2012, 9(2): 413-430. doi: 10.3934/mbe.2012.9.413

The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States, United States, United States

2. 

Department of Forestry and Natural Resources, Purdue University, West Lafayette, IN 47907, United States

Received  June 2011 Revised  November 2011 Published  March 2012

When a new pandemic influenza strain has been identified, mass-production of vaccines can take several months, and antiviral drugs are expensive and usually in short supply. Social distancing measures, such as school closures, thus seem an attractive means to mitigate disease spread. However, the transmission of influenza is seasonal in nature, and as has been noted in previous studies, a decrease in the average transmission rate in a seasonal disease model may result in a larger final size. In the studies presented here, we analyze a hypothetical pandemic using a SIR epidemic model with time- and age-dependent transmission rates; using this model we assess and quantify, for the first time, the the effect of the timing and length of widespread school closures on influenza pandemic final size and average peak time.
    We find that the effect on pandemic progression strongly depends on the timing of the start of the school closure. For instance, we determine that school closures during a late spring wave of an epidemic can cause a pandemic to become up to 20% larger, but have the advantage that the average time of the peak is shifted by up to two months, possibly allowing enough time for development of vaccines to mitigate the larger size of the epidemic. Our studies thus suggest that when heterogeneity in transmission is a significant factor, decisions of public health policy will be particularly important as to how control measures such as school closures should be implemented.
Citation: Sherry Towers, Katia Vogt Geisse, Chia-Chun Tsai, Qing Han, Zhilan Feng. The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 413-430. doi: 10.3934/mbe.2012.9.413
References:
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W. J. Alonso, C. Viboud, L. Simonsen, E. W. Hirano, L. Z. Daufenbach and M. A. Miller, Seasonality of influenza in Brazil: A traveling wave from the Amazon to the Subtropics,, Am. J. Epidemiol., 165 (2007), 1434. doi: 10.1093/aje/kwm012. Google Scholar

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[3]

N. Bacaër and E. H. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. Google Scholar

[4]

N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality,, Bulletin of Mathematical Biology, 71 (2009), 1954. doi: 10.1007/s11538-009-9433-7. Google Scholar

[5]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9. Google Scholar

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N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar

[7]

S. Cauchemez, N. M. Ferguson, C. Wachtel, A. Tegnell, G. Saour, B. Duncan and A. Nicoll, Closure of schools during an influenza pandemic,, Lancet. Infect. Dis., 9 (2009), 473. doi: 10.1016/S1473-3099(09)70176-8. Google Scholar

[8]

S. Cauchemez, A.-J. Valleron, P.-Y. Boëlle, A. Flahault and N. M. Ferguson, Estimating the impact of school closure on influenza transmission from Sentinel data,, Nature, 452 (2008), 750. doi: 10.1038/nature06732. Google Scholar

[9]

V. Colizza, A. Barrat, M. Barthelemy, A.-J. Valleron and A. Vespignani, Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions,, PLoS Med., 4 (2007). doi: 10.1371/journal.pmed.0040013. Google Scholar

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S. Y. Del Valle, P. D. Stroud and S. M. Mniszewski, Dynamic contact patterns and social structure,, in, (2009), 201. Google Scholar

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S. F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases,, Emerg. Infec. Dis., 7 (2001), 369. Google Scholar

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J. Dushoff, J. B. Plotkin, C. Viboud, D. J. Earn and L. Simonsen, Mortality due to influenza in the United States,, American Journal of Epidemiology, 163 (2006), 181. Google Scholar

[13]

Z. Feng, S. Towers and Y. Yang, Modeling the effects of vaccination and treatment on pandemic influenza,, American Association of Pharmaceutical Science Journal, 13 (2011), 427. Google Scholar

[14]

N. M. Ferguson, D. A. T. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic,, Nature, 442 (2006), 448. doi: 10.1038/nature04795. Google Scholar

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N. M. Ferguson, A .P. Galvani and R. M. Bush, Ecological and immunological determinants of influenza evolution,, Nature, 422 (2003), 428. doi: 10.1038/nature01509. Google Scholar

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C. Fraser, et al., Pandemic potential of a strain of influenza A(H1N1): Early findings,, Science, 324 (2009), 1557. doi: 10.1126/science.1176062. Google Scholar

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T. C. Germann, K. Kadau, I. M. Longini and C. A. Macken, Mitigation strategies for pandemic influenza in the United States,, PNAS, 103 (2006), 5935. doi: 10.1073/pnas.0601266103. Google Scholar

[18]

R. J. Glass, L. M. Glass, W. E. Beyeler and J. H. Min, Targeted social distancing design for pandemic influenza,, Emerg. Infect. Dis., 12 (2006), 1671. doi: 10.3201/eid1211.060255. Google Scholar

[19]

M. Z. Gojovic, B. Sander, D. Fisman, M. D. Krahn and C. T. Bauch, Modelling mitigation strategies for pandemic (H1N1) 2009,, CMAJ, 181 (2009), 673. doi: 10.1503/cmaj.091641. Google Scholar

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N. Halder, J. Kelso and G. Milne, Analysis of the effectiveness of interventions used during the 2009 A/H1N1 influenza pandemic,, BMC Public Health, 10 (2010). doi: 10.1186/1471-2458-10-168. Google Scholar

[21]

S. D. Holmberg, C. M. Layton, G. S. Ghneim and D. K. Wagener, State plans for containment of pandemic influenza,, Emerg. Infect. Dis., 12 (2006), 1414. doi: 10.3201/eid1209.060369. Google Scholar

[22]

L. Kahn, Pandemic influenza school closure policies,, Emerg. Infect. Dis., 13 (2007), 344. doi: 10.3201/eid1302.061109. Google Scholar

[23]

J. K. Kelso, G. J. Milne and H. Kelly, Simulation suggests that rapid activation of social distancing can arrest epidemic development due to a novel strain of influenza,, BMC Public Health, 9 (2009). Google Scholar

[24]

B. Y. Lee, S. T. Brown, P. Cooley, M. A. Potter, W. D. Wheaton, R. E. Voorhees, S. Stebbins, J. J. Grefenstette, S. M. Zimmer, R. K. Zimmerman, T. M. Assi, R. R. Bailey, D. K. Wagener and D. S. Burke, Simulating school closure strategies to mitigate an influenza epidemic,, Journal of Public Health Management and Practice, 16 (2010), 252. Google Scholar

[25]

E. Lofgren, N.H . Fefferman, Y. N. Naumov, J. Gorski and E. N. Naumova, Influenza seasonality: Underlying causes and modeling theories,, J. Virol., 81 (2007), 5429. doi: 10.1128/JVI.01680-06. Google Scholar

[26]

A. C. Lowen, S. Mubareka, J. Steel and P. Palese, Influenza virus transmission is dependent on relative humidity and temperature,, PLoS Pathogens, 4 (2007), 151. doi: 10.1371/journal.ppat.0030151. Google Scholar

[27]

J. Medlock and A. P. Galvani, Optimizing influenza vaccine distribution,, Science, 325 (2009), 1705. doi: 10.1126/science.1175570. Google Scholar

[28]

G. J. Milne, J. K. Kelso, H. A. Kelly, S. T. Huband and J. McVernon, A small community model for the transmission of infectious diseases: Comparison of school closure as an intervention in individual-based models of an influenza pandemic,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0004005. Google Scholar

[29]

J. Mossong, N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikolajczyk, M. Massari, S. Salmaso, G. Scalia Tomba, J. Wallinga, J. Heijne, M. Sadkowska-Todys, M. Rosinska and W. J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases,, PLoS Med, 5 (2008). doi: 10.1371/journal.pmed.0050074. Google Scholar

[30]

B. Sander, A. Nizam, L. P. Garrison, M. J. Postma, M. E. Halloran and I. M. Longini, Economic evaluation of influenza pandemic mitigation strategies in the United States using a stochastic microsimulation transmission model,, Value in Health, 12 (2009). doi: 10.1111/j.1524-4733.2008.00437.x. Google Scholar

[31]

S. Towers and Z. Feng, Pandemic H1N1 influenza: Predicting the course of pandemic and assessing the efficacy of the planned vaccination programme in the United States,, Eurosurveillance, 14 (2009). Google Scholar

[32]

S. Towers, K. Vogt Geisse, Y. Zheng and Z. Feng, Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model,, Journal of Theoretical Biology, 289 (2011), 259. doi: 10.1016/j.jtbi.2011.08.011. Google Scholar

[33]

, U.S. 2000 Census., Available from: \url{http://www.census.gov}., (). Google Scholar

[34]

W. Wang and X.-Q. Zhang, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar

[35]

World Health Organization, Nonpharmaceutical interventions for pandemic influenza, national and community measures,, Emerg. Infec. Dis., 12 (2006), 88. Google Scholar

[36]

Y. Yang, Jonathan D. Sugimoto, M. E. Halloran, N. E. Basta, D. L. Chao, L. Matrajt, G. Potter, E. Kenah and I. M. Longini Jr., The transmissibility and control of pandemic influenza A(H1N1) virus,, Science, 326 (2009), 729. doi: 10.1126/science.1177373. Google Scholar

show all references

References:
[1]

W. J. Alonso, C. Viboud, L. Simonsen, E. W. Hirano, L. Z. Daufenbach and M. A. Miller, Seasonality of influenza in Brazil: A traveling wave from the Amazon to the Subtropics,, Am. J. Epidemiol., 165 (2007), 1434. doi: 10.1093/aje/kwm012. Google Scholar

[2]

R. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1991). Google Scholar

[3]

N. Bacaër and E. H. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,, J. Math. Biol., 62 (2011), 741. Google Scholar

[4]

N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality,, Bulletin of Mathematical Biology, 71 (2009), 1954. doi: 10.1007/s11538-009-9433-7. Google Scholar

[5]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9. Google Scholar

[6]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar

[7]

S. Cauchemez, N. M. Ferguson, C. Wachtel, A. Tegnell, G. Saour, B. Duncan and A. Nicoll, Closure of schools during an influenza pandemic,, Lancet. Infect. Dis., 9 (2009), 473. doi: 10.1016/S1473-3099(09)70176-8. Google Scholar

[8]

S. Cauchemez, A.-J. Valleron, P.-Y. Boëlle, A. Flahault and N. M. Ferguson, Estimating the impact of school closure on influenza transmission from Sentinel data,, Nature, 452 (2008), 750. doi: 10.1038/nature06732. Google Scholar

[9]

V. Colizza, A. Barrat, M. Barthelemy, A.-J. Valleron and A. Vespignani, Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions,, PLoS Med., 4 (2007). doi: 10.1371/journal.pmed.0040013. Google Scholar

[10]

S. Y. Del Valle, P. D. Stroud and S. M. Mniszewski, Dynamic contact patterns and social structure,, in, (2009), 201. Google Scholar

[11]

S. F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases,, Emerg. Infec. Dis., 7 (2001), 369. Google Scholar

[12]

J. Dushoff, J. B. Plotkin, C. Viboud, D. J. Earn and L. Simonsen, Mortality due to influenza in the United States,, American Journal of Epidemiology, 163 (2006), 181. Google Scholar

[13]

Z. Feng, S. Towers and Y. Yang, Modeling the effects of vaccination and treatment on pandemic influenza,, American Association of Pharmaceutical Science Journal, 13 (2011), 427. Google Scholar

[14]

N. M. Ferguson, D. A. T. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic,, Nature, 442 (2006), 448. doi: 10.1038/nature04795. Google Scholar

[15]

N. M. Ferguson, A .P. Galvani and R. M. Bush, Ecological and immunological determinants of influenza evolution,, Nature, 422 (2003), 428. doi: 10.1038/nature01509. Google Scholar

[16]

C. Fraser, et al., Pandemic potential of a strain of influenza A(H1N1): Early findings,, Science, 324 (2009), 1557. doi: 10.1126/science.1176062. Google Scholar

[17]

T. C. Germann, K. Kadau, I. M. Longini and C. A. Macken, Mitigation strategies for pandemic influenza in the United States,, PNAS, 103 (2006), 5935. doi: 10.1073/pnas.0601266103. Google Scholar

[18]

R. J. Glass, L. M. Glass, W. E. Beyeler and J. H. Min, Targeted social distancing design for pandemic influenza,, Emerg. Infect. Dis., 12 (2006), 1671. doi: 10.3201/eid1211.060255. Google Scholar

[19]

M. Z. Gojovic, B. Sander, D. Fisman, M. D. Krahn and C. T. Bauch, Modelling mitigation strategies for pandemic (H1N1) 2009,, CMAJ, 181 (2009), 673. doi: 10.1503/cmaj.091641. Google Scholar

[20]

N. Halder, J. Kelso and G. Milne, Analysis of the effectiveness of interventions used during the 2009 A/H1N1 influenza pandemic,, BMC Public Health, 10 (2010). doi: 10.1186/1471-2458-10-168. Google Scholar

[21]

S. D. Holmberg, C. M. Layton, G. S. Ghneim and D. K. Wagener, State plans for containment of pandemic influenza,, Emerg. Infect. Dis., 12 (2006), 1414. doi: 10.3201/eid1209.060369. Google Scholar

[22]

L. Kahn, Pandemic influenza school closure policies,, Emerg. Infect. Dis., 13 (2007), 344. doi: 10.3201/eid1302.061109. Google Scholar

[23]

J. K. Kelso, G. J. Milne and H. Kelly, Simulation suggests that rapid activation of social distancing can arrest epidemic development due to a novel strain of influenza,, BMC Public Health, 9 (2009). Google Scholar

[24]

B. Y. Lee, S. T. Brown, P. Cooley, M. A. Potter, W. D. Wheaton, R. E. Voorhees, S. Stebbins, J. J. Grefenstette, S. M. Zimmer, R. K. Zimmerman, T. M. Assi, R. R. Bailey, D. K. Wagener and D. S. Burke, Simulating school closure strategies to mitigate an influenza epidemic,, Journal of Public Health Management and Practice, 16 (2010), 252. Google Scholar

[25]

E. Lofgren, N.H . Fefferman, Y. N. Naumov, J. Gorski and E. N. Naumova, Influenza seasonality: Underlying causes and modeling theories,, J. Virol., 81 (2007), 5429. doi: 10.1128/JVI.01680-06. Google Scholar

[26]

A. C. Lowen, S. Mubareka, J. Steel and P. Palese, Influenza virus transmission is dependent on relative humidity and temperature,, PLoS Pathogens, 4 (2007), 151. doi: 10.1371/journal.ppat.0030151. Google Scholar

[27]

J. Medlock and A. P. Galvani, Optimizing influenza vaccine distribution,, Science, 325 (2009), 1705. doi: 10.1126/science.1175570. Google Scholar

[28]

G. J. Milne, J. K. Kelso, H. A. Kelly, S. T. Huband and J. McVernon, A small community model for the transmission of infectious diseases: Comparison of school closure as an intervention in individual-based models of an influenza pandemic,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0004005. Google Scholar

[29]

J. Mossong, N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikolajczyk, M. Massari, S. Salmaso, G. Scalia Tomba, J. Wallinga, J. Heijne, M. Sadkowska-Todys, M. Rosinska and W. J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases,, PLoS Med, 5 (2008). doi: 10.1371/journal.pmed.0050074. Google Scholar

[30]

B. Sander, A. Nizam, L. P. Garrison, M. J. Postma, M. E. Halloran and I. M. Longini, Economic evaluation of influenza pandemic mitigation strategies in the United States using a stochastic microsimulation transmission model,, Value in Health, 12 (2009). doi: 10.1111/j.1524-4733.2008.00437.x. Google Scholar

[31]

S. Towers and Z. Feng, Pandemic H1N1 influenza: Predicting the course of pandemic and assessing the efficacy of the planned vaccination programme in the United States,, Eurosurveillance, 14 (2009). Google Scholar

[32]

S. Towers, K. Vogt Geisse, Y. Zheng and Z. Feng, Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model,, Journal of Theoretical Biology, 289 (2011), 259. doi: 10.1016/j.jtbi.2011.08.011. Google Scholar

[33]

, U.S. 2000 Census., Available from: \url{http://www.census.gov}., (). Google Scholar

[34]

W. Wang and X.-Q. Zhang, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar

[35]

World Health Organization, Nonpharmaceutical interventions for pandemic influenza, national and community measures,, Emerg. Infec. Dis., 12 (2006), 88. Google Scholar

[36]

Y. Yang, Jonathan D. Sugimoto, M. E. Halloran, N. E. Basta, D. L. Chao, L. Matrajt, G. Potter, E. Kenah and I. M. Longini Jr., The transmissibility and control of pandemic influenza A(H1N1) virus,, Science, 326 (2009), 729. doi: 10.1126/science.1177373. Google Scholar

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