected to further limit 326 the epidemic. Finally, we have assumed a closed population with no mixing 327 outside of households or classes and full adherence by the population. We 328 should account for further external contacts, which we would expect to 329 15 . CC-BY 4.0 International license It 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 March 1, 2021. ; https://doi.org/10.1101/2021.02.26.21252501 doi: medRxiv preprint change the quantitative values found here, we would expect the fundamental 331 findings -that smaller class sizes lead to smaller epidemics with less variation 332 and that the patterns will vary according to the target measure -will remain. 333

Population biology of infectious diseases

Population biology of infectious diseases: Part 340 I

Gavenčiak distancing, face masks, and eye protection to prevent person-to-person

Many thanks to Alexander Fletcher for input while developing the stochastic 335 simulation code.