2011, 8(1): 1-20. doi: 10.3934/mbe.2011.8.1

Pandemic influenza: Modelling and public health perspectives

1. 

Department of Mathematics, University of Manitoba, Winnipeg, MB

2. 

Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

3. 

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

4. 

Department of Community Health Sciences, University of Manitoba, Winnipeg, MB, R3E 0W3, Canada

5. 

Public Health Agency of Canada and Dalla Lana School of Public Health, University of Toronto, Toronto, ON, M5T 3M7, Canada

6. 

Institute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba, Canada, R3B 1Y6

7. 

Institute for Biodiagnostics, National Research Council of Canada, Winnipeg, MB, R3B 1Y6, Canada

8. 

Department of Health Policy, Management and Evaluation, University of Toronto, Toronto, ON, M5T 3M6, Canada

9. 

Dalla Lana School of Public Health, University of Toronto, Toronto, ON, M5T 3M7, Canada

10. 

Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

11. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3

12. 

MITACS Centre for Disease Modelling, York University Institute for Health Research, Toronto, ON, M3J 1P3, Canada

13. 

Public Health Agency of Canada, Ottawa, ON, K1A 0K9, Canada

Received  April 2010 Revised  September 2010 Published  January 2011

We describe the application of mathematical models in the study of disease epidemics with particular focus on pandemic influenza. We outline the general mathematical approach and the complications arising from attempts to apply it for disease outbreak management in a real public health context.
Citation: Julien Arino, Chris Bauch, Fred Brauer, S. Michelle Driedger, Amy L. Greer, S.M. Moghadas, Nick J. Pizzi, Beate Sander, Ashleigh Tuite, P. van den Driessche, James Watmough, Jianhong Wu, Ping Yan. Pandemic influenza: Modelling and public health perspectives. Mathematical Biosciences & Engineering, 2011, 8 (1) : 1-20. doi: 10.3934/mbe.2011.8.1
References:
[1]

M. E. Alexander, S. M. Dietrich, Y. Hua and S. M. Moghadas, A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance,, J. Theor. Biol., 259 (2009), 253. doi: 10.1016/j.jtbi.2009.03.029.

[2]

N. Arinaminpathy and A. R. McLean, Antiviral treatment for the control of pandemic influenza: Some logistical constraints,, J. Roy. Soc. Interface, 5 (2008), 545. doi: 10.1098/rsif.2007.1152.

[3]

J. Arino, C. S. Bowman and S. M. Moghadas, Antiviral resistance during pandemic influenza: Implications for stockpiling and drug use,, BMC Infect. Dis., 9 (2009), 8. doi: 10.1186/1471-2334-9-8.

[4]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic,, J. Roy. Soc. Interface, 3 (2006), 453. doi: 10.1098/rsif.2006.0112.

[5]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment,, J. Theor. Biol., 253 (2008), 118. doi: 10.1016/j.jtbi.2008.02.026.

[6]

J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics,, Math. Biosc., 206 (2007), 46. doi: 10.1016/j.mbs.2005.09.002.

[7]

C. T. Bauch, J. Lloyd-Smith, M. Coffee and A. Galvani, Dynamically modeling SARS and respiratory EIDS: Past, present, future,, Epidemiology, 16 (2005), 791. doi: 10.1097/01.ede.0000181633.80269.4c.

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D. Bernoulli, Essai d'une nouvelle analyse de la mortalité causée par la petite verole,, Mem. Math. Phys. Acad. R. Sci. Paris, (1766), 1.

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S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV Model as an example,, Int. Stat. Rev., 62 (1994), 229.

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M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in U.S. cities,, Proc Nat. Acad Sci U.S.A, 104 (2007), 7588. doi: 10.1073/pnas.0611071104.

[11]

F. Brauer, Age of infection models and the final size relation,, Math. Biosc. & Eng., 5 (2008). doi: 10.3934/mbe.2008.5.681.

[12]

F. Brauer, Compartmental models in epidemiology,, in Mathematical Epidemiology (F. Brauer, (2008), 19.

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F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models,, Math. Biosc. & Eng., 7 (2010), 1. doi: 10.3934/mbe.2010.7.1.

[14]

P. Caley, D. J. Philp and K. McCracken, Quantifying social distancing arising from pandemic influenza,, J. Roy. Soc. Interface, 5 (2008), 631. doi: 10.1098/rsif.2007.1197.

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CDC, Drug susceptibility of swine-origin influenza A (H1N1) viruses, April 2009,, MMWR, 58 (2009), 433.

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CDC, Oseltamivir-resistant 2009 pandemic influenza A (H1N1) virus infection in two summer campers receiving prophylaxis - North Carolina, 2009,, MMWR, 58 (2009), 969.

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G. Chowell, P. W. Fenimore, M. Castillo-Garsow and C. Castillo - Chavez, SARS outbreaks in Ontario, Hong Kong, and Singapore: The role of diagnosis and isolation as a control mechanism,, J. Theor. Biol., 224 (2003), 1. doi: 10.1016/S0022-5193(03)00228-5.

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J. Dushoff, J. B. Plotkin, S. A. Levin and D. J. Earn, Dynamical resonance can account for seasonality of influenza epidemics,, Proc. Natl. Acad. Sci. USA, 101 (2004), 16915. doi: 10.1073/pnas.0407293101.

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W. J. Edmunds, G. F. Medley and D. J. Nokes, Evaluating the cost-effectiveness of vaccination programmes: A dynamic perspective,, Stat. Med., 18 (1999), 3263. doi: 10.1002/(SICI)1097-0258(19991215)18:23<3263::AID-SIM315>3.0.CO;2-3.

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J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: mathematical and computational explorations,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0003955.

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N. M. Ferguson, D. A. T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia,, Nature, 437 (2005), 209. doi: 10.1038/nature04017.

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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,, Proc. Nat. Acad. Sci. U.S.A., 103 (2006), 5935. doi: 10.1073/pnas.0601266103.

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

[33]

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M. E. Halloran, N. M. Ferguson, S. Eubank, I. M. Longini, D. A. Cummings, B. Lewis, S. Xu, C. Fraser, A. Vullikanti, T. C. Germann et al, Modeling targeted layered containment of an influenza pandemic in the United States,, Proc. Nat. Acad. Sci. U.S.A, 105 (2008), 4639.

[35]

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[36]

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[37]

K. Khan, J. Arino, W. Hu, P. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Lieuw, A. Chan and M. Gardam, Spread of a novel influenza A (H1N1) virus via global airline transportation,, New England J. Med., 361 (2009), 212. doi: 10.1056/NEJMc0904559.

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M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, G. Gopalakrisna, S. K. Chew, C. C. Tam, M. H. Samore, D. Fisman and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome,, Science, 300 (2003), 1966. doi: 10.1126/science.1086616.

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show all references

References:
[1]

M. E. Alexander, S. M. Dietrich, Y. Hua and S. M. Moghadas, A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance,, J. Theor. Biol., 259 (2009), 253. doi: 10.1016/j.jtbi.2009.03.029.

[2]

N. Arinaminpathy and A. R. McLean, Antiviral treatment for the control of pandemic influenza: Some logistical constraints,, J. Roy. Soc. Interface, 5 (2008), 545. doi: 10.1098/rsif.2007.1152.

[3]

J. Arino, C. S. Bowman and S. M. Moghadas, Antiviral resistance during pandemic influenza: Implications for stockpiling and drug use,, BMC Infect. Dis., 9 (2009), 8. doi: 10.1186/1471-2334-9-8.

[4]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic,, J. Roy. Soc. Interface, 3 (2006), 453. doi: 10.1098/rsif.2006.0112.

[5]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment,, J. Theor. Biol., 253 (2008), 118. doi: 10.1016/j.jtbi.2008.02.026.

[6]

J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics,, Math. Biosc., 206 (2007), 46. doi: 10.1016/j.mbs.2005.09.002.

[7]

C. T. Bauch, J. Lloyd-Smith, M. Coffee and A. Galvani, Dynamically modeling SARS and respiratory EIDS: Past, present, future,, Epidemiology, 16 (2005), 791. doi: 10.1097/01.ede.0000181633.80269.4c.

[8]

D. Bernoulli, Essai d'une nouvelle analyse de la mortalité causée par la petite verole,, Mem. Math. Phys. Acad. R. Sci. Paris, (1766), 1.

[9]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV Model as an example,, Int. Stat. Rev., 62 (1994), 229.

[10]

M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in U.S. cities,, Proc Nat. Acad Sci U.S.A, 104 (2007), 7588. doi: 10.1073/pnas.0611071104.

[11]

F. Brauer, Age of infection models and the final size relation,, Math. Biosc. & Eng., 5 (2008). doi: 10.3934/mbe.2008.5.681.

[12]

F. Brauer, Compartmental models in epidemiology,, in Mathematical Epidemiology (F. Brauer, (2008), 19.

[13]

F. Brauer, C. Castillo-Chavez and Z. Feng, Discrete epidemic models,, Math. Biosc. & Eng., 7 (2010), 1. doi: 10.3934/mbe.2010.7.1.

[14]

P. Caley, D. J. Philp and K. McCracken, Quantifying social distancing arising from pandemic influenza,, J. Roy. Soc. Interface, 5 (2008), 631. doi: 10.1098/rsif.2007.1197.

[15]

CDC, Drug susceptibility of swine-origin influenza A (H1N1) viruses, April 2009,, MMWR, 58 (2009), 433.

[16]

CDC, Oseltamivir-resistant 2009 pandemic influenza A (H1N1) virus infection in two summer campers receiving prophylaxis - North Carolina, 2009,, MMWR, 58 (2009), 969.

[17]

G. Chowell, P. W. Fenimore, M. Castillo-Garsow and C. Castillo - Chavez, SARS outbreaks in Ontario, Hong Kong, and Singapore: The role of diagnosis and isolation as a control mechanism,, J. Theor. Biol., 224 (2003), 1. doi: 10.1016/S0022-5193(03)00228-5.

[18]

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

[19]

N. J. Cox, S. E. Tamblyn and T. Tam, Influenza pandemic planning,, Vaccine, 21 (2003), 1801.

[20]

V. T. Covello, Communicating right to know information on chemical risks,, Environ. Sci. Technol., 23 (1989), 1444. doi: 10.1021/es00070a002.

[21]

T. Day, A. Park, N. Madras, A. B. Gumel and J. Wu, When is quarantine a useful control strategy for emerging infectious diseases?,, Am J Epidemiol., 163 (2006), 479. doi: 10.1093/aje/kwj056.

[22]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases,", Wiley, (2000).

[23]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models,, J. Roy. Soc. Interface, 7 (2010), 873. doi: 10.1098/rsif.2009.0386.

[24]

J. Dushoff, J. B. Plotkin, S. A. Levin and D. J. Earn, Dynamical resonance can account for seasonality of influenza epidemics,, Proc. Natl. Acad. Sci. USA, 101 (2004), 16915. doi: 10.1073/pnas.0407293101.

[25]

W. J. Edmunds, G. F. Medley and D. J. Nokes, Evaluating the cost-effectiveness of vaccination programmes: A dynamic perspective,, Stat. Med., 18 (1999), 3263. doi: 10.1002/(SICI)1097-0258(19991215)18:23<3263::AID-SIM315>3.0.CO;2-3.

[26]

J. M. Epstein, J. Parker, D. Cummings and R. A. Hammond, Coupled contagion dynamics of fear and disease: mathematical and computational explorations,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0003955.

[27]

N. M. Ferguson, D. A. T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia,, Nature, 437 (2005), 209. doi: 10.1038/nature04017.

[28]

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.

[29]

C. Fraser, S. Riley, R. M. Anderson and N. M. Ferguson, Factors that make an infectious disease outbreak controllable,, Proc. Nat. Acad. Sci. USA, 101 (2004), 6146. doi: 10.1073/pnas.0307506101.

[30]

C. Fraser, C. A. Donnelly, S. Cauchemez, W. P. Hanage, M. D. Van Kerkhove, T. D. Hollingsworth, J. Griffin, R. F. Baggaley, H. E. Jenkins, E. J. Lyons, T. Jombart, W. R. Hinsley, N. C. Grassly, F. Balloux, A. C. Ghani and N. M. Ferguson, Pandemic potential of a strain of influenza A (H1N1): Early findings,, Science, 324 (2009), 1557. doi: 10.1126/science.1176062.

[31]

T. C. Germann, K. Kadau, I. M. Longini and C. A. Macken, Mitigation strategies for pandemic influenza in the United States,, Proc. Nat. Acad. Sci. U.S.A., 103 (2006), 5935. doi: 10.1073/pnas.0601266103.

[32]

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

[33]

A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. E. Alexander, S. Ardal, J. Wu and B. M. Sahai, Modeling strategies for controlling SARS outbreaks in Toronto, Hong Kong, Singapore and Beijing,, Proc. Roy. Soc. London, 271 (2004), 2223. doi: 10.1098/rspb.2004.2800.

[34]

M. E. Halloran, N. M. Ferguson, S. Eubank, I. M. Longini, D. A. Cummings, B. Lewis, S. Xu, C. Fraser, A. Vullikanti, T. C. Germann et al, Modeling targeted layered containment of an influenza pandemic in the United States,, Proc. Nat. Acad. Sci. U.S.A, 105 (2008), 4639.

[35]

E. Hansen, T. Day, J. Arino, J. Wu and S. M. Moghadas, Strategies for use of oseltamivir and zanamivir during pandemic outbreaks,, Can. J. Infect. Dis. Med. Microb., (2010).

[36]

W. O. Kermack and A. G McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Royal Soc. London, 115 (1927), 700. doi: 10.1098/rspa.1927.0118.

[37]

K. Khan, J. Arino, W. Hu, P. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Lieuw, A. Chan and M. Gardam, Spread of a novel influenza A (H1N1) virus via global airline transportation,, New England J. Med., 361 (2009), 212. doi: 10.1056/NEJMc0904559.

[38]

Q. M. Le, H. F. Wertheim, N. D. Tran, H. R. van Doorn, T. H. Nguyen and P. Hornby, Vietnam H1N1 Investigation Team, A community cluster of oseltamivir - resistant cases of 2009 H1N1 influenza,, New England. J. Medicine, 362 (2010), 86. doi: 10.1056/NEJMc0910448.

[39]

M. Lipsitch, T. Cohen, M. Murray and B. R. Levin, Antiviral resistance and the control of pandemic influenza,, PLoS Medicine, 4 (2007).

[40]

M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, G. Gopalakrisna, S. K. Chew, C. C. Tam, M. H. Samore, D. Fisman and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome,, Science, 300 (2003), 1966. doi: 10.1126/science.1086616.

[41]

J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp and W. M. Getz, Superspreading and the effect of individual variation on disease emergence,, Nature, 438 (2005), 355. doi: 10.1038/nature04153.

[42]

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