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Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks
Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza
1.  Department of Mathematics, University of Florida, Gainesville, FL 32611, United States 
2.  Department of Mathematics, Texas A&M University, College Station, TX 77843, United States 
3.  Department of Mathematics, Spelman College, Atlanta, GA 30314, United States 
4.  School of Human Evolution and Social Change, Mathematical, Computational and Modeling Science Center, Arizona State University, Tempe, AZ 85287, United States, United States 
5.  Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287 
References:
[1] 
L. Altman, "Many Swine Flu Cases Have no Fever," New York Times, 2009. 
[2] 
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. 
[3] 
I. G. Barr, J. McCauley, N. Cox, R. Daniels, O. G. Engelhardt, K. Fukuda, G. Grohmann, A. Hay, A. Kelso, A. Klimov, T. Odagiri, D. Smith, C. Russell, M. Tashiro, R. Webby, J. Wood, Z. Ye and W. Zhang, Epidemiological, antigenic and genetic characteristics of seasonal influenzaA(H1N1), A(H3N2) and B influenza viruses: Basis for the WHO recommendation on the composition of influenza vaccines for use in the 2009–2010 Northern Hemisphere season, Vaccine, 28 (2010), 11561167. doi: 10.1016/j.vaccine.2009.11.043. 
[4] 
H. Behnke, Optimal control of deterministic epidemics, Optimal Control Application Methods, 21 (2000), 269285. doi: 10.1002/oca.678. 
[5] 
W. I. B. Beveridge, "Influenza: The Last Great Plague. An Unfinished Story of Discovery," Prodist, 1977. 
[6] 
F. Brauer, Z. Feng and C. CastilloChavez, Discrete epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 115. doi: 10.3934/mbe.2010.7.1. 
[7] 
C. CastilloChavez, H. Hethcote, V. Andreason, S. A. Levin and W. M. Liu, Crossimmunity in the dynamics of homogeneous and heterogeneous populations, Mathematical Ecology, (1988), 303316. 
[8] 
Centers for Disease Control and Prevention (CDC), Key facts about seasonal influenza,, \url{http://www.cdc.gov/flu/keyfacts.htm}., (). 
[9] 
Centers for Disease Control and Prevention (CDC), Monitoring influenza activity, including 2009 H1N1, (2009), Monitoring Influenza Activity, Including 2009 h1n1. 
[10] 
Centers for Disease Control and Prevention (CDC), Serum crossreactive antibody response to a novel influenza A(H1N1) virus after vaccination with seasonal influenza vaccine, MMWR Morb Mortal Wkly Rep, 58 (2009), 521524. 
[11] 
G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France, and Australia: Transmission an prospects for control, Epidem. Infect., 136 (2008), 852864. doi: 10.1017/S0950268807009144. 
[12] 
G. Chowell, S. M. Bertozzi, M. A. Colchero, H. LopezGatell, C. AlpucheAranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674679. doi: 10.1056/NEJMoa0904023. 
[13] 
Brian Coburn, "Multispecies Influenza Models with Recombination," Ph.D thesis, University of Miami in Coral Gables,FL, 2009. 
[14] 
R. Couch and J. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (2002), 529549. doi: 10.1146/annurev.mi.37.100183.002525. 
[15] 
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," John Wiley & Sons, Ltd., Chichester, 2000. 
[16] 
D. J. D. Earn, J. Dushoff and S. A. Levin, Ecology and evolution of the flu, Trends Ecol. Evol., 17 (2002), 334340. doi: 10.1016/S01695347(02)025028. 
[17] 
S. EchevarríaZuno, J. M. MejíaAranguré, A. V. MarObeso, C. GrajalesMuñiz, E. RoblesPérez, M. GonzálezLeón, M. C. OrtegaAlvarez, C. GonzalezBonilla, R. A. RascónPacheco and V. H. BorjaAburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 20722079. doi: 10.1016/S01406736(09)61638X. 
[18] 
A. EstevesJaramillo, S. B. Omer and E. GonzalezDiaz, Acceptance of a vaccine against novel influenza A (H1N1) virus among health care workers in two major cities in Mexico, Archives of Medical Research, 40 (2009), 705711. doi: 10.1016/j.arcmed.2010.01.004. 
[19] 
W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," SpringerVerlag, New York, 1994. 
[20] 
FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010),, \url{http://www.flu.gov/individualfamily/vaccination/supply.html}., (). 
[21] 
M. A. HerreraValdez, M. CruzAponte and C. CastilloChavez, Multiple waves for the same pandemic: Local transportation and social distancing explain the dynamics of the A/H1N1 epidemic during 2009 in Mexico, (2010). 
[22] 
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a twostrain tuberculosis model, Discrete and Continuous Dynamical SystemsSeries B, 2 (2002), 473482. doi: 10.3934/dcdsb.2002.2.473. 
[23] 
P. Y. Lee, D. B. Matchar, D. A. Clements, J. Huber, J. D. Hamilton and E. D. Peterson, Economic analysis of influenza vaccination and antiviral treatment for healthy working adults, Ann. Intern. Med., 137 (2002), 225231. 
[24] 
S. Lee, G. Chowell and C. CastilloChavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136150. doi: 10.1016/j.jtbi.2010.04.003. 
[25] 
S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, 2007. 
[26]  
[27] 
H. Nishiura, C. CastilloChavez, M. Safan and G. Chowell, Transmission potential of the new Influenza A(H1N1) virus and its agespecificity in Japan, Eurosurveillance, 14 (2009), 14. 
[28] 
M. Nuno, G. Chowell and A. B. Gumel, Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, Journal of The Royal Society Interface, 4 (2007), 505521. doi: 10.1098/rsif.2006.0186. 
[29] 
J. Plotkin, J. Dushoff and S. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 62636268. doi: 10.1073/pnas.082110799. 
[30] 
L. S. Pontryagin, R. V. Boltyanski, R. V. Gamkrelidge and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," John Wiley and Sons, N.Y., 1962. 
[31] 
Prevent Influenza Now! Sponsored by the National Influenza Vaccine Summit, Influenza vaccine availability tracking system (IVATS),, \url{http://www.preventinfluenza.org/ivats/}., (). 
[32] 
C. E. Shoichet, Mexico still waiting for most swine flu vaccines, (2010), http://www.boston.com/business/articles/2010/01/13/mexico_still_waiting_for_most_swine _flu_vaccines. 
[33] 
E. Spackman, D. Stallknecht, R. Slemons, K. Winker, D. L. Suarez, M. Scott and D. E. Swayne, Phylogenetic analyses of type A influenza genes in natural reservoir species in North America reveals genetic variation, Virus research, 114 (2005), 89100. doi: 10.1016/j.virusres.2005.05.013. 
[34] 
R. Stengel, Optimal control and estimation,, http://www.princeton.edu/ stengel/MAE546.html., (). 
[35] 
T. Suess, U. Buchholz, S. Dupke, R. Grunow, M. an der Heiden, A. Heider, B. Biere, B. Schweiger, W. Haas and G. Krause, Shedding and transmission of novel influenza virus A/H1N1 infection in households—Germany, 2009, American Journal of Epidemiology, 171 (2010), 11571164. doi: 10.1093/aje/kwq071. 
[36] 
J. K. Taubenberger, D. M. Morens, 1918 influenza: the mother of all pandemics, Emerging Infectious Diseases, (2009), http://www.cdc.gov/ncidod/EID/vol12no01/050979.htm. 
[37] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[38] 
World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 20012002 season, Wkly. Epidemiol. Rec., 76 (2001), 5861. 
show all references
References:
[1] 
L. Altman, "Many Swine Flu Cases Have no Fever," New York Times, 2009. 
[2] 
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. 
[3] 
I. G. Barr, J. McCauley, N. Cox, R. Daniels, O. G. Engelhardt, K. Fukuda, G. Grohmann, A. Hay, A. Kelso, A. Klimov, T. Odagiri, D. Smith, C. Russell, M. Tashiro, R. Webby, J. Wood, Z. Ye and W. Zhang, Epidemiological, antigenic and genetic characteristics of seasonal influenzaA(H1N1), A(H3N2) and B influenza viruses: Basis for the WHO recommendation on the composition of influenza vaccines for use in the 2009–2010 Northern Hemisphere season, Vaccine, 28 (2010), 11561167. doi: 10.1016/j.vaccine.2009.11.043. 
[4] 
H. Behnke, Optimal control of deterministic epidemics, Optimal Control Application Methods, 21 (2000), 269285. doi: 10.1002/oca.678. 
[5] 
W. I. B. Beveridge, "Influenza: The Last Great Plague. An Unfinished Story of Discovery," Prodist, 1977. 
[6] 
F. Brauer, Z. Feng and C. CastilloChavez, Discrete epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 115. doi: 10.3934/mbe.2010.7.1. 
[7] 
C. CastilloChavez, H. Hethcote, V. Andreason, S. A. Levin and W. M. Liu, Crossimmunity in the dynamics of homogeneous and heterogeneous populations, Mathematical Ecology, (1988), 303316. 
[8] 
Centers for Disease Control and Prevention (CDC), Key facts about seasonal influenza,, \url{http://www.cdc.gov/flu/keyfacts.htm}., (). 
[9] 
Centers for Disease Control and Prevention (CDC), Monitoring influenza activity, including 2009 H1N1, (2009), Monitoring Influenza Activity, Including 2009 h1n1. 
[10] 
Centers for Disease Control and Prevention (CDC), Serum crossreactive antibody response to a novel influenza A(H1N1) virus after vaccination with seasonal influenza vaccine, MMWR Morb Mortal Wkly Rep, 58 (2009), 521524. 
[11] 
G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France, and Australia: Transmission an prospects for control, Epidem. Infect., 136 (2008), 852864. doi: 10.1017/S0950268807009144. 
[12] 
G. Chowell, S. M. Bertozzi, M. A. Colchero, H. LopezGatell, C. AlpucheAranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674679. doi: 10.1056/NEJMoa0904023. 
[13] 
Brian Coburn, "Multispecies Influenza Models with Recombination," Ph.D thesis, University of Miami in Coral Gables,FL, 2009. 
[14] 
R. Couch and J. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (2002), 529549. doi: 10.1146/annurev.mi.37.100183.002525. 
[15] 
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," John Wiley & Sons, Ltd., Chichester, 2000. 
[16] 
D. J. D. Earn, J. Dushoff and S. A. Levin, Ecology and evolution of the flu, Trends Ecol. Evol., 17 (2002), 334340. doi: 10.1016/S01695347(02)025028. 
[17] 
S. EchevarríaZuno, J. M. MejíaAranguré, A. V. MarObeso, C. GrajalesMuñiz, E. RoblesPérez, M. GonzálezLeón, M. C. OrtegaAlvarez, C. GonzalezBonilla, R. A. RascónPacheco and V. H. BorjaAburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 20722079. doi: 10.1016/S01406736(09)61638X. 
[18] 
A. EstevesJaramillo, S. B. Omer and E. GonzalezDiaz, Acceptance of a vaccine against novel influenza A (H1N1) virus among health care workers in two major cities in Mexico, Archives of Medical Research, 40 (2009), 705711. doi: 10.1016/j.arcmed.2010.01.004. 
[19] 
W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," SpringerVerlag, New York, 1994. 
[20] 
FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010),, \url{http://www.flu.gov/individualfamily/vaccination/supply.html}., (). 
[21] 
M. A. HerreraValdez, M. CruzAponte and C. CastilloChavez, Multiple waves for the same pandemic: Local transportation and social distancing explain the dynamics of the A/H1N1 epidemic during 2009 in Mexico, (2010). 
[22] 
E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a twostrain tuberculosis model, Discrete and Continuous Dynamical SystemsSeries B, 2 (2002), 473482. doi: 10.3934/dcdsb.2002.2.473. 
[23] 
P. Y. Lee, D. B. Matchar, D. A. Clements, J. Huber, J. D. Hamilton and E. D. Peterson, Economic analysis of influenza vaccination and antiviral treatment for healthy working adults, Ann. Intern. Med., 137 (2002), 225231. 
[24] 
S. Lee, G. Chowell and C. CastilloChavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136150. doi: 10.1016/j.jtbi.2010.04.003. 
[25] 
S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, 2007. 
[26]  
[27] 
H. Nishiura, C. CastilloChavez, M. Safan and G. Chowell, Transmission potential of the new Influenza A(H1N1) virus and its agespecificity in Japan, Eurosurveillance, 14 (2009), 14. 
[28] 
M. Nuno, G. Chowell and A. B. Gumel, Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, Journal of The Royal Society Interface, 4 (2007), 505521. doi: 10.1098/rsif.2006.0186. 
[29] 
J. Plotkin, J. Dushoff and S. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 62636268. doi: 10.1073/pnas.082110799. 
[30] 
L. S. Pontryagin, R. V. Boltyanski, R. V. Gamkrelidge and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," John Wiley and Sons, N.Y., 1962. 
[31] 
Prevent Influenza Now! Sponsored by the National Influenza Vaccine Summit, Influenza vaccine availability tracking system (IVATS),, \url{http://www.preventinfluenza.org/ivats/}., (). 
[32] 
C. E. Shoichet, Mexico still waiting for most swine flu vaccines, (2010), http://www.boston.com/business/articles/2010/01/13/mexico_still_waiting_for_most_swine _flu_vaccines. 
[33] 
E. Spackman, D. Stallknecht, R. Slemons, K. Winker, D. L. Suarez, M. Scott and D. E. Swayne, Phylogenetic analyses of type A influenza genes in natural reservoir species in North America reveals genetic variation, Virus research, 114 (2005), 89100. doi: 10.1016/j.virusres.2005.05.013. 
[34] 
R. Stengel, Optimal control and estimation,, http://www.princeton.edu/ stengel/MAE546.html., (). 
[35] 
T. Suess, U. Buchholz, S. Dupke, R. Grunow, M. an der Heiden, A. Heider, B. Biere, B. Schweiger, W. Haas and G. Krause, Shedding and transmission of novel influenza virus A/H1N1 infection in households—Germany, 2009, American Journal of Epidemiology, 171 (2010), 11571164. doi: 10.1093/aje/kwq071. 
[36] 
J. K. Taubenberger, D. M. Morens, 1918 influenza: the mother of all pandemics, Emerging Infectious Diseases, (2009), http://www.cdc.gov/ncidod/EID/vol12no01/050979.htm. 
[37] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[38] 
World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 20012002 season, Wkly. Epidemiol. Rec., 76 (2001), 5861. 
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