The ABM represented each individual person living in the state of Pennsylvania and was similar in design to previously described models of Allegheny County, Pennsylvania [7,18], and the Washington, DC metropolitan area [19,20]. A geospatially explicit human agent database, termed a synthetic population, represented the state of Pennsylvania in the year 2000. Each agent was assigned to a household, so that at the census tract level the synthetic population contained realistic distributions of households, and agent demographics. This database comprised:

• synthesized households and persons,

• public and private K-12 schools, and

• workplaces including hospital and clinics.

In addition school-aged agents were linked to the schools they attend and workers were linked to their workplace of employment (see [19] for additional details). Specifically, the synthetic population data included assignments of students and teachers to schools, facilitating simulation of school closure policies. Figure 1 shows the location and size of all of the schools in the state of Pennsylvania. The model was calibrated using empirical data from the 2009 H1N1 epidemic and historical epidemiologic studies of epidemic influenza. Important to this work and consistent with other influenza modeling efforts, schools are a critical place for the spread of the disease [21,22]. Figure 2 shows the percent of incident infections for school-aged children versus the rest of the population as a result of the model without school closures. It shows that across several basic reproductive rates, R₀ (the average number of secondary infections produced by an infected individual in a completely susceptible population), that the percent of infections for school-aged children is much higher than that of others in the population. The US Census Bureau's 2005-2009 American Community Survey results show that there were insignificant changes to the age distribution in Pennsylvania between 2000 and 2009 [23].

To account for the spectrum of potential influenza transmission characteristics and dynamics, our simulation runs explored the effects of varying the R₀, which is widely used as a measure of transmissibility. Estimates of the R₀ for the 2009 H1N1 epidemic ranged from 1.2 - 1.7 [24-30]. Values of R₀ equal to 1.2, 1.6, and 2.0 were explored in this study.

The school closure strategy modeled here is an individual school closure; implying that each school in the system is self-monitoring and closed for the specified duration when a certain in-school clinical incidence was reached (i.e. number of symptomatic cases detected in the school). In consulting with public health officials in Pennsylvania during the 2009 H1N1 pandemic, this scheme may be consistent with how school closure would be implemented in the State during an epidemic, and sensitivity analysis of other closure triggering mechanisms showed no significant change in the results. Our model varied two school closure policy parameters: the duration of closure (from 1 to 8 weeks) and the number of detected symptomatic cases that trigger a school to close (from 1 to 30 cases). These two parameters were varied independently to simulate different policy scenarios. School closure duration of 8 weeks have been shown in previous research to significantly decrease the overall attack rate [7], and longer school closure durations were not considered as they become logistically impractical to implement. Consistent with surveillance data collected during the 2009 H1N1 epidemic by the CDC, it is assumed that when the epidemic starts, schools are open. On weekends, students do not go to school and instead have increased activity in their neighborhoods and communities. The peak of the epidemic occurred well before possible extended holidays such as Thanksgiving and Christmas and so these were not considered.

For each scenario, the results presented are the average of 20 stochastic simulation runs, which is sufficient to obtain statistically significant results from computed confidence intervals. The simulations were all performed in parallel on the Intel Xeon based supercomputer, Axon, at the Pittsburgh Supercomputing Center and required approximately 15 minutes to complete using 20 compute cores. Hence, model outputs could be obtained very quickly in response to a crisis.

A Monte Carlo cost-benefit simulation model used the results from the ABM to estimate the cost-benefit of school closure in mitigating an influenza epidemic [31]. Table 1 lists the model's input parameters along with their distributions and sources. All costs and benefits were expressed in 2010 U.S. dollars. The model determined the net cost of implementing school closure using the following formula:(1)

where a positive net cost meant that implementing school closure resulted in a net cost to society and a negative cost meant that implementing school closure resulted in net cost savings to society.

Figures 3, 4, and 5 present the cost of disease, cost of school closure and total costs as a function of school closure duration and the R₀ of the epidemic for the state of Pennsylvania. As the R₀ increased, the total number of influenza cases increased, which resulted in higher costs of disease (Figure 3). Consistent with a previous study, school closure of 1 to 4 weeks had a very modest effect on the epidemic. However, an school closure of 8 weeks in duration significantly decreased the number of cases [7]. With an 8 week school closure policy, the median cost of disease was $323.6 million (95% range: $122.0 - $734.7 million) for R₀ = 1.2, $953.9 million (95% range: $370.5 - $2,290 million) for R₀ = 1.6, and $1,263 million (95% range: $384.4 - $2,451 million) for R₀ = 2.0. Varying the trigger by which schools decided to close (the number of symptomatic influenza detected in the school) had little effect on the cost of disease.

Figure 4 shows the cost of school closure with varying durations. Since the number of symptomatic cases detected in the school determined whether the school closed, varying the R₀ could potentially have changed the number of schools that closed during an epidemic. However, this effect was not substantial, as the cost of school closure remained fairly constant across the entire range of R₀ values explored. The median cost of school closure of 8 weeks was $21,277 million (95% range: $8,131 - $45,896 million) for R₀ = 1.2, $21,248 million (95% range: $7,989 - $44,989 million) for R₀ = 1.6, and $22,103 million (95% range: $7,973 - $45,697 million) for R₀ = 2.0. The cost of school closure remained fairly constant across different numbers of symptomatic cases used to trigger school closure.

Figure 5 shows the total costs (the sum of the costs of disease and school closure) for various values of school closure durations and R₀'s. It was notable that the cost of school closure contributed the vast majority of the total costs so that the cost of school closure primarily drove overall costs. The median total cost for an 8 week school closure policy was $21,600 million (95% values: $8,254 - $46,666 million) for R₀ = 1.2, $22,201 million (95% values: $8,360 - $47,279 million) for R₀ = 1.6, and $22,366 million (95% values: $8,445 - $48,696 million) for R₀ = 2.0.

The net cost for the various durations of school closure and R₀'s are shown in Figure 6. All of the scenarios explored exhibited net cost due to the substantially greater cost of the school closure versus the reduction in costs that resulted from mitigating the influenza epidemic. For school closures of 8 weeks duration, the median net cost was $21,028 million (95% values: $8,040 - $45,309 million) for R₀ = 1.2, $21,093 million (95% values: $7,936 - $44,656 million) for R₀ = 1.6, and $20,976 (95% values: $7,923 - $45,490 million) for R₀ = 2.0.

A potential factor to consider from the 2009 H1N1 epidemic is that the case fatality rate (CFR) was relatively low, especially compared with the estimates used in the preparedness planning for avian influenza [1]. To examine the potential sensitivity of the model to increased CFR, rates of 10 and 100 times that of the estimated CFR for 2009 H1N1 were explored. Figure 7 shows the net costs for varying durations of school closure with the 2009 H1N1 CFR as well as 10 and 100 times the original value. As expected, increases in CFR lower the net cost as the benefit of the school closure prevents mortality. This effect, however, is minor when compared to the cost of closing schools, and so the net costs do not drop significantly.

Table 2 shows the cost per influenza case averted for each of the scenarios explored. In the cases of R₀ = 1.2 and 1.6, the cost per case averted increased from 1 week to 2 weeks of school closures as a consequence of the mitigation not having been effective. For school closure of 2 weeks and longer, and 1 week or longer in the case of R₀ = 2.0, there was a decrease in the cost per case averted since longer school closures resulted in increased mitigation of the epidemic. The median cost per case averted for school closure of 8 weeks was $14,185 (95% values: $5,423 - $30,565) for R₀ = 1.2, $25,253 (95% values: $9501 - $53,461) for R₀ = 1.6, and $23,483 (95% values: $8,870 - $50,926) for R₀ = 2.0.

The cost per case averted varied in surprising ways depending on the trigger for closure. For example, it was shown in previous work simulating school closure for Allegheny County, that increasing the number of detected symptomatic cases needed to trigger a school closure could have a positive effect in reducing the number of total infections during an epidemic [7]. There appeared to be a dependence on timing the school closure mitigation so that it spanned the peak of the epidemic in order to gain the maximum mitigation benefit, which resulted in a decrease in the number of cases as the trigger parameter was increased. Once the optimal number of triggering cases was reached, any further increases in the trigger yielded decreasing benefit. In the cost per case averted, we saw a similar trend when R₀'s of 1.6 and 2.0 were simulated, which had optimal median costs at 20 trigger cases ($17,269) and 3 trigger cases ($22,714) respectively. For an epidemic with an R₀ of 1.2, allowing schools to close when there was one symptomatic case produced a low cost per case averted, with a median cost of $11,260 per case. In this instance, the epidemic curve was relatively broad which indicated a slower build up to the peak of the epidemic. Closing schools when there was one symptomatic case detected allowed mitigation to have a substantial effect prior to the influenza taking significant hold in the population. At a trigger of 3 cases there was an increase in the cost per case averted; the cost per case averted trended downward until 20 cases.

All computer models are simplifications of reality and cannot account for every possible factor or interaction [38,39]. Rather than dictate courses of action, models do provide valuable information to decision makers about possible scenarios and relationships. An influenza epidemic and the resulting circumstances may not necessarily conform to the data and assumptions in any given model. Our model drew from referenced sources and previously published models. The economic model focused on health care costs and productivity losses directly resulting from absenteeism and mortality and did not include other possible negative externalities (e.g., lack of school lunch programs during school closure). Additionally, wages are an often used but imperfect proxy for productivity lost. The economic model does not include an attempt to account for persons who may be able to work-from-home, which may reduce the productivity loss estimates of school closure but is unlikely to change to conclusions of this research.