The costs of keeping schools open during the COVID-19 pandemic

There is a trade-off between restrictions on the education sector and other economic sectors in the control of SARS-CoV-2 transmission. Here we integrate a dynamic model of SARS-CoV-2 transmission with a 63-sector economic model reecting sectoral heterogeneity in transmission and economic interdependence between sectors. We identify control strategies which optimize economic production while keeping schools and universities operational, and constraining infections such that emergency hospital capacity is not exceeded. We estimate an economic gain of between £163bn (24%) and £205bn (31%) for the United Kingdom compared to a blanket lockdown of non-essential activities over six months, depending on hospital capacity. Sectors identied as priorities for closures are contact-intensive, produce few crucial inputs for other sectors and/or are less economically productive. Partial closures over some months are required for retail trade, hospitality, accommodation, creative activities, arts, entertainment, and personal services including hairdressing and beauty treatments under most scenarios. contact in separately for and Contacts on by by sector-specic proportions of homeworkers. in community contacts in the whilst education, transport and hospitality-related contact proportional to the of sector and are also age-dependent. basic reproductive number R 0 ; effectiveness of UK’s rst δ LD ; Transmissibility from the tted basic reproductive number R and pre-lockdown contact patterns using the next-generation eigenvalue method. 27 Worker-to-worker contact rates are derived survey conducted 2012 France, 28 and other parameters are obtained from separate epidemiological model tted to UK see S2. is to capture post-lockdown dampening impact NPIs individuals’ behavior on transmissions as R , capturing combined effect NPIs (other closures) are dicult to estimate empirically, including social distancing in social and work testing-and-tracing, shielding of travel restrictions, and limits to social gatherings. tted value adjust less


Introduction
Closures of schools, universities, and workplaces are a key non-pharmaceutical intervention (NPI) in the control of the COVID-19 pandemic, 1,2 and were implemented by many countries in the rst half of 2020 when infections were rising rapidly. By mid-2020, UNESCO 3 estimated that around 60% of the world's students had their education disrupted by national closures of educational institutions during the pandemic. Even short periods of missed education can have grave consequences for educational development, 4-6 reduce lifetime earning potential, 7 and damage social and psychological development of children and young adults. 8 School closures are also associated with lost income and productivity of carers who cannot work because of childcare responsibilities. 9 The high economic and social costs of school and university closures have led most countries to re-start education activities in the second half of 2020, 10 even if this increases SARS-CoV-2 transmissions and puts heavy pressure on health services. 11 The New York Times commented that France and many other European nations are managing to keep infections under control while keeping schools open. 12 The Prime Minister of the United Kingdom stated in December that children must be kept in schools, even if that puts pressure on the hospitality sector and other parts of the economy. 13 To maintain control, European countries have to tighten NPIs in other areas of society, most notably through closure of businesses deemed non-essential for day-to-day life. Lockdowns of non-essential businesses are also associated with high economic and social costs, [14][15][16][17] and they are a crude lever if implemented as a blanket policy across the whole economy. Economic sectors differ greatly in the infection risk that they pose to both workers and consumers, in their potential to implement effective social distancing measures, and in the contribution they make to Gross Domestic Product (GDP). It is vital to determine how lockdowns can be netuned to prevent health services from being overwhelmed, whilst allowing educational institutions to stay open and minimizing economic costs associated with business closures.
Conventional GDP measures do not fully capture the bene ts of health and education services. 18 We therefore present here a closure strategy which identi es an optimal six-month trajectory of selective opening and closing of 63 sectors while keeping educational institutions operational and infections under control.
We maximize GDP while limiting the spread of SARS-CoV-2 over six months, with periodic decisions on economic con gurations -the extent to which each sector is open. We have hardly any empirical evidence on how to optimally design lockdown policies during pandemics. Our ndings are generated by a modelling study that necessarily rests on many assumptions, but they provide urgently needed guidance to policy makers on how to design policies that balance key societal objectives.

Methods
In our model we assume that relatively contact-light sectors which employ fewer workers carry fewer infections back into the community when they are open compared to more contact-intensive sectors with more workers (see Supplement for a detailed description of the model). Partial or full opening and closing of sectors are assumed to give rise to proportionate changes in the sector's active workforce, community transmission, and the associated impact on disease transmission. We then use the model to identify the set of sector closures over time that maximizes GDP, whilst keeping the education sector operational and containing daily hospital occupancy of COVID-19 patients within the maximum spare emergency hospital capacity (H max ). We consider three different assumptions for H max (either 12,000, 18,000, or 24,000 beds available, noting that at the peak in April 2020 18,000 COVID-19 patients occupied beds). We further constrain the effective reproductive number R t to be at or below 1 at the end of six months, ensuring that residual infections do not cause a rapid epidemic just beyond the intervention period.
Our strategy does not necessarily prioritize the sectors of the economy that contribute most to GDP relative to the spread of infection. Instead, it respects interdependencies between sectors, all of which rely, to some extent, on inputs from other sectors to produce their nal and intermediate outputs; a sector that is nominally opened may not be able to function properly if its supply chain is interrupted. 19,20 We obtained data on interdependencies between all 63 sectors from the most recent United Kingdom (UK) Input-Output (IO) table for 2016, 21 see table S1 for education. We use recent data on the workforce, 22 and on those working-from-home. 23 We specify a lower bound to production, given by the economic con guration that allows essential services to operate. Speci cally, economic activity is sustained throughout to at least 80% (healthcare 100%) of those observed during the UK's rst lockdown in March-May 2020, to allow for uncertainty regarding the exact values and changes in production processes. 24,25 The upper bound is given by the level of pre-pandemic production and assumes that the demand for goods and services does not exceed pre-pandemic levels. The chosen closure of sectors under optimal solutions is in uenced by at least ve sector characteristics and their interplay: the size of workforce, workplace contact rates, effect on community contact rates, GDP contribution and interdependence with other sectors.
We use a deterministic Susceptible-Exposed-Infectious-Removed (SEIR) model of SARS-CoV-2 transmission to project the spread of infection in the workplace, the education sector, households and the community as sectors are opened and closed to varying degrees. The SEIR model accommodates sectoral heterogeneity in risks of infection via three contact matrices: worker-to-worker, community, and consumer-to-worker. Community contacts occur with respect to four age groups: 0-4, 5-19, 20-64 and 65+. This includes contacts made in and between households, outside, in education institutions, on public transport, and in hospitality venues. All working-age adults, whether actively working or not, are subject to contacts in the community. Community contact in education is estimated separately for age groups 0-4 and 5-19. Contacts on public transport are estimated by sector and are reduced by sector-speci c proportions of homeworkers. We incorporate age-heterogeneity in community contacts occurring in the hospitality sector, whilst education, transport and hospitality-related contact rates are proportional to the extent of sector opening and are also age-dependent. We calibrate the model via a least-squares t by comparison with English hospital occupancy data from 20 th March to 30 th June 2020 by varying four parameters: 26 the basic reproductive number R 0 ; effectiveness of the UK's rst lockdown captured by a parameter δ LD ; epidemic start time and lockdown onset. Transmissibility is calculated from the tted basic reproductive number R 0 and pre-lockdown contact patterns using the next-generation eigenvalue method. 27 Worker-to-worker contact rates are derived from a survey conducted in 2012 in France, 28 and all other parameters are obtained from a separate epidemiological model tted to UK data, 29 see table S2. A scalar multiplier δ is used to capture the post-lockdown dampening impact of NPIs and individuals' behavior on transmissions as represented by R t , capturing the combined effect of NPIs (other than closures) that are di cult to estimate empirically, including social distancing in social and work environments, facemasks, testing-and-tracing, shielding of the vulnerable, travel restrictions, and limits to social gatherings. The tted value of δ LD over the rst lockdown represents the lower bound (optimistic) estimate of δ. For forward projections, we adjust δ to re ect less stringent NPIs and weaker adherence.

Modelling Scenarios
We maximize GDP while containing the spread of SARS-CoV-2 over six months, with bimonthly decisions on economic con gurations from 1 st September 2020. We incorporate ve types of constraints: (a) interdependencies of supply and demand between sectors; (b) in each sector, economic activity is sustained throughout to at least 80% of lockdown values (healthcare 100%); (c) the demand for goods and services does not exceed pre-pandemic levels; (d) hospital occupancy remains within capacity H max throughout the intervention period; (e) at the end of the intervention period.
We then calculate GDP, total disease incidence, and hospital occupancy for ve scenarios: Outcomes from scenarios A and B provide the schedule of sector closures that maximizes GDP, subject to the respective constraints, whilst LDA, LDB and FO provide benchmark scenarios. We use the 'Global search' with derivative-based base algorithm 'fmincon' in MATLAB's 'global optimization' toolbox for the optimization.

Results
The strategy that maximizes GDP while keeping infections within constraints (Scenario A) allows the potential closure of all economic sectors, including the education sector. If emergency hospital capacity for COVID-19 patients is constrained at H max =18,000, the optimal solution lets infections increase in September and October, then from November imposes more stringent economic con gurations to remain within the epidemiological constraints ( Figure 1A, gures S1A H max =12,000, S2A H max =24,000). This strategy of GDP maximization results solely in the closure of the education sector ( Figures 1B, S1B impact on short-term GDP as measured in national accounts. Our analysis considers economic production over six months, and not any longer-term economic bene ts of keeping schools and universities open which are likely substantial. The GDP achieved by Scenario A is £877bn over six months (H max =18,000, gure 3A), higher than the £660bn of a blanket lockdown (Scenario LDA), but lower than the £889bn achieved with a fully open economy (Scenario FO). However, Scenario FO results in high incidence and deaths. Scenario FO also means that around 68,000 COVID-19 patients would require hospital treatment at the projected peak in January 2021, compared to 18,000 patients under Scenario A.
Optimizing GDP while keeping education activities operational Scenario LDB requires all sectors to close at levels observed during the rst lockdown except the education sector, which would operate at 80% of prepandemic activity. LDB would keep maximum hospital occupancy at around 10,000 at the peak ( Figure 3B), ~56% of peak occupancy albeit not as low as LDA that allows the education sector to close as other sectors ( Figure 3A). This is because expanding the activity of the education sector increases transmission. The economic output achieved following a strategy that optimizes GDP while the education sector is operational is £863bn over six months (H max =18,000, Figure 3B,  Table S6). Sector closures can be less stringent if decision makers are prepared to let the level of infections (and hospitalizations and deaths) increase, and if they invest in an expansion of emergency hospital capacity. The gain in GDP for Scenario B when hospital capacity is increased from 12,000 to 18,000 is £30bn over six months, and £12bn for an increase from 18,000 to 24,000 ( Figure 3B, Table S5). This gain occurs because the increase in hospital capacity by 6,000 beds allows for a more open economy ( Figures S3B, 2B, S4B). Over the rst lockdown, the UK managed to increase capacity to 18,000 beds by cancelling many elective surgeries, using private hospital capacity, deploying retired medical and nursing staff, constructing eld hospitals and re-organizing care. there would be no GVA loss for accommodation & food and retail, and losses of less than £2bn each for the other sectors. If we allow hospitalizations and infections to increase there will be more deaths, which of course are of central importance to decision makers but which we do not calculate as part of this study. Instead, they are implicit in the level of hospital capacity chosen by decision-makers.

Sensitivity analyses
The projections from all scenarios rest on the assumption that other NPIs are relatively stringent, i.e. that interventions such as wearing facemasks, social distancing, reduced mixing of households, self-isolation, and others are implemented and well adhered to. The tted value of δ is 0.54, which re ects the reduction in R t achieved by NPIs over the rst lockdown. For all scenarios, we allow for about 11% less stringency in NPIs in the period after May 2020 by setting δ=0.6. We nd that small increases in δ ranging from 0.61 to 0.64 for Scenario B (H max =18,000) result in much stricter closures required to keep within the epidemiological constraints (Tabs. S4 versus S7), and a substantial associated GVA loss ( Figure 5, The projections are also sensitive to assumptions on contact rates in the community and the education sector ( Figure S5). We varied contact rates by 5% standard deviation around their sector-speci c means assuming contact rates are independently and normally distributed. We nd that much of the uncertainty in projected hospital occupancy arises from contact rates in the community and education sectors, rather than the other economic sectors.
At baseline, we assume that the proportion of workers homeworking stays constant at sector-speci c values observed over the rst lockdown. If we decrease these proportions to 20% fewer workers homeworking ( Figure S6), hospital occupancy increases from a maximum peak occupancy of 18,000 to over 25,000 (Scenario B) and 28,000 (Scenario A).
Children may be less susceptible to infection than adults [30][31][32] though evidence is con icting. 33,34 We evaluated the outcomes for scenario B (H max =18,000) if children under the age of 16 have a 50% lower susceptibility to infection than adults. After re tting the model, we nd that schools make a somewhat lower contribution to transmission dynamics (Fig S7). Some economic sectors therefore need to close more strictly at a greater loss to GVA compared to the assumption of equal susceptibility, although the sectors recommended for closure are the same.
Lastly, we evaluated outcomes when changes to the economic con guration are allowed every month instead of two months ( Figure S8). The reduction in GVA loss is modest (£487 million, table S5), and may not justify the upheaval associated with more frequent changes in policy.

Conclusion
We examine the extent to which economic activity can be sustained whilst educational institutions are kept open. If a differentiated sectoral closure strategy is followed, whereby certain economic sectors are partially closed over a six-month period, a GDP gain of between £163bn (24%) and £205bn (31%) over six months can be secured (depending on spare hospital capacity) compared to a blanket lockdown of all non-essential services. Differentiated sectoral closures that keep hospital occupancy at a set maximum (between 12,000 and 24,000) throughout the period are compared with a fully open economy that is projected to cause about 68,000 COVID-19 patients requiring hospital care at its peak. Activities that require partial closure in various months over autumn and winter 2020/21 are accommodation & food services including restaurants and bars, retail, creative and arts, entertainment, sports, amusement, recreation, and activities of membership organizations. To achieve the same outcomes, sectoral closures need to be much stricter if adherence to other NPIs such as social distancing is weak. Decision makers can reoptimize for a new intervention period before the end of six months if objectives change or new data become available.
A few studies focus on the trade-off between economic and public health impact of COVID-19 and compare optimal control strategies by economic sector. 19,20,[35][36][37][38] Most studies simplify the economy by allocating sectors into two categories, with the exception of two studies 19,20 which model interdependencies. All past studies specify simpli ed epidemiological models that do not consider latent period, asymptomatic infection, and differences in severity which are important for realistic projections. All but two models 35 Our analysis has important limitations. We use contact data classi ed by economic sector of employment. 28 Although the survey is likely to be representative of many high-income countries, such data would ideally be tailored directly to the country and the detailed sectors under scrutiny. Additionally, the sector in which the survey respondent is employed is reported only at a high level and does not give further information on what type of work the individual performs.
There are large variations in physical proximity by occupation type. 39 The most recent available IO table is from 2016 and does not re ect recent changes in the economy. In line with usual IO methodology, we assume Leontief production functions with constant returns to scale. 40 With heterogeneous sectors it will always be the case that partial opening of a sector may be able to focus on subsectors that are highly productive or have low reliance on inputs from other sectors that remain closed. However, policymakers may nd it di cult to formulate granular opening/closure policies focusing on economic activities within sectors and instead be forced towards blanket sectoral policies. A further concern is that -with constraints on supplies -some producers may change to alternative suppliers. However, relatively xed production processes mean that there is likely to be limited scope for changing the sector in which the supplies are produced. Producers may instead seek to solve supply chain problems by importing inputs for which there are domestic shortages. We have built in certain exibility in production processes by allowing some tolerance in maximum and minimum levels of economic activity.
We make no allowance for changes in prices or demand for nal products. To some extent, if demand changes for a sector's produce are expected and can be quanti ed, this can be incorporated by imposing an exogenous change to the relevant economic constraint. We do not model the effect of vaccinations, because they are unlikely to signi cantly impact transmission of SARS-CoV-2 within our intervention horizon. We also do not model the impact of the new SARS-CoV-2 lineage B.1.1.7 on transmissibility as evidence is only emerging now. 29 Vaccinations and new variants will be considered in future applications of DAEDALUS. Lastly and importantly, we have made only rudimentary efforts to adjust for NPIs put in place to reduce transmission risk and voluntary behavior change. The major challenge is identifying the likely nature and magnitude of changes in demand, NPIs and behavior in the absence of available data, although estimates may be forthcoming as evidence from countries' experiences becomes available.
Our study attempts to minimize the deleterious effects to the broader economy of protecting education and health services, a policy trade-off that is being considered worldwide. While the world is waiting for effective vaccines deployed at scale, the policy challenge will be how to  Figure 1 Optimal economic con guration under scenario A (GDP maximization), hospital capacity 18,000 beds, January 2020 to February 2021. (A) Projected incidence and hospital occupancy (B) Economic con guration across 63 sectors. Scenario A maximizes GDP via successive bi-monthly opening and closing of 63 sectors over a six-months intervention period, subject to epidemiological and economic constraints; any economic sector including education may close to 80% of observed minimum levels during the UK's rst lockdown (March-May 2020), but not lower, in order to sustain essential services; (A) shows projected daily infection incidence and daily hospital occupancy from January 2020 to February 2021; emergency hospital capacity for the treatment of COVID-19 patients is constrained at 18,000 beds (2nd grey line from below); (B) illustrates the optimal economic con gurations (extent of bi-monthly sector closures) under Scenario A GDP maximization; sector divisions are listed on the vertical axis (see table S4 for sector descriptions), and months on the horizontal axis; PRE is the pre-pandemic period, LD is the rst lockdown March-May 2020 in the UK, based on available data for closures of higher-level sector categories; period 1 is September-October 2020, period 2 is November-December 2020, period 3 is January-February 2021; openings vary between fully open as prepandemic (yellow, 1) to closed (blue, 0); scenario recommends partial closure of education sector (See table S3). Figure 2 personal service activities'. Hmax=24,000: light blue sector from 'Government, health & education' from left to right is 56 'Education' (loss of £9.7bn); dark red sectors from 'Other services' from left to right are 59 'Creative, arts and entertainment activities; libraries, archives, museums and other cultural activities; gambling and betting activities', 60 'Sports activities and amusement and recreation activities', 61 'Activities of membership organisations' 63 'Other personal service activities'. δ=0.64 (relatively weak NPIs): Light grey sector is 'Construction'; dark blue sectors from 'Distribution, transport, hotels and restaurants' are 28 'Wholesale and retail trade and repair of motor vehicles and motorcycles', 29 'Wholesale trade, except of motor vehicles and motorcycles' (high loss of £12bn), 30 'Retail trade, except of motor vehicles and motorcycles' (high GVA loss of nearly £21bn), 31 'Land transport and transport via pipelines', and 36 'Accommodation and food services' (high GVA loss of over £26bn); orange sector from 'Information and communication' is 40 'Computer programming, consultancy and related activities; information service activities; yellow sector from 'Financial and insurance' is 41 'Financial service activities, except insurance and pension funding'; purple sector is 44 'Real estate activities'; dark green sectors are 46 'Legal and accounting activities; activities of head o ces; management consultancy activities', and 54 'Security and investigation activities; services to buildings and landscape activities; o ce administrative, o ce support and other business support activities'; light blue sectors from 'Government, health & education' are 55 'Public administration and defence; compulsory social security', 56 'Education', and 57 'Human health activities' and 58 'Social work activites'; dark red sectors from 'Other services' are 59 'Creative, arts and entertainment activities; libraries, archives, museums and other cultural activities; gambling and betting activities', and 63 'Other personal service activities'.

Supplementary Files
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