Bulletin of Mathematical Biology

, Volume 74, Issue 4, pp 958–980

# Modeling Optimal Age-Specific Vaccination Strategies Against Pandemic Influenza

Original Article

## Abstract

In the context of pandemic influenza, the prompt and effective implementation of control measures is of great concern for public health officials around the world. In particular, the role of vaccination should be considered as part of any pandemic preparedness plan. The timely production and efficient distribution of pandemic influenza vaccines are important factors to consider in mitigating the morbidity and mortality impact of an influenza pandemic, particularly for those individuals at highest risk of developing severe disease. In this paper, we use a mathematical model that incorporates age-structured transmission dynamics of influenza to evaluate optimal vaccination strategies in the epidemiological context of the Spring 2009 A (H1N1) pandemic in Mexico. We extend previous work on age-specific vaccination strategies to time-dependent optimal vaccination policies by solving an optimal control problem with the aim of minimizing the number of infected individuals over the course of a single pandemic wave. Optimal vaccination policies are computed and analyzed under different vaccination coverages (21%–77%) and different transmissibility levels ($$\mathcal{R}_{0}$$ in the range of 1.8–3). The results suggest that the optimal vaccination can be achieved by allocating most vaccines to young adults (20–39 yr) followed by school age children (6–12 yr) when the vaccination coverage does not exceed 30%. For higher $$\mathcal{R}_{0}$$ levels ($$\mathcal{R}_{0}>=2.4$$), or a time delay in the implementation of vaccination (>90 days), a quick and substantial decrease in the pool of susceptibles would require the implementation of an intensive vaccination protocol within a shorter period of time. Our results indicate that optimal age-specific vaccination rates are significantly associated with $$\mathcal{R}_{0}$$, the amount of vaccines available and the timing of vaccination.

### Keywords

Influenza pandemic A/H1N1 pandemic Optimal control Age-specific vaccination

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### References

1. Aaby, K., Abbey, R., Herrmann, J., Treadwell, M., Jordan, C., & Wood, K. (2006). Embracing computer modeling to address pandemic influenza in the 21st century. J. Public Health Manag. Pract., 12(4), 365–372. Google Scholar
2. Behncke, H. (2000). Optimal control of deterministic pandemics. Optim. Control Appl. Methods, 21, 269–285.
3. Blayneh, K., Cao, Y., & Kwon, H. (2009). Optimal control of vector-borne disease: treatment and prevention. Discrete Contin. Dyn. Syst., Ser. B, 11(3), 587–611.
4. Centers For Disease Control and Prevention (2009). Large-scale vaccination clinic output and staffing estimates: An example. www.cdc.gov/h1n1flu/vaccination/pdf/A-Wortley-H1N1-sample-clinic.pdf.
5. Cho, B., Hicks, K., Honeycutt, A., Hupert, N., Khavjou, O., Messonnier, M., & Washington, M. (2011). A tool for the economic analysis of mass prophylaxis operations with an application to H1N1 influenza vaccination clinics. J. Public Health Manag. Pract., 17, E22–E28. Google Scholar
6. Chowell, G., Miller, M. A., & Viboud, C. (2008). Seasonal influenza in the United States, France and Australia: transmission and prospects for control. Epidemiol. Infect., 136, 852–864. Google Scholar
7. Chowell, G., Viboud, C., Wang, X., Bertozzi, S., & Miller, M. (2009). Adaptive vaccination strategies to mitigate pandemic influenza: Mexico as a case study. PLoS ONE, 12, e8164.
8. Chowell, G., Bertozzi, S. M., Colchero, M. A., Lopez-Gatell, H., Alpuche-Aranda, C., Hernandez, M., & Miller, M. A. (2009). Severe respiratory disease concurrent with the circulation of H1N1 influenza. N. Engl. J. Med., 361, 674–679.
9. Chowell, G., Echevarría-Zuno, S., Viboud, C., Simonsen, L., Tamerius, J., Miller, M. A., & Borja-Aburto, V. (2011). Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico. PLoS Med., 8(5), e1000436.
10. Edmunds, W. J., O’Callaghan, C. J., & Nokes, D. J. (1997). Who mixes with whom? A method to determine the contact patterns of adults that may lead to the spread of airborne infections. Proc. Biol. Sci., 264(1384), 949–957.
11. Ferguson, N. M., Cummings, D. T., Fraser, C., Cajka, J. C., Cooley, P. C., & Burke, D. S. (2006). Strategies for mitigating an influenza pandemic. Nature, 442(7101), 448–452.
12. Fleming, W. H., & Rishel, R. W. (1975). Deterministic and stochastic optimal control. New York: Springer.
13. Gani, R., Hughes, H., Fleming, D., Grifin, T., Medlock, J., & Leach, S. (2005). Potential impact of antiviral use during influenza pandemic. Emerg. Infect. Dis., 11, 1355–1362.
14. Fedson, D. S. (2003). Pandemic influenza and the global vaccine supply. Clin. Infect. Dis., 36(12), 1562–1563.
15. Germann, T. C., Kadau, K., Longini, I. M., & Macken, C. A. (2006). Mitigation strategies for pandemic influenza in the United States. Proc. Natl. Acad. Sci. USA, 103(15), 5935–5940.
16. Goodwin, K., Viboud, C., & Simonsen, L. (2006). Antibody response to influenza vaccination in the elderly: a quantitative review. Vaccine, 24(8), 1159–1169.
17. Gostin, L., & Berkman, B. (2007). Pandemic influenza: ethics, law, and the public’s health. Adm. Law Rev., 59(1), 121–175. Google Scholar
18. Hansen, E., & Day, T. Optimal control of pandemics with limited resources J. Math. Biol. doi:10.1007/s00285-010-0341-0.
19. Herrera-Valdez, M., Cruz-Aponte, M., & Castillo-Chavez, C. (2011). Multiple outbreaks for the same pandemic: local transportation and social distancing explain the different “waves” of A-H1N1pdm cases observed in Mexico during 2009. Math. Biosci. Eng., 8, 21–48.
20. Health Industry Distributors Association: 2008–2009 influenza vaccine production and distribution 2009 [www.flusupplynews.com/documents/09 FluBrief 000.pdf].
21. Hill, A. N., & Longini, I. M. Jr. (2003). The critical vaccination fraction for heterogeneous pandemic models. Math. Biosci., 181(1), 85–106.
22. Jung, E., Lenhart, S., & Feng, Z. (2002). Optimal control of treatments in a two strain tuberculosis model. Discrete Contin. Dyn. Syst., Ser. B, 2, 473–482.
23. Knipl, D. H., & Rost, G. (2011). Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks. Math. Biosci. Eng., 8(1), 123–139.
24. Kotalik, J. (2005). Preparing for an influenza pandemic: ethical issues. Bioethics, 19(4), 422–431.
25. Lee, S., Chowell, G., & Castillo-Chavez, C. (2010). Optimal control of influenza pandemics: the role of antiviral treatment and isolation. J. Theor. Biol., 265, 136–150.
26. Lee, S., Morales, R., & Castillo-Chavez, C. (2011). A note on the use of influenza vaccination strategies when supply is limited. Math. Biosci. Eng., 8(1), 171–182.
27. Lenhart, S., & Workman, J. T. (2007). Optimal control applied to biological models. CRC Mathematical and Computational Biology series. London: Chapman & Hall.
28. Lipsitch, M., Riley, S., Cauchemez, S., Ghani, A. C., & Ferguson, N. M. (2009). Managing and reducing uncertainty in an emerging influenza pandemic. N. Engl. J. Med., 361, 112–115.
29. Macroepidemiology of Influenza Vaccination Study Group (2005). The Macro-epidemiology of influenza vaccination in 56 countries, 1997–2003. Vaccine, 23(44), 5133–5143.
30. Merler, S., Ajelli, M., & Rizzo, C. (2009). Age-prioritized use of antivirals during an influenza pandemic. BMC Infect. Dis., 9, 117. doi:10.1186/1471-2334-9-117.
31. Medlock, J., Meyers, L. A., & Galvani, A. (2009). Optimizing allocation for a delayed influenza vaccination campaign PLoS Curr Influenza RRN1134. Google Scholar
32. Miller, M., Viboud, C., Balinska, M., & Simonsen, L. (2009). The signature features of influenza pandemics—implications for policy. N. Engl. J. Med., 360(25), 2595–2598.
33. Mossong, J., et al. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med., 5(3), e74.
34. Mylius, S. D., et al. (2008). Optimal allocation of pandemic influenza vaccine depends on age, risk and timing. Vaccine, 26(29–30), 3742–3749.
35. Nishiura, H., Castillo-Chavez, C., Safan, M., & Chowell, G. (2009). Transmission potential of the new influenza (h1N1) virus and its age-specificity in Japan. Euro Surveill., 14(22), 1–4. Google Scholar
36. Nuno, M., Chowell, G., & Gumel, A. B. (2007). Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: scenarios for the US, UK and the Netherlands. J. R. Soc. Interface, 224(14), 505–521.
37. Oliver Wyman Group and Program for Appropriate Technology in Health: Influenza vaccine strategies for broad global access, key findings and project methodology (2007). www.path.org/files/VAC_infl_publ_rpt_10-07.pdf.
38. Oshitani, H., Kamigaki, T., & Suzuki, A. (2008). Major issues and challenges of influenza pandemic preparedness in developing countries. Emerg. Infect. Dis., 14(6), 875–880.
39. Patel, R., Longini, I. M. Jr., & Halloran, M. E. (2005). Finding optimal vaccination strategies for pandemic influenza using genetic algorithms. J. Theor. Biol., 234(2), 201–212.
40. Peterborough County-city health unit pandemic influenza plan, Annex A: Mass vaccination plan (2010). http://pcchu.peterborough.on.ca/IC/IC-pandemic-plan.html.
41. Phillips, F., & Williamson, J. (2005). Local health department applies incident management system for successful mass influenza clinics. J. Public Health Manag. Pract., 11(4), 269. Google Scholar
42. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The mathematical theory of optimal processes. New Jersey: Wiley.
43. Rowthorn, R. E., Laxminarayan, R., & Gilligan, C. A. (2009). Optimal control of pandemics in metapopulations. Proc. R. Soc. doi:10.1098/?rsif.2008.0402. Google Scholar
44. Stohr, K. (2010). Vaccinate before the next pandemic? Nature, 465, 13.
45. Tennenbaum, S. (2008). Simple criteria for finding (nearly) optimal vaccination strategies. J. Theor. Biol., 250(4), 673–683.
46. Tracht, S. M., Del Valle, S. Y., & Hyman, J. M. (2010). Mathematical modeling of the effectiveness of facemasks in reducing the spread of novel influenza A (H1N1). PLoS ONE, 105(2), e9018.
47. Tuite, A. R., Fisman, D. N., Kwong, J. C., & Greer, A. L. (2010). Optimal pandemic influenza vaccine allocation strategies for the Canadian population. PLoS ONE, 5(5), e10520. doi:10.1371/journal.pone.0010520.
48. Ulmer, J., & Liu, M. (2002). Ethical issues for vaccines and immunization. Nat. Rev. Immunol., 2, 291–296.
49. Valadez, B. Aplicadas, Solo 10% de las Dosis Contra el A/H1N1 (2010). http://www.milenio.com/node/368812.
50. Wallinga, J., Teunis, P., & Kretzschmar, M. (2006). Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents. Am. J. Epidemiol., 164(10), 936–944.
51. Washington, M. (2009). Evaluating the capability and cost of a mass influenza and pneumococcal vaccination clinic via computer simulation. Med. Decis. Mak., 29(4), 414–423.