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Dynamic modelling of costs and health consequences of school closure during an influenza pandemic

  • Yiting Xue1, 2Email author,
  • Ivar Sønbø Kristiansen3 and
  • Birgitte Freiesleben de Blasio1, 2
BMC Public Health201212:962

DOI: 10.1186/1471-2458-12-962

Received: 22 June 2012

Accepted: 16 October 2012

Published: 9 November 2012

Abstract

Background

The purpose of this article is to evaluate the cost-effectiveness of school closure during a potential influenza pandemic and to examine the trade-off between costs and health benefits for school closure involving different target groups and different closure durations.

Methods

We developed two models: a dynamic disease model capturing the spread of influenza and an economic model capturing the costs and benefits of school closure. Decisions were based on quality-adjusted life years gained using incremental cost-effectiveness ratios. The disease model is an age-structured SEIR compartmental model based on the population of Oslo. We studied the costs and benefits of school closure by varying the age targets (kindergarten, primary school, secondary school) and closure durations (1–10 weeks), given pandemics with basic reproductive number of 1.5, 2.0 or 2.5.

Results

The cost-effectiveness of school closure varies depending on the target group, duration and whether indirect costs are considered. Using a case fatality rate (CFR) of 0.1-0.2% and with current cost-effectiveness threshold for Norway, closing secondary school is the only cost-effective strategy, when indirect costs are included. The most cost-effective strategies would be closing secondary schools for 8 weeks if R 0 =1.5, 6 weeks if R 0 =2.0, and 4 weeks if R 0 = 2.5. For severe pandemics with case fatality rates of 1-2%, similar to the Spanish flu, or when indirect costs are disregarded, the optimal strategy is closing kindergarten, primary and secondary school for extended periods of time. For a pandemic with 2009 H1N1 characteristics (mild severity and low transmissibility), closing schools would not be cost-effective, regardless of the age target of school children.

Conclusions

School closure has moderate impact on the epidemic’s scope, but the resulting disruption to society imposes a potentially great cost in terms of lost productivity from parents’ work absenteeism.

Keywords

Influenza pandemic School closure Costs Benefits Simulation

Background

Influenza pandemics occur at irregular intervals and cause significant mortality and morbidity as well as substantial economic losses [1]. School closure is a possible strategy for mitigating transmission during the early phase of a pandemic when vaccine is not yet available. School closure has three main consequences: reducing the total disease burden, postponing the peak of infection and lowering the peak prevalence of the disease. Postponing the pandemic increases the time available for strain-specific vaccine production and distribution, and allows for more time to prepare for the peak workload in health care settings. Lowering the peak of the pandemic reduces the risk for overloading of health services and shortage of health care personnel due to influenza sickness.

Schools are thought to play a special role in transmission due to high contact rates among school children combined with higher susceptibility among children compared with adults. During the A(H1N1) pandemic in 2009, the estimated infection rate among school children was significantly higher than that of the general population [2]. However, extended school closure is costly and may cause significant disruption to local communities by keeping working parents away from work and reducing school children’s learning time. Quantifying the costs and benefits of school closure might help inform pandemic policy making.

There is currently no consensus about the expected health benefits of school closure [3]. Previous studies have investigated the impact of school closure either by analysing data from previous pandemics and epidemics or by computer simulation. The historical data approach includes studies of the 1918 influenza pandemic and suggests that school closure, combined with other interventions, lowered the disease burden and that the timing and duration of such interventions mattered [4, 5]. A 2009 study of eight European countries indicated that during holidays and weekends the social contact patterns of children and the basic reproductive number were reduced by almost a quarter [6]. However, little effect on transmission was observed during a two-week kindergarten and primary school closure in Hong Kong in 2008 [7]. The estimated impact of school closure from computer simulations varies widely depending on model assumptions about how children contribute to influenza transmission, virus transmissibility and illness threshold when school closure is triggered [812]. Only a limited number of studies have explored the cost of school closure. Two studies focused on productivity loss of care-taking parents suggest that school closure for 12 weeks may cost 0.2-1% of GDP in the UK [13], and 4 weeks closure 0.1-0.3% of GDP in the US [14]. To reduce the economic loss from working parents, reactive short-term (1–4 weeks) school closure has been studied, where schools are shut when ICU units reach peak demand [15], but the optimal timing of such interventions may be difficult. Some studies have combined cost estimates with micro-simulation models [1619] or dynamic compartmental models [20]. While the assumptions used in the studies differ, the general picture in the cost-effectiveness is that school closure may be effective under high transmissibility, and/or high severity. Some of the studies were based on the characteristics of the 2009 H1N1 pandemic. Halder and co-workers [16] found that productivity losses due to sick leave and taking care of children when schools are closed were the dominating part of cost. A similar result was obtained in a study by Brown and co-workers [17] suggesting that the cost of school closure may far outweigh the cost saved from reducing the disease burden when the severity is low, regardless of the transmissibility.

In this study, we estimated potential costs and health benefits of school closure when implemented before substantial transmission of influenza among children has occurred (proactive school closure). We combined the cost estimates with a dynamic epidemiological transmission model, and determined the optimal closure strategy based on incremental cost-effectiveness ratios. Our study complements previous work on school closure by focusing on the age of the target school children, covering several scenarios for transmissibility, closure duration and severity. The study may be useful for public health authorities and may inform preparedness planning for future influenza pandemics.

Methods

Background

We modelled the impact of school closure in the context of a local community, using the capital city of Norway, Oslo, as the study setting. The city has a population size of 587 000, covering 12% of the Norwegian population. The unemployment rate is low (3.4%) and women’s participation in the labour force is high (70% of women aged 15–74 are employed) [21]. The education system is composed of primary school for children aged 6 to 12 years and secondary school for children aged 13 to 18 years. The attendance rate in kindergarten is approximately 90% for children aged 1 to 5 [21].

The disease model

We considered a closed population of size N=587 000, ignoring demography (births, deaths and immigration) since influenza epidemics are of very short duration. We divided the population into six age groups (i=1-6): 1–5 years (6.7%), 6–12 years (7.2%), 13–18 years (6.9%), 19–39 years (36.6%), 40–64 years (30.5%) and 65+ years (12.2%). We modelled a pandemic influenza using a deterministic dynamic SEIR (Susceptible-Exposed-Infected-Recovered) model [22]. People in each age group are divided into four mutually exclusive compartments: susceptible, infected symptomatically, infected asymptomatically, and recovered with immunity/dead from influenza (Figure 1). People progress from one compartment to another at the rates determined by the contact pattern and characteristics of the virus.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2458-12-962/MediaObjects/12889_2012_Article_4737_Fig1_HTML.jpg
Figure 1

The dynamic influenza transmission model.

A susceptible individual (S i ) becomes infected according to the age-specific force of infection λ i . Newly infected individuals first enter the exposed state (E i ) where they are infected, but not yet contagious, before developing either symptomatic infection (IS i ) or asymptomatic infection (IA i ). To obtain more realistic distributions of the exposed and infectious periods, we divided these periods into n i stages, where the progression from each stage occurs at a rate r i  = n i /D i , where D i is the mean duration of period i = E, IS, IA. This gives gamma distributed waiting times with shape parameters k = n i and scale parameters θ = D i /n i . The mean duration of the exposed period was set to 1/σ = 1.9 days (17;18) and modelled in n E  = 3 stages. Individuals in the last exposed stage were assumed to be infectious with infectivity 50% compared to the infectivity of symptomatic infection, as viral shedding increases after one day following transmission [23]. We assumed that a proportion p=0.67 will become symptomatically infected while a proportion (1-p)=0.33 develop asymptomatic infection [24, 25]. The average duration of the symptomatic infectious period was set to 1/γ c =7 days for children (i=1, 2) and 1/γ a  = 5 days for adolescents/adults (i=3-6) [23, 24, 26] and modelled in n I  = 5 stages. Infectivity during the stages was set at 100%, 100%, 50%, 30% and 15% in accordance with data showing that viral transmission peaks during the early period after symptoms develop [23, 27]. We assumed that asymptomatic infections are 50% as infectious per contact as symptomatic infections [23], but with similar duration and infectivity profile as symptomatic infections. However, other studies have found that asymptomatically infected individuals might be less important for transmission [28]. At the end of the infectious stage, people either recover or are removed from the system due to death. Individuals who have recovered from infection (R i ) are assumed be protected from re-infection during the course of the simulation. The system can be described by a set of differential equations for each age group i=1-6:
d S i d t = S i λ i d E 1 i d t = S i λ i n E σ E 1 i d E li d t = n E σ E l 1 i n E σ E li l = 2 , 3 d I A 1 i d t = ( 1 p ) n E σ E 3 i n I γ i I A 1 i d I A mi d t = n I γ i I A m 1 i n I γ i I A mi m = 2…5 d I S 1 i d t = p n E σ E 3 i n I γ i I S 1 i d I S ni d t = n I γ i I S n 1 i n I γ i I S ni n = 2…5 d R i d t = n I γ i I A 5 i + I S 5 i λ i = j = 1 6 β ij α E E j + k = 1 5 α IA k I A kj + α IS k I S kj

Where λ i is the per capita force of infection for a susceptible individual in age group i to become infected and β ij is the transmission rate from age group j to age group i The age-specific force of infection λ i is a product of age-specific contact rates, the prevalence of the infectious people (I i ) and the probability of transmission given contact (q). We obtained the contact rates based on conversational data from a study in the Netherlands [29]. We employed a WAIFW matrix (“Who-acquires-infection-from-whom” matrix) based on the contact rates between age groups. The basic reproductive number (R 0) was calculated as the largest eigenvalue in the next generation matrix (23). The basic reproductive number is “the average number of secondary cases arising from an average primary case in an entire susceptible population” [22]. Through varying the value of q, we can produce the desired R 0 .

The differential equations were solved numerically using a fourth-order Runge–Kutta method with adaptable step size in Matlab 2009. It is unclear whether cross-immunity from past exposure to influenza will provide protection against a future pandemic strain. We assumed that the population was fully susceptible to the novel pandemic strain at the beginning of the simulation. Transmission was initiated at day t i =1 by moving a proportion of 10-6 of susceptible in each age class into the exposed class. The simulation was run for a period of t=250 days.

The transmissibility of a future pandemic strain is a major source of uncertainty. For this reason, we tested the model with three different basic reproductive numbers R 0 =1.5, 2.0 and 2.5. The school closure intervention was initiated when the prevalence of symptomatic infections had reached 1% of the population and was assumed to have full impact from this point in time. In the baseline scenario (scenario A), we assumed a 90% reduction in contacts among isolated children/adolescents with individuals in their own age group and a 25% decrease in contacts with other age groups. We did not consider changes in the contact patterns of affected parents taking care of children at home in this baseline scenario.

One-way sensitivity analysis

To account for some of the uncertainty in the model, we performed additional simulations varying assumptions about: the behaviours of care-taking parents, the behaviours of dismissed student during school closure and the case fatality rate (CFR).

In Scenario B, we introduced a 50% reduction in same age contacts among care-taking parents absent from work; in Scenario C we reduced the same age contact of dismissed children by 50% instead of 90% in the base case, and by 10% with other age groups instead of 25% to simulate low compliance among affected children; in Scenario D we increased the case fatality rate (CFR) by a factor of 10 compared to the baseline scenarios, using CFR of 1-2% in children and adults below 65 years similar to the level observed during the Spanish flu [30]; in Scenario E we reduced the CFR by a factor of 10 relative to the baseline scenarios, using CFR of 0.01-0.02% to simulate a mild pandemic. Finally, in Scenario F we modelled a pandemic with similar characteristics as the 2009 H1N1 pandemic. In these simulations, we assumed an R 0 of 1.3. 60% of the populations in the 65+ year old age group and 10% of the 40–64 year old age group were assumed to have prior immunity. We also reduced the case fatality rate in accordance with Norwegian data showing that approximately 30 people died from H1N1 influenza (http://www.fhi.no/dokumenter/6cbae0eece.pdf).

The economic model

The costs of school closure comprised parents’ productivity losses and students’ loss of learning. Avoided costs resulted from less use of health care resources, less loss of productivity and less use of energy in school buildings. Health benefits were expressed as gained quality-adjusted life-years (QALYs). Productivity loss due to illness and health benefits were included for cases of mortality and cases of morbidity. We used 2008 data (US$1.00=NOK7.00 [21]) for all economic calculations. All future costs and health outcomes were discounted by 4% as recommended by the Ministry of Health.

Costs of school closure

Absence from school means lost learning hours and potentially permanent loss of learning and income [31, 32]. We searched the literature and databases, and contacted experts in education and educational economics. We were unable to identify any studies that directly address the issue of learning consequences of school closure. We assumed that this was the case only for students in upper secondary schools while children in kindergarten, primary and lower secondary school have no loss of learning from some weeks’ school closure. Most schools in Norway are public and free of charge, but some private schools offer upper secondary school education. Here, the tuition fee for one school year comprising 40 weeks was $8143, which is equivalent to $203 per week. We used this amount as an estimate of the value of lost learning.

School closure will keep working parents at home to care for children who are affected by the intervention. We assumed that students over 12 years do not need parental care during school closures. Similar to Sadique’s study [13], we assumed that only one parent is needed to care for children in a single household during school closure. Consequently, we distinguished between children living together with a single parent and with two parents. The percentages of both parents working were 66% among married couples with children and 78% among co-habitant couples with children (personal communication with Statistics Norway, 12 March, 2010). The percentage of working single parents was assumed to be the same as the percentage of working people in the same gender group (90% for men and 85% for women) [21]. We multiplied these percentages by the number of married couples, co-habitant couples and single parents, respectively. The sum of the products was taken as the number of individuals who would be absent from work during school closure.

We estimated the productivity losses from parents’ work absenteeism by multiplying the number of individuals that would need to be away from work during school closure with the number of days when schools are closed under different scenarios. The value of one day’s work was set equal to the national average wage rate (US$290 per day) plus 40%, which accounts for the value of productivity that is not returned to the worker as wages, including employer tax, payment for holiday and pension contributions.

Reduction of total cost due to school closure

The model outcome for symptomatically infected was divided into four types: mild cases who receive no medical care, moderate cases who receive outpatient service, severe cases who are hospitalized and fatal cases. Since the severity of a future pandemic is unknown, we used estimates of case fatality rates and health outcomes based on data from previous pandemics [33] (Table 1). We assumed that people with asymptomatic infection incur no economic costs, and therefore they were ignored in the economic analyses. The medical costs were estimated as the sum of mild, moderate and severe cases, multiplied by their respective unit costs. The unit costs were taken from a recent study of influenza costs in Norway [34].
Table 1

Model parameters

 

Mean

Distribution

Parameter

References

Demographic data

Population by age

   

15

  1--5 years old

6.63%

   

  6—12 years old

7.17%

   

  13—19 years old

6.86%

   

  20—39 years old

36.65%

   

  40—64 years old

30.46%

   

  65+ years old

12.24%

   

Percentage of adult population affected by school closure:

   

15

  kindergarten (1-5 years old)

4.54%

   

  kindergarten/primary school (1-11 years old)

10%

   

Disease parameters

  Basic reproductive number (R 0 )

1.5, 2.0, 2.5

  

31; 32; 8

  Mean duration of exposed period

1.9 days

  

17; 18

  Mean duration of infectious period

7 days (<12 years) 5 days (12+ years)

  

17; 18; 19

  Proportion asymptomatic (p)

33%

   

  Infectivity (last exposed stage)

50%

  

19

  Infectivity (in the five infectious stages)

100%, 100%, 50%, 30%, 15%

  

19;20

Mixing assumptions

Scenario A (baseline)

    

  Reduction in contact rate between dismissed   children of same/other age groups

90%/25%

   

  Reduction in contact rate among care-taking   parents and same age group

0%

   

Scenario B

  Reduction in contact rate between dismissed   children of same/other age groups

90%/25%

   

  Reduction in contact rate among care-taking   parents and same age group

50%

   

Scenario C

  Reduction in contact rate between dismissed   children of same/other age groups

50%/10%

   

  Reduction in contact rate among care-taking   parents and same age group

0%

   

Disease outcomes

Outcomes per 1000 cases by age groupsa

   

25

  Outpatient

(534, 389, 497)

Uniform

((494-574), (369-410), (487-506))

 

  Inpatient

(4, 8, 29)

 

((1-8), (2-13), (21-37))

 

  Death

(1, 2, 13)

 

((0-2),(0-4),(11-15))

 

Economic parameters

  Cost of energy saving (1000 US$)

1 439

Gammab

α=16; β=90

Oslo Municipality

  Cost of lost learning (1000 US$)

25 797

Gamma

α=16; β=1 612

Bjørknes private school

  Proportion of productivity loss catching up

15%

Uniform

range [0: 30%]

 

  Average cost per self-care person (US$)

43

Normal

σ=3.57

26

  Average cost per out-patient (US$)

59

Normal

σ=4.92

Den norske legeforening

  Average cost per in-patient (US$)

5 211

Normal

σ=434

26

  Average wage per day (US$)

290

Normal

σ=24

15

aAge groups were grouped by 1—18 years old, 19—64 years old and 65+years old.

b f x ; k , θ = x k 1 e x / θ θ k Γ k where Γ is the Gamma function.

Loss of productivity associated with influenza has two components: the loss of working hours for the symptomatically infected and the loss of potential productivity for the fatal cases. Productivity losses due to morbidity were valued in the same way as parents’ work absenteeism. Productivity losses due to mortality were valued according to the remaining life expectancy at the relevant ages, discounted by 4% and with the assumption that people participate in the work force until age 65.

The avoided school heating cost was estimated using data from the Educational Buildings and Property Department in Oslo municipality.

Health benefits

Assuming that school closure will reduce the number of symptomatic and fatal influenza cases, we expressed the health benefits from school closure in terms of quality-adjusted life years (QALYs). For those who are symptomatically infected, we used utility scores from a Canadian study [35]. These utility scores represent the utility people have on each of the seven days since the onset (0 for worst possible health and 1 for normal health). The utilities are 0.41, 0.47, 0.58, 0.67, 0.73, 0.78 and 0.81 for day 1 to day 7, respectively. For those who died due to the illness, the QALY loss was calculated from the remaining life expectancy at the age of death predicted by the disease model and the discount factor.

Intervention strategy scenarios

We explored the costs and benefits of intervention policies with different durations (from 1 to 10 weeks) and for different target groups (closing kindergarten alone, primary school alone, secondary school alone, kindergarten and primary school or all three).

Uncertainty in cost-effectiveness estimates

To quantify the uncertainty in the cost-effectiveness ratios, we performed a probabilistic sensitivity analysis (number of simulations=1000) on the selected strategy for R 0 = 1.5, 2.0 and 2.5, incorporating the uncertainty in the demographic parameters, disease parameters, disease outcomes and economic parameters (Table 1). In addition, we reduced the work loss of care-taking parents by 0-30% (uniform distribution) assuming that some children were cared for by relatives or other persons, or that part of their work loss could be carried out through work from home or through work at a later time. The results were presented graphically by means of cost-effectiveness acceptability curves (Additional file 1: e-Figure 1).

Results

Epidemiological impact of school closure

Figures 2, 3 show the epidemiological results of school closure. In the absence of intervention, our baseline model predicts 216 000, 300 000 and 340 000 symptomatic infections in the Oslo population for R 0 =1.5, 2.0 and 2.5 pandemics, corresponding to clinical attack rates (AR) of 37%, 51% or 58%, respectively (Table 2). The relative effectiveness of the interventions increased with lower R 0 values but required longer closure time to achieve the health benefits (Figure 3). School closure lowers the attack rate with up to 7-22%, 4-13% and 2-9% with R 0 =1.5, 2.0 or 2.5; these reductions are achieved after approximately 10, 8 and 7 weeks of closure (Figure 3). The peak prevalence of symptomatic infections was reduced correspondingly with up to 7-36%, 6-26% and 5-20%. To reach maximum reduction, school closure must be maintained for some weeks and beyond the point in time when the mitigated pandemic passes through its natural peak (Additional file 1: e-Figure 2). If schools are re-opened earlier, the pandemic will rebound. This will also happen if the intervention stops in the wake of the pandemic, provided the effective reproductive number of the un-mitigated pandemic is still above 1. Consequently, the maximum delay of the peak occurred for intermediate closure durations. The peak was delayed by up to 8–10 days (R 0 =1.5), and to 4–5 days for R 0 =2.0, 2.5. To avoid restarting the epidemic, we found that closure must be effective for at least 3–4 week for R 0 =1.5, and 2–3 weeks when the transmissibility is higher.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2458-12-962/MediaObjects/12889_2012_Article_4737_Fig2_HTML.jpg
Figure 2

Epidemic curves showing the prevalence of symptomatic infections for unmitigated pandemic versus implementing a 12-week school closure with R 0 =1.5, 2.0 and 2.5.

https://static-content.springer.com/image/art%3A10.1186%2F1471-2458-12-962/MediaObjects/12889_2012_Article_4737_Fig3_HTML.jpg
Figure 3

The relative attack rate compared to an unmitigated pandemic as function of school closure duration (number of closure weeks).

Table 2

Disease outcomes given R 0 =1.5, 2.0 and 2.5

School closure of 12 weeks

R0=1.5

R0=2.0

R0=2.5

 

outp.

inp.

deaths

AR(%)

outp.

inp.

deaths

AR(%)

outp.

inp.

deaths

AR(%)

No intervention

92779

1929

584

37

128932

2738

844

51

146088

3150

983

58

Scenario A (baseline)

            

K

87388

1846

560

35

123904

2673

825

49

141642

3098

968

56

P

83081

1779

540

33

121245

2638

815

49

140075

3080

962

56

S

85718

1813

550

34

123784

2665

822

49

142328

3101

968

57

K+P

77605

1692

514

31

115823

2566

793

47

135161

3022

945

54

K+P+S

69989

1559

474

29

109800

2477

767

44

130661

2962

927

53

 

SENSITIVITY ANALYSIS

 

R0=1.5

R0=2.0

R0=2.5

 

outp.

inp.

deaths

AR(%)

outp.

inp.

deaths

AR(%)

outp.

inp.

deaths

AR(%)

Scenario B

            

K

85200

1798

546

34

122669

2645

817

49

140911

3082

963

56

P

79765

1707

519

32

119377

2597

803

48

138986

3056

955

55

S

85718

1813

550

34

123784

2665

822

49

142328

3101

968

57

K+P

71608

1559

475

29

112224

2487

770

45

133028

2975

932

53

K+P+S

64030

1423

434

26

105671

2387

740

43

128221

2910

912

52

Scenario C

            

K

89954

1885

572

36

126502

2707

835

50

144110

3127

976

57

P

87441

1847

560

35

125354

2691

830

50

143677

3121

974

57

S

89346

1873

568

36

126696

2706

835

50

144607

3131

977

57

K+P

84498

1801

547

34

122774

2657

820

49

141574

3097

967

56

K+P+S

80744

1738

528

33

120292

2621

810

48

139934

3075

961

56

Outp= outpatient. Inp= inpatient. AR=attack rate.

Scenario A is the base case scenario; scenario B included a 50% reduction in contacts among care-taking parents absent from work based on scenario A; scenario C reduced the compliance to 50% from scenario A.

The baseline scenarios gave an estimated 93 000–147 000 outpatient visits, 1 900–3 100 hospitalizations and 590–990 deaths (Table 2). The simulation runs showed that a 12-week school closure would reduce the attack rate by up to 22%, 14% and 7% for R 0 =1.5, 2.0 and 2.5 pandemics. The reductions in disease outcomes followed the reductions in attack rate, with slightly higher reductions in outpatients (6%–25%) and slightly lower reductions in inpatients and deaths (4%–20%) for a 12-week closure with R 0 =1.5, 2.0 or 2.5 in the base case.

Economic impact

Without school closure, the total health care costs would be $21 million, $29 million and $33 million, productivity losses due to mortality would be $313 million, $428 million and $480 million and productivity losses due to morbidity $102 million, $139 million and $155 million, for basic reproductive numbers of 1.5, 2.0 and 2.5 (Tables 3, 4 and 5). Depending on the type and duration of school closure, the cost of lost learning would be $0–32 million, while the cost of lost productivity were in the range of $0–630 million, and reduction in school heating costs varied between $0.18 and 5.4 million. The total influenza related costs would range from $435 million to $1285 million from the societal perspective (Tables 3, 4 and 5).
Table 3

Cost and health outcome according to type and duration of school closure when R 0 =1.5

Target school

Duration (weeks)

Cost of lost learning ($1000)

Lost productivity due to school closure ($1000)

Energy savings ($1000)

Health care costs ($1000)

Lost productivity due to fatal cases ($1000)

Lost productivity due to sickness ($1000)

Total cost ($1000)

QALY gains (compared to no intervention)

Cost per QALY (compared to no intervention)

ICER

0

0

0

0

0

20 591

312 958

101 576

435 125

0

  

3

6

19 350

0

1 080

19 557

298 239

97 846

433 912

507

−2 395

 

3

7

22 575

0

1 260

19 410

296 139

97 312

434 175

579

−1 641

3 648

3

5

16 125

0

900

19 766

301 213

98 600

434 804

404

−796

Dominated

3

8

25 800

0

1 440

19 318

294 825

96 978

435 481

624

570

28 929

3

4

12 900

0

720

20 008

304 661

99 474

436 323

286

4 193

Dominated

3

1

3 225

0

180

20 509

311 792

101 278

436 625

40

37 316

Dominated

3

9

29 025

0

1 620

19 264

294 064

96 784

437 517

650

3 679

77 819

3

3

9 675

0

540

20 235

307 897

100 293

437 560

174

13 962

Dominated

3

2

6 450

0

360

20 403

310 287

100 897

437 678

92

27 727

Dominated

3

10

32 250

0

1 800

19 237

293 672

96 684

440 043

664

7 412

187 991

2

1

0

26 795

188

20 495

311 614

101 261

459 977

47

531 474

 

1

1

0

36 194

174

20 530

312 120

101 383

470 054

29

1 193 056

 

2

2

0

53 591

376

20 385

310 094

100 909

484 603

100

496 745

 

4

1

0

62 989

362

20 440

310 857

101 085

495 009

73

817 647

 

5

1

3 225

62 989

542

20 367

309 817

100 816

496 672

109

564 499

 

1

2

0

72 388

348

20 453

311 044

101 138

504 674

67

1 039 847

 

2

3

0

80 386

564

20 210

307 651

100 342

508 024

185

395 026

 

2

4

0

107 181

752

19 950

304 039

99 504

529 922

310

305 814

 

1

3

0

108 582

522

20 323

309 257

100 729

538 369

129

798 460

 

2

5

0

133 976

940

19 645

299 793

98 516

550 991

457

253 333

 

4

2

0

125 979

724

20 263

308 401

100 517

554 436

159

751 538

 

5

2

6 450

125 979

1 084

20 109

306 200

99 944

557 598

234

522 386

 

1

4

0

144 776

696

20 148

306 836

100 173

571 237

214

636 557

 

2

6

0

160 772

1 128

19 356

295 758

97 575

572 332

597

229 715

 

2

7

0

187 567

1 316

19 117

292 419

96 793

594 579

713

223 637

 

1

5

0

180 970

870

19 971

304 374

99 606

604 051

300

563 630

 

4

3

0

188 968

1 086

19 989

304 590

99 632

612 092

291

607 281

 

5

3

9 675

188 968

1 626

19 736

300 952

98 673

616 377

416

435 391

 

2

8

0

214 362

1 504

18 961

290 230

96 279

618 327

789

232 250

 

1

6

0

217 164

1 044

19 812

302 170

99 095

637 197

377

536 585

 

2

9

0

241 157

1 692

18 858

288 799

95 942

643 065

838

248 040

 

4

4

0

251 957

1 448

19 608

299 290

98 394

667 801

476

489 135

 

2

10

0

267 953

1 880

18 803

288 023

95 759

668 657

865

269 909

 

1

7

0

253 358

1 218

19 703

300 654

98 742

671 239

429

549 816

 

5

4

12 900

251 957

2 168

19 239

293 960

96 967

672 854

658

361 133

 

1

8

0

289 552

1 392

19 639

299 767

98 536

706 101

460

588 611

 

4

5

0

314 946

1 810

19 151

292 916

96 892

722 096

697

411 700

 

5

5

16 125

314 946

2 710

18 630

285 363

94 847

727 200

955

305 707

 

1

9

0

325 746

1 566

19 605

299 302

98 427

741 514

477

642 905

 

4

6

0

377 936

2 172

18 702

286 631

95 396

776 493

915

373 065

 

1

10

0

361 940

1 740

19 585

299 025

98 363

777 173

486

703 475

 

5

6

19 350

377 936

3 252

18 018

276 690

92 681

781 422

1 255

275 997

 

4

7

0

440 925

2 534

18 330

281 412

94 143

832 277

1 096

362 429

 

5

7

22 575

440 925

3 794

17 426

268 283

90 555

835 970

1 544

259 543

 

4

8

0

503 914

2 896

18 058

277 580

93 217

889 873

1 228

370 185

 

5

8

25 800

503 914

4 336

16 965

261 700

88 870

892 913

1 771

258 490

 

4

9

0

566 903

3 258

17 885

275 152

92 627

949 309

1 312

391 782

 

5

9

29 025

566 903

4 878

16 627

256 878

87 624

952 179

1 937

266 957

 

4

10

0

629 893

3 620

17 789

273 802

92 297

1 010 161

1 359

423 098

 

5

10

32 250

629 893

5 420

16 386

253 424

86 726

1 013 259

2 056

281 259

 

Note: The maximum willingness to pay is set to be NOK 500,000 or US$71,500 based on the government guidance28. The most cost-effective option is shown with bold font.

Table 4

Cost and health outcome according to type and duration of school closure when R 0 =2.0

Target school

Duration (weeks)

Cost of lost learning ($1000)

Lost productivity due to school closure ($1000)

Energy savings ($1000)

Health care costs ($1000)

Lost productivity due to fatal cases ($1000)

Lost productivity due to sickness ($1000)

Total cost ($1000)

QALY gains (compared to no intervention)

Cost per QALY (compared to no intervention)

ICER

0

0

0

0

0

28 890

428 137

138 654

595 682

   

3

4

12 900

0

720

28 215

419 135

136 843

596 374

321

2 155

 

3

5

16 125

0

900

28 049

416 920

136 411

596 604

400

2 306

2 921

3

1

3 225

0

180

28 846

427 542

138 529

597 961

21

106 854

 

3

3

9 675

0

540

28 491

422 813

137 570

598 009

190

12 224

Dominated

3

6

19 350

0

1 080

27 985

416 062

136 245

598 562

431

6 686

64 224

3

2

6 450

0

360

28 732

426 018

138 216

599 056

76

44 470

Dominated

3

7

22 575

0

1 260

27 964

415 780

136 190

601 248

441

12 628

267 404

3

8

25 800

0

1 440

27 957

415 695

136 173

604 186

444

19 161

975 711

3

9

29 025

0

1 620

27 955

415 672

136 169

607 201

445

25 907

3 654 485

3

10

32 250

0

1 800

27 955

415 664

136 167

610 236

445

32 714

11 358 909

2

1

0

26 795

188

28 844

427 532

138 545

621 528

22

1 179 444

 

1

1

0

36 194

174

28 853

427 657

138 575

631 105

17

2 029 542

 

2

2

0

53 591

376

28 735

426 112

138 297

646 358

73

691 761

 

4

1

0

62 989

362

28 810

427 096

138 472

657 005

38

1 622 566

 

5

1

3 225

62 989

542

28 769

426 546

138 354

659 342

58

1 107 041

 

1

2

0

72 388

348

28 752

426 362

138 363

665 517

65

1 082 646

 

2

3

0

80 386

564

28 481

422 818

137 732

668 853

192

380 871

 

2

4

0

107 181

752

28 108

417 981

136 924

689 442

366

256 079

 

1

3

0

108 582

522

28 535

423 576

137 909

698 080

166

618 250

 

2

5

0

133 976

940

27 795

413 930

136 260

711 021

512

225 466

 

4

2

0

125 979

724

28 618

424 608

138 047

716 527

128

944 416

 

5

2

6 450

125 979

1 084

28 494

422 945

137 687

720 470

187

665 879

 

1

4

0

144 776

696

28 275

420 231

137 363

729 949

287

467 968

 

2

6

0

160 772

1 128

27 636

411 869

135 925

735 074

585

238 117

 

2

7

0

187 567

1 316

27 576

411 079

135 797

760 703

614

268 896

 

1

5

0

180 970

870

28 108

418 079

137 011

763 298

365

459 401

 

4

3

0

188 968

1 086

28 179

418 930

137 085

772 075

333

529 302

 

5

3

9 675

188 968

1 626

27 906

415 233

136 270

776 425

465

388 791

 

2

8

0

214 362

1 504

27 557

410 834

135 758

787 007

622

307 365

 

1

6

0

217 164

1 044

28 044

417 253

136 876

798 292

395

513 250

 

2

9

0

241 157

1 692

27 551

410 765

135 746

813 528

625

348 587

 

4

4

0

251 957

1 448

27 555

410 851

135 722

824 637

624

366 634

 

5

4

12 900

251 957

2 168

27 050

403 997

134 214

827 950

868

267 629

 

1

7

0

253 358

1 218

28 022

416 976

136 830

833 968

405

588 668

 

2

10

0

267 953

1 880

27 550

410 744

135 743

840 109

626

390 639

 

1

8

0

289 552

1 392

28 017

416 902

136 818

869 897

407

672 986

 

4

5

0

314 946

1 810

27 034

404 084

134 582

878 837

868

326 324

 

5

5

16 125

314 946

2 710

26 272

393 758

132 340

880 731

1 234

230 988

 

1

9

0

325 746

1 566

28 015

416 881

136 815

905 892

408

759 953

 

4

6

0

377 936

2 172

26 753

400 417

133 965

936 898

999

341 470

 

5

6

19 350

377 936

3 252

25 798

387 502

131 192

938 526

1 457

235 261

 

1

10

0

361 940

1 740

28 015

416 876

136 814

941 904

408

847 747

 

4

7

0

440 925

2 534

26 645

399 011

133 728

997 774

1 050

383 078

 

5

7

22 575

440 925

3 794

25 597

384 843

130 703

1 000 849

1 552

261 045

 

4

8

0

503 914

2 896

26 611

398 573

133 654

1 059 855

1 065

435 700

 

5

8

25 800

503 914

4 336

25 521

383 838

130 518

1 065 255

1 588

295 719

 

4

9

0

566 903

3 258

26 600

398 435

133 631

1 122 311

1 070

492 048

 

5

9

29 025

566 903

4 878

25 494

383 489

130 454

1 130 488

1 600

334 181

 

4

10

0

629 893

3 620

26 597

398 391

133 623

1 184 883

1 072

549 693

 

5

10

32 250

629 893

5 420

25 486

383 386

130 435

1 196 030

1 604

374 276

 
Table 5

Cost and health outcome according to type and duration of school closure when R 0 =2.5

Target school

Duration (weeks)

Cost of lost learning ($1000)

Lost productivitydue to school closure ($1000)

Energy savings ($1000)

Health care costs ($1000)

Lost productivity due to fatal cases ($1000)

Lost productivity due to sickness ($1000)

Total cost ($1000)

QALY gains (compared to no intervention)

Cost per QALY (compared to no intervention)

ICER

0

0

0

0

0

32 961

479 607

155 079

667 646

   

3

1

3 225

0

180

32 928

479 185

155 005

670 162

16

160 991

 

3

3

9 675

0

540

32 544

474 295

154 205

670 179

195

12 994

 

3

4

12 900

0

720

32 367

472 045

153 864

670 456

277

10 150

3 380

3

2

6 450

0

360

32 801

477 565

154 728

671 184

75

47 003

Dominated

3

5

16 125

0

900

32 318

471 424

153 771

672 739

299

17 011

101 226

3

6

19 350

0

1 080

32 308

471 296

153 752

675 626

304

26 248

620 315

3

7

22 575

0

1 260

32 306

471 271

153 749

678 641

305

36 059

3 386 921

3

8

25 800

0

1 440

32 306

471 267

153 748

681 681

305

46 005

20 007 697

3

9

29 025

0

1 620

32 306

471 266

153 748

684 725

305

55 979

126 703 892

3

10

32 250

0

1 800

32 306

471 266

153 748

687 770

305

65 955

289 245 859

2

1

0

26 795

188

32 929

479 210

155 022

693 768

15

1 764 356

 

1

1

0

36 194

174

32 929

479 216

155 031

703 195

15

2 442 399

 

2

2

0

53 591

376

32 811

477 750

154 823

718 598

69

738 185

 

4

1

0

62 989

362

32 899

478 844

154 977

729 347

28

2 168 767

 

5

1

3 225

62 989

542

32 871

478 484

154 908

731 936

42

1 538 432

 

1

2

0

72 388

348

32 797

477 600

154 839

737 276

75

934 187

 

2

3

0

80 386

564

32 504

473 932

154 337

740 595

210

348 001

 

2

4

0

107 181

752

32 174

469 835

153 856

762 295

359

263 354

 

1

3

0

108 582

522

32 532

474 332

154 456

769 379

196

520 297

 

2

5

0

133 976

940

32 013

467 836

153 631

786 516

432

275 032

 

4

2

0

125 979

724

32 671

476 022

154 614

788 561

133

907 803

 

5

2

6 450

125 979

1 084

32 558

474 572

154 337

792 811

187

670 647

 

1

4

0

144 776

696

32 315

471 663

154 148

802 206

294

457 624

 

2

6

0

160 772

1 128

31 970

467 302

153 571

812 487

452

320 722

 

1

5

0

180 970

870

32 242

470 771

154 046

837 160

327

518 523

 

2

7

0

187 567

1 316

31 962

467 200

153 560

838 973

455

376 283

 

4

3

0

188 968

1 086

32 117

469 165

153 760

842 924

386

453 852

 

5

3

9 675

188 968

1 626

31 806

465 143

153 021

846 987

533

336 228

 

2

8

0

214 362

1 504

31 960

467 181

153 558

865 557

456

434 015

 

1

6

0

217 164

1 044

32 226

470 573

154 023

872 943

334

614 247

 

2

9

0

241 157

1 692

31 960

467 177

153 557

892 159

456

492 180

 

4

4

0

251 957

1 448

31 531

461 895

152 891

896 827

653

351 186

 

5

4

12 900

251 957

2 168

30 971

454 663

151 625

899 947

916

253 572

 

1

7

0

253 358

1 218

32 224

470 539

154 019

908 922

335

719 170

 

2

10

0

267 953

1 880

31 960

467 176

153 557

918 766

456

550 486

 

1

8

0

289 552

1 392

32 223

470 532

154 019

944 934

336

825 928

 

4

5

0

314 946

1 810

31 242

458 297

152 470

955 145

784

366 760

 

5

5

16 125

314 946

2 710

30 530

449 119

150 906

958 916

1 118

260 600

 

1

9

0

325 746

1 566

32 223

470 531

154 019

980 953

336

933 116

 

4

6

0

377 936

2 172

31 161

457 290

152 353

1 016 567

821

425 217

 

1

10

0

361 940

1 740

32 223

470 531

154 019

1 016 972

336

1 040 374

 

5

6

19 350

377 936

3 252

30 395

447 425

150 689

1 022 542

1 179

300 960

 

4

7

0

440 925

2 534

31 145

457 092

152 330

1 078 958

828

496 882

 

5

7

22 575

440 925

3 794

30 360

446 990

150 633

1 087 690

1 195

351 506

 

4

8

0

503 914

2 896

31 142

457 055

152 326

1 141 540

829

571 548

 

5

8

25 800

503 914

4 336

30 354

446 910

150 623

1 153 266

1 198

405 402

 

4

9

0

566 903

3 258

31 142

457 047

152 325

1 204 158

829

646 844

 

5

9

29 025

566 903

4 878

30 353

446 893

150 621

1 218 917

1 199

459 966

 

4

10

0

629 893

3 620

31 141

457 045

152 324

1 266 783

829

722 298

 

5

10

32 250

629 893

5 420

30 352

446 889

150 620

1 284 584

1 199

514 689

 

Health benefits from school closure would range from 15 QALYs to 2056 QALYs depending on R 0, the age target group and the duration of school closure (Tables 3, 4 and 5). Our results indicate that in the baseline scenario, closing secondary schools for 8, 6 and 4 weeks, when R 0 is 1.5, 2.0 and 2.5 respectively, is the most cost-effective strategy when indirect costs are accounted for. Closing secondary schools is cost-effective given a wide range of cost-effective threshold ratios, as shown by cost-effectiveness acceptability curves (Additional file 1: e-Figure 1). The strategy of closing secondary was also cost-effective for varying closure durations (data not shown).

Sensitivity analyses

The sensitivity analyses confirm that closing secondary schools is the optimal strategy from a societal perspective, unless the case fatality rate (CFR) is very high.

Scenario B: Reduced (adult-adult) contact among care-taking parents. We found increased effect of school closure relative to the baseline scenarios. The estimated reduction in the attack rate compared to an unmitigated pandemic was 8-30%, 4-16%, and 3-10%, for R 0 =1.5, 2.0 and 2.5 pandemics, respectively (Table 2). The corresponding optimal strategies were closing secondary schools with durations of 8 weeks, 6 weeks and 4 weeks, identical to the findings in the baseline scenario (Additional file 1: e-Table 1; I-III).

Scenario C: Reduced compliance of dismissed children/students to stay at home. The simulations showed an overall small effect of school closure. The estimated maximum reduction in the attack rate compared to an unmitigated pandemic ranged between 3-11%, 2-6% and 2-3% for R 0 =1.5, 2.0 and 2.5, respectively (Table 2). The optimal strategies were closing secondary schools for 7, 4, and 3 weeks (Additional file 1: e-Table 2; I-III), indicating a shorter optimal period of one week compared with the baseline model for R 0 =1.5 and 2.5.

Scenario D: Increasing the case fatality rate by a factor of 10. This means increasing the severity of the pandemics to levels similar to those observed during the Spanish Flu [36]. In this case, the optimal strategies were closing kindergartens, primary and secondary schools for 9 weeks if R 0 =1.5, 7 weeks if R 0 =2.0, and 5 weeks if R 0 = 2.5 (Additional file 1: e-Table 3).

Scenario E: Decreasing the case fatality rate by a factor of 10. In this case, when R 0 =1.5, closing secondary school for 6 weeks is most cost-effective. Otherwise, there is no cost-effective strategy among the strategies we examined (Additional file 1: e-Table 4).

Scenario F: Pandemic with 2009 H1N1 characteristics. The results show that the added cost of school closure was higher than not closing schools, regardless of the age target of school children. Consequently school closure would not have been cost-effective during the 2009 H1N1 pandemic (Additional file 1: e-Table 5).

Discussion

Our study shows that school closure during influenza pandemic has a moderate impact on the total disease burden. The cost-effectiveness of school closure varies considerably across different strategies with different target groups and durations. Generally we found that for R 0 =1.5, 2.0 and 2.5 pandemics with case fatality rates of 0.1-0.2%, only those strategies involving closure of secondary schools were cost-effective from a societal point of view. The study shows that optimal school closure depends on the transmissibility and severity of the pandemic and may provide guidance to local policy planning. The optimal duration of closing secondary schools is shorter (4 weeks) with R 0 =2.5 compared to 8 weeks with R 0 =1.5. In contrast, school closure involving primary schools and kindergartens incur substantial economic costs due to lost productivity of care-taking parents. Consequently, most school closure strategies cannot be considered cost-effective (Tables 3, 4 and 5) at current values of quality adjusted life-years in Norway [37]. However, school closure involving children in need of parental care may be indicated when case fatality rates are high, for instance in the event of a future pandemic with an avian (H5N1) virus.

We also simulated a pandemic with characteristics of the 2009 H1N1 pandemic. Our results suggest that school closure as a single intervention would not have been cost-effective during the recent pandemic. This finding is in agreement with results by Brown and co-workers [17], who found that the net costs of school closure during the 2009 H1N1 pandemic would have been substantially higher than the cost savings from preventing influenza disease. However, other studies indicate that school closure might have been cost-effective, despite the low severity and low transmissibility of the 2009 H1N1 pandemic. Halder and co-workers [16] found that short-duration school closure of 2 to 4 weeks would be relatively cost-effective while in general school closure intervention as a single strategy would be less efficient than strategies involving widespread use of antivirals, and Araz and co-workers found that a 0.5% prevalence closure trigger followed by a 12 week closure would be cost-effective [20].

Our findings are similar to other computer simulation studies [810, 17, 36] and a surveillance data study from Hong Kong [7], all of which indicate that the impact of school closure on the pandemic is modest. In general we found that school closure peak timing was delayed with only few days compared with that of an unmitigated pandemic. The delay increased with lower transmissibility. The maximum delay was observed for intermediate closure durations, when the epidemic re-started influenced by the higher transmissibility of the unmitigated pandemic (R eff  > 1). A micro-simulation study by Lee and co-workers [9] also show that intermediate duration closure produces the longest delays. However, their observed delay for long closure duration was longer: 4–8 days for system wide school closure for R 0 =1.4-2.4. One possible explanation for the shorter delay in our study is that we assume that the whole population is interacting, while we did not model the individual transmission processes. In addition, individuals in our model generally mix most with individuals in their own age group. Therefore, there is a tendency that the epidemic in school children develops “independently” of how the epidemic develops in the other age groups, and school closure has only small impact on the disease burden in the population that is not directly affected by the intervention. We have performed additional simulations using a lower closure trigger of 0.5% instead of the 1% assumed in the baseline scenario (results not shown). These simulations show that an earlier trigger increases the maximum delay by approximately one third, while the peak timing during long duration closure increased only little.

Our approach is analogous to a recent study by Araz and co-workers [20], using a dynamic compartmental model combined with calculations of incremental cost-effectiveness ratios to select the preferred policy. They studied pandemics with transmissibility in the range R 0 =1.1-2.1, using various closure triggers and fixed school closure durations of 1–24 weeks or prevalence-based reopening triggers. They found that in low transmissibility scenarios, early triggers combined with long closure duration of 12–24 weeks were preferred, regardless of severity; for high transmissibility scenarios, later triggers combined with 8–18 weeks closure were preferred. In comparison, our selected strategies involved much shorter closure durations of 4–8 weeks. One reason for this large discrepancy could be that they used early triggers. In addition, their model has a very long serial interval of 9 days, whereas our model has a serial interval of approximately 4 days due to the infectious profile, which we believe is more in agreement with data [38].

The present work highlights the potential importance of school closure among students who do not need parental care. The benefit of school closure interventions targeting this group appears to have escaped notice in the literature. Our results suggest that closing secondary school alone can decrease the peak prevalence of symptomatic infection by 10–20% while incurring no loss of productivity for parents. Hence, school closure for children over 12 years could have important implications for the functioning of the healthcare system during the surge of a pandemic, when the capacity of health services may be pressured. We note that in Norway laptop computers are mandatory equipment in secondary schools and an organized computer network (“Fronter”) for communication between students and teachers in primary and secondary schools is already in place. It would therefore be possible to plan for sustained teaching and learning during an extended school closure, making secondary school closure even more cost-effective. However, for the strategy to be effective, it is important that students actually follow the recommendations and isolate themselves. This may be difficult to achieve for extended periods of time.

The health-economic evaluation in this study was based on estimates of age-specific health-outcome from previous pandemics [26]. If we scale up the results in the baseline scenarios for R 0 =1.5-2.5 pandemics to the national level (Oslo comprises approximately 12% of Norwegian population), our results correspond to 16 000–26 000 hospitalizations and 4 900–8 200 deaths in Norway with an attack rate ranging from 37-58%. In comparison, the yearly influenza epidemics (attack rate of 5-10%) results in approximately 2 700 cases of hospitalizations [34] and approximately 1 000 deaths [39]. Adjusting for the difference in attack rates, this indicates that our results are in reasonable agreement with findings from the seasonal epidemics; however, the numbers are difficult to compare because the seasonal epidemics primarily affect the elderly population.

Our study has several limitations. Firstly, the age-specific contact rate data were adopted from a Dutch study, as no Norwegian data on social mixing is currently available. The contact pattern in Norway may differ, in particular due to the high attendance rates in kindergarten and high employment rate of women. Secondly, the effect of school closure on the contact pattern in the population is not well documented in the literature and is uncertain. However, our choices were guided by observation from weekends and holidays and previous school closures in Oslo due to strikes, etc. Thirdly, the cost of lost learning is uncertain. We used tuition fees as a proxy for the value of learning, but private schools are primarily used by people with higher incomes and the tuition fee may therefore overstate the value of lost learning. Fourthly, productivity losses may be overestimated because some parents who are away from work may be absent anyway because they have influenza themselves. Fifthly, energy savings in schools during school closure may be partly off-set by higher energy use in homes. However, energy in Norway is cheap and only small proportions of households have day-time energy saving systems according to the governmental energy saving organization. Lastly, we have considered school closure as a single strategy. Combining school closure with other interventions such as use of antiviral medications or other social distancing measures might change the conclusions about optimal duration of school closure, and the target group.

Conclusions

School closure has moderate impact on influenza disease and may incur substantial economic costs in terms of lost productivity from care-taking parents absent from work. Closing secondary schools, assuming children above 12 years would not need parental care, is a cost-effective strategy from a societal perspective. With the current willingness to pay in Norway, closing kindergartens and primary schools is not a cost-effective policy to mitigate an influenza pandemic, unless the case fatality rates are high. Reliable information on influenza mortality is therefore of primary importance to inform decision-making on school closure. Finally, we note that the perspective of the policy maker is crucial for optimal design of school closure. If the policy maker disregards productivity losses, the optimal strategy is to close as many school as possible for as long time as possible.

Funding source

Yiting Xue was supported by the Norwegian Research Council through project number 177401/V50 and Birgitte Freiesleben de Blasio was supported by the Norwegian Research Council through project number 166056/V50.

Declarations

Acknowledgements

We are grateful to Kirsten E. Dybendal at Statistics Norway for providing detailed data on population statistics and to Gianpaolo Scalia Tomba at the Department of Mathematics, University of Rome for suggestions in the disease modelling. Arna Desser at Department of Health Management and Health Economics, University of Oslo, has provided valuable suggestions on heath economics and helped to improve the language.

Authors’ Affiliations

(1)
Department of Biostatistics, Institute of Basic Medical Sciences, University of Oslo
(2)
Department of Infectious Disease Epidemiology, Division of Infectious Disease Control, Norwegian Institute of Public Health
(3)
Department of Health Management and Health Economics, Institute of Health and Society, University of Oslo

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  40. Pre-publication history

    1. The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2458/12/962/prepub

Copyright

© Xue et al.; licensee BioMed Central Ltd. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.