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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

I. M. Longini Jr., M. E. Halloran, A. Nizam and Y. Yang, Containing pandemic influenza with antiviral agents,, Am. J. Epidem., 159 (2004), 623. doi: 10.1093/aje/kwh092. Google Scholar

[43]

I. M. Longini Jr., A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. T. Cummings and M. E. Halloran, Containing pandemic influenza at the source,, Science, 309 (2004), 623. Google Scholar

[44]

I. M. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. T. Cummings and M. E. Halloran, Containing pandemic influenza at the source,, Science, 309 (2005), 1083. doi: 10.1126/science.1115717. Google Scholar

[45]

A. C. Lowen, J. Steel, S. Murbareka and P. Palese, High temperature ($30\,^{\circ} C$) blocks aerosol but not contact transmission of influenza,, J. Virol., 82 (2008), 5650. doi: 10.1128/JVI.00325-08. Google Scholar

[46]

J. Ma and D. J. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease,, Bull. Math. Biol., 68 (2006), 679. doi: 10.1007/s11538-005-9047-7. Google Scholar

[47]

S. Merler, P. Poletti, M. Ajelli, B. Caprile and P. Manfredi, Coinfection can trigger multiple pandemic waves,, J. Theor. Biol., 254 (2008), 499. doi: 10.1016/j.jtbi.2008.06.004. Google Scholar

[48]

L. A. Meyers, Contact network epidemiology: Bond percolation applied to infectious disease prediction and control,, Bull. Am. Math. Soc., 44 (2007), 63. doi: 10.1090/S0273-0979-06-01148-7. Google Scholar

[49]

L. A. Meyers, B. Pourbohloul, M. E. J. Newman, D. M. Skowronski and R. C. Brunham, Network theory and SARS; predicting outbreak diversity,, J. Theor. Biol., 232 (2005), 71. doi: 10.1016/j.jtbi.2004.07.026. Google Scholar

[50]

J. C. Miller, B. Davoudi, R. Meza, A. C. Slim and B. Pourbohloul, Epidemics with general generation interval distribution,, J. Theor. Biol., 262 (2010), 107. doi: 10.1016/j.jtbi.2009.08.007. Google Scholar

[51]

S. M. Moghadas, C. S. Bowman, G. Röst and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza,, PLoS ONE, 3 (2008). Google Scholar

[52]

S. M. Moghadas, N. Pizzi, J. Wu and P. Yan, Managing public health crises: The role of models in pandemic preparedness,, Influenza and Other Respiratory Viruses, 3 (2008), 75. doi: 10.1111/j.1750-2659.2009.00081.x. Google Scholar

[53]

J. Mossong, N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikalajczyk, M. Massari, S. Salmoso, G. Scalia Tomba, J. Wallinga, J. Heijna, M. Sadkowska-Tadus, M. Rosinski and W. J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases,, PLoS Medicine, 5 (2008). Google Scholar

[54]

M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016128. Google Scholar

[55]

M. E. J. Newman, A. L. Barabási and D. J. Watts, "The Structure and Dynamics of Networks,", Princeton University Press, (2006). Google Scholar

[56]

R. Olinsky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics,, J. Math. Biol., 56 (2008), 827. doi: 10.1007/s00285-007-0140-4. Google Scholar

[57]

P. Palese, Influenza; Old and new threats,, Nature Med., 10 (2004). doi: 10.1038/nm1141. Google Scholar

[58]

Pan-infORM, Modelling an influenza pandemic: A guide for the perplexed,, Can. Med. Assoc. J., 181 (2009), 171. doi: 10.1503/cmaj.090885. Google Scholar

[59]

P. Poletti, B. Caprile, M. Ajelli, A. Puliese and S. Merler, Spontaneous behavioural changes in response to epidemics,, J. Theor. Biol., 260 (2009), 31. doi: 10.1016/j.jtbi.2009.04.029. Google Scholar

[60]

B. Pourbohloul, A. Ahued, B. Davoudi, R. Meza, L. A. Meyers, D. M. Skowronski, I. Villase\ nor, F. Galván, P. Cravioto, D. J. D. Earn, J. Dushoff, D. Fisman, W. J. Edmunds, N. Hupert, S. V. Scarpino, J. Trujillo, M. Lutzow, J. Morales, A. Contreras, C. Ch, Initial human transmission dynamics of the pandemic (H1N1) 2009 virus in North America,, Influenza and Other Respiratory Viruses, 3 (2009), 215. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

I. M. Longini Jr., M. E. Halloran, A. Nizam and Y. Yang, Containing pandemic influenza with antiviral agents,, Am. J. Epidem., 159 (2004), 623. doi: 10.1093/aje/kwh092. Google Scholar

[43]

I. M. Longini Jr., A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. T. Cummings and M. E. Halloran, Containing pandemic influenza at the source,, Science, 309 (2004), 623. Google Scholar

[44]

I. M. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. T. Cummings and M. E. Halloran, Containing pandemic influenza at the source,, Science, 309 (2005), 1083. doi: 10.1126/science.1115717. Google Scholar

[45]

A. C. Lowen, J. Steel, S. Murbareka and P. Palese, High temperature ($30\,^{\circ} C$) blocks aerosol but not contact transmission of influenza,, J. Virol., 82 (2008), 5650. doi: 10.1128/JVI.00325-08. Google Scholar

[46]

J. Ma and D. J. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease,, Bull. Math. Biol., 68 (2006), 679. doi: 10.1007/s11538-005-9047-7. Google Scholar

[47]

S. Merler, P. Poletti, M. Ajelli, B. Caprile and P. Manfredi, Coinfection can trigger multiple pandemic waves,, J. Theor. Biol., 254 (2008), 499. doi: 10.1016/j.jtbi.2008.06.004. Google Scholar

[48]

L. A. Meyers, Contact network epidemiology: Bond percolation applied to infectious disease prediction and control,, Bull. Am. Math. Soc., 44 (2007), 63. doi: 10.1090/S0273-0979-06-01148-7. Google Scholar

[49]

L. A. Meyers, B. Pourbohloul, M. E. J. Newman, D. M. Skowronski and R. C. Brunham, Network theory and SARS; predicting outbreak diversity,, J. Theor. Biol., 232 (2005), 71. doi: 10.1016/j.jtbi.2004.07.026. Google Scholar

[50]

J. C. Miller, B. Davoudi, R. Meza, A. C. Slim and B. Pourbohloul, Epidemics with general generation interval distribution,, J. Theor. Biol., 262 (2010), 107. doi: 10.1016/j.jtbi.2009.08.007. Google Scholar

[51]

S. M. Moghadas, C. S. Bowman, G. Röst and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza,, PLoS ONE, 3 (2008). Google Scholar

[52]

S. M. Moghadas, N. Pizzi, J. Wu and P. Yan, Managing public health crises: The role of models in pandemic preparedness,, Influenza and Other Respiratory Viruses, 3 (2008), 75. doi: 10.1111/j.1750-2659.2009.00081.x. Google Scholar

[53]

J. Mossong, N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikalajczyk, M. Massari, S. Salmoso, G. Scalia Tomba, J. Wallinga, J. Heijna, M. Sadkowska-Tadus, M. Rosinski and W. J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases,, PLoS Medicine, 5 (2008). Google Scholar

[54]

M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016128. Google Scholar

[55]

M. E. J. Newman, A. L. Barabási and D. J. Watts, "The Structure and Dynamics of Networks,", Princeton University Press, (2006). Google Scholar

[56]

R. Olinsky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics,, J. Math. Biol., 56 (2008), 827. doi: 10.1007/s00285-007-0140-4. Google Scholar

[57]

P. Palese, Influenza; Old and new threats,, Nature Med., 10 (2004). doi: 10.1038/nm1141. Google Scholar

[58]

Pan-infORM, Modelling an influenza pandemic: A guide for the perplexed,, Can. Med. Assoc. J., 181 (2009), 171. doi: 10.1503/cmaj.090885. Google Scholar

[59]

P. Poletti, B. Caprile, M. Ajelli, A. Puliese and S. Merler, Spontaneous behavioural changes in response to epidemics,, J. Theor. Biol., 260 (2009), 31. doi: 10.1016/j.jtbi.2009.04.029. Google Scholar

[60]

B. Pourbohloul, A. Ahued, B. Davoudi, R. Meza, L. A. Meyers, D. M. Skowronski, I. Villase\ nor, F. Galván, P. Cravioto, D. J. D. Earn, J. Dushoff, D. Fisman, W. J. Edmunds, N. Hupert, S. V. Scarpino, J. Trujillo, M. Lutzow, J. Morales, A. Contreras, C. Ch, Initial human transmission dynamics of the pandemic (H1N1) 2009 virus in North America,, Influenza and Other Respiratory Viruses, 3 (2009), 215. Google Scholar

[1]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159

[2]

Eunha Shim. Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1615-1634. doi: 10.3934/mbe.2013.10.1615

[3]

Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469

[4]

Diána H. Knipl, Gergely Röst. Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks. Mathematical Biosciences & Engineering, 2011, 8 (1) : 123-139. doi: 10.3934/mbe.2011.8.123

[5]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[6]

Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045

[7]

Eunha Shim. Prioritization of delayed vaccination for pandemic influenza. Mathematical Biosciences & Engineering, 2011, 8 (1) : 95-112. doi: 10.3934/mbe.2011.8.95

[8]

Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006

[9]

Fred Brauer. Age-of-infection and the final size relation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 681-690. doi: 10.3934/mbe.2008.5.681

[10]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[11]

Olivia Prosper, Omar Saucedo, Doria Thompson, Griselle Torres-Garcia, Xiaohong Wang, Carlos Castillo-Chavez. Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza. Mathematical Biosciences & Engineering, 2011, 8 (1) : 141-170. doi: 10.3934/mbe.2011.8.141

[12]

Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635

[13]

Christopher S. Bowman, Julien Arino, S.M. Moghadas. Evaluation of vaccination strategies during pandemic outbreaks. Mathematical Biosciences & Engineering, 2011, 8 (1) : 113-122. doi: 10.3934/mbe.2011.8.113

[14]

Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067

[15]

Sunmi Lee, Romarie Morales, Carlos Castillo-Chavez. A note on the use of influenza vaccination strategies when supply is limited. Mathematical Biosciences & Engineering, 2011, 8 (1) : 171-182. doi: 10.3934/mbe.2011.8.171

[16]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229

[17]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457

[18]

Marco Arieli Herrera-Valdez, Maytee Cruz-Aponte, Carlos Castillo-Chavez. Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different "waves" of A-H1N1pdm cases observed in México during 2009. Mathematical Biosciences & Engineering, 2011, 8 (1) : 21-48. doi: 10.3934/mbe.2011.8.21

[19]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999

[20]

Eunha Shim. A note on epidemic models with infective immigrants and vaccination. Mathematical Biosciences & Engineering, 2006, 3 (3) : 557-566. doi: 10.3934/mbe.2006.3.557

[Back to Top]