### 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.

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:

\begin{array}{l}\frac{d{S}_{i}}{dt}=-{S}_{i}{\lambda}_{i}\\ \frac{d{E}_{1i}}{dt}={S}_{i}{\lambda}_{i}-{n}_{E}\sigma {E}_{1i}\\ \frac{d{E}_{\mathit{li}}}{dt}={n}_{E}\sigma {E}_{\left(l-1\right)i}-{n}_{E}\sigma {E}_{\mathit{li}}\phantom{\rule{0.5em}{0ex}}\forall l=2,3\\ \frac{dI{A}_{1i}}{dt}=(1-p){n}_{E}\sigma {E}_{3i}-{n}_{I}{\gamma}_{i}I{A}_{1i}\phantom{\rule{1em}{0ex}}\\ \frac{dI{A}_{\mathit{mi}}}{dt}={n}_{I}{\gamma}_{i}I{A}_{\left(m-1\right)i}-{n}_{I}{\gamma}_{i}I{A}_{\mathit{mi}}\phantom{\rule{0.5em}{0ex}}\forall m=\text{2\u20265}\phantom{\rule{0.5em}{0ex}}\\ \frac{dI{S}_{1i}}{dt}=p{n}_{E}\sigma {E}_{3i}-{n}_{I}{\gamma}_{i}I{S}_{1i}\phantom{\rule{0.5em}{0ex}}\\ \frac{dI{S}_{\mathit{ni}}}{dt}={n}_{I}{\gamma}_{i}I{S}_{\left(n-1\right)i}-{n}_{I}{\gamma}_{i}I{S}_{\mathit{ni}}\phantom{\rule{0.5em}{0ex}}\forall n=\text{2\u20265}\phantom{\rule{3.75em}{0ex}}\\ \frac{d{R}_{i}}{dt}={n}_{I}{\gamma}_{i}\left(I{A}_{5i}+I{S}_{5i}\right)\\ {\lambda}_{i}={\displaystyle \sum _{j=1}^{6}{\beta}_{\mathit{ij}}\left({\alpha}_{E}{E}_{j}+{\displaystyle \sum _{k=1}^{5}\left({\alpha}_{\mathit{IA}}\left(k\right)I{A}_{\mathit{kj}}+{\alpha}_{\mathit{IS}}\left(k\right)I{S}_{\mathit{kj}}\right)}\right)}\end{array}

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].

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).