# American Institute of Mathematical Sciences

2011, 8(1): 141-170. doi: 10.3934/mbe.2011.8.141

## 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

Received  June 2010 Revised  September 2010 Published  January 2011

The lessons learned from the 2009-2010 H1N1 influenza pandemic, as it moves out of the limelight, should not be under-estimated, particularly since the probability of novel influenza epidemics in the near future is not negligible and the potential consequences might be huge. Hence, as the world, particularly the industrialized world, responded to the potentially devastating effects of this novel A-H1N1 strain with substantial resources, reminders of the recurrent loss of life from a well established foe, seasonal influenza, could not be ignored. The uncertainties associated with the reported and expected levels of morbidity and mortality with this novel A-H1N1 live in a backdrop of $36,000$ deaths, over 200,000 hospitalizations, and millions of infections (20% of the population) attributed to seasonal influenza in the USA alone, each year. So, as the Northern Hemisphere braced for the possibility of a potentially "lethal" second wave of the novel A-H1N1 without a vaccine ready to mitigate its impact, questions of who should be vaccinated first if a vaccine became available, came to the forefront of the discussion. Uncertainty grew as we learned that the vaccine, once available, would be unevenly distributed around the world. Nations capable of acquiring large vaccine supplies soon became aware that those who could pay would have to compete for a limited vaccine stockpile. The challenges faced by nations dealing jointly with seasonal and novel A-H1N1 co-circulating strains under limited resources, that is, those with no access to novel A-H1N1 vaccine supplies, limited access to the seasonal influenza vaccine, and limited access to antivirals (like Tamiflu) are explored in this study. One- and two-strain models are introduced to mimic the influenza dynamics of a single and co-circulating strains, in the context of a single epidemic outbreak. Optimal control theory is used to identify and evaluate the "best" control policies. The controls account for the cost associated with social distancing and antiviral treatment policies. The optimal policies identified might have, if implemented, a substantial impact on the novel H1N1 and seasonal influenza co-circulating dynamics. Specifically, the implementation of antiviral treatment might reduce the number of influenza cases by up to 60% under a reasonable seasonal vaccination strategy, but only by up to 37% when the seasonal vaccine is not available. Optimal social distancing policies alone can be as effective as the combination of multiple policies, reducing the total number of influenza cases by more than 99% within a single outbreak, an unrealistic but theoretically possible outcome for isolated populations with limited resources.
Citation: 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
##### References:
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Infect., 136 (2008), 852-864. doi: 10.1017/S0950268807009144. [12] G. Chowell, S. M. Bertozzi, M. A. Colchero, H. Lopez-Gatell, C. Alpuche-Aranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674-679. doi: 10.1056/NEJMoa0904023. [13] Brian Coburn, "Multi-species 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), 529-549. 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), 334-340. doi: 10.1016/S0169-5347(02)02502-8. [17] S. Echevarría-Zuno, J. M. Mejía-Aranguré, A. V. Mar-Obeso, C. Grajales-Muñiz, E. Robles-Pérez, M. González-León, M. C. Ortega-Alvarez, C. Gonzalez-Bonilla, R. A. Rascón-Pacheco and V. H. Borja-Aburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 2072-2079. doi: 10.1016/S0140-6736(09)61638-X. [18] A. Esteves-Jaramillo, S. B. Omer and E. Gonzalez-Diaz, 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), 705-711. doi: 10.1016/j.arcmed.2010.01.004. [19] W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," Springer-Verlag, New York, 1994. [20] FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010), http://www.flu.gov/individualfamily/vaccination/supply.html. [21] M. A. Herrera-Valdez, M. Cruz-Aponte and C. Castillo-Chavez, 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 two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. 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), 225-231. [24] S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136-150. 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] E. Malkin, Flu? What flu?, The New York Times, http://www.nytimes.com/2009/05/09/business/global/09peso.html. [27] H. Nishiura, C. Castillo-Chavez, M. Safan and G. Chowell, Transmission potential of the new Influenza A(H1N1) virus and its age-specificity in Japan, Eurosurveillance, 14 (2009), 1-4. [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), 505-521. 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), 6263-6268. 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), 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), 89-100. 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), 1157-1164. 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/vol12no01/05-0979.htm. [37] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [38] World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 2001-2002 season, Wkly. Epidemiol. Rec., 76 (2001), 58-61.

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), 1156-1167. doi: 10.1016/j.vaccine.2009.11.043. [4] H. Behnke, Optimal control of deterministic epidemics, Optimal Control Application Methods, 21 (2000), 269-285. 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. Castillo-Chavez, Discrete epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 1-15. doi: 10.3934/mbe.2010.7.1. [7] C. Castillo-Chavez, H. Hethcote, V. Andreason, S. A. Levin and W. M. Liu, Cross-immunity in the dynamics of homogeneous and heterogeneous populations, Mathematical Ecology, (1988), 303-316. [8] Centers for Disease Control and Prevention (CDC), Key facts about seasonal influenza, 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 cross-reactive antibody response to a novel influenza A(H1N1) virus after vaccination with seasonal influenza vaccine, MMWR Morb Mortal Wkly Rep, 58 (2009), 521-524. [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), 852-864. doi: 10.1017/S0950268807009144. [12] G. Chowell, S. M. Bertozzi, M. A. Colchero, H. Lopez-Gatell, C. Alpuche-Aranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674-679. doi: 10.1056/NEJMoa0904023. [13] Brian Coburn, "Multi-species 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), 529-549. 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), 334-340. doi: 10.1016/S0169-5347(02)02502-8. [17] S. Echevarría-Zuno, J. M. Mejía-Aranguré, A. V. Mar-Obeso, C. Grajales-Muñiz, E. Robles-Pérez, M. González-León, M. C. Ortega-Alvarez, C. Gonzalez-Bonilla, R. A. Rascón-Pacheco and V. H. Borja-Aburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 2072-2079. doi: 10.1016/S0140-6736(09)61638-X. [18] A. Esteves-Jaramillo, S. B. Omer and E. Gonzalez-Diaz, 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), 705-711. doi: 10.1016/j.arcmed.2010.01.004. [19] W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," Springer-Verlag, New York, 1994. [20] FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010), http://www.flu.gov/individualfamily/vaccination/supply.html. [21] M. A. Herrera-Valdez, M. Cruz-Aponte and C. Castillo-Chavez, 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 two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. 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), 225-231. [24] S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136-150. 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] E. Malkin, Flu? What flu?, The New York Times, http://www.nytimes.com/2009/05/09/business/global/09peso.html. [27] H. Nishiura, C. Castillo-Chavez, M. Safan and G. Chowell, Transmission potential of the new Influenza A(H1N1) virus and its age-specificity in Japan, Eurosurveillance, 14 (2009), 1-4. [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), 505-521. 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), 6263-6268. 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), 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), 89-100. 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), 1157-1164. 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/vol12no01/05-0979.htm. [37] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [38] World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 2001-2002 season, Wkly. Epidemiol. Rec., 76 (2001), 58-61.
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