# Maximizing the effectiveness of a pediatric vaccine formulary while prohibiting extraimmunization

## Abstract

The growing complexity of the United States Recommended Childhood Immunization Schedule has resulted in as many as five required injections during a single well-baby office visit. To reduce this number, vaccine manufacturers have developed combination vaccines that immunize against several diseases in a single injection. At the same time, a growing number of parents are challenging the safety and effectiveness of vaccinating children. They are also particularly concerned about the use of combination vaccines, since they believe that injecting a child with multiple antigens simultaneously may overwhelm a child’s immune system. Moreover, combination vaccines make it more likely that extraimmunization (i.e., administering more than the required amount of vaccine antigens) occurs, resulting in greater concerns by parents and vaccine wastage costs borne by an already strained healthcare system. This paper formulates an integer programming model that solves for the maximum number of vaccines that can be administered without any extraimmunization. An exact dynamic programming algorithm and a randomized heuristic for the integer programming model is formulated and the heuristic is shown to be a randomized ξ-approximation algorithm. Computational results are reported on three sets of test problems, based on existing and future childhood immunization schedules, to demonstrate their computational effectiveness and limitations. Given that future childhood immunization schedules may need to be solved for each child, on a case-by-case basis, the results reported here may provide a practical and valuable tool for the public health community.

## Keywords

Pediatric immunization Pediatric vaccines Randomized approximation algorithms Discrete optimization Dynamic programming Heuristics Integer programming## Notes

### Acknowledgements

The authors would like to thank the three anonymous referees for their excellent comments and input, which have resulted in a significantly improved manuscript. The authors would also like to thank Bruce G. Weniger, M.D., M.P.H., Assistant Chief for Vaccine Development, Vaccine Safety and Development Branch, National Immunization Program, Centers for Disease Control and Prevention, for his longstanding encouragement and feedback on this line of research; his input on this work has been invaluable and most appreciated. This research has been supported in part by the National Science Foundation (DMI-0457176, DMI-0456945). The third author has also been supported in part by the Air Force Office of Scientific Research (FA9550-07-1-0232). The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, National Science Foundation, or the United States Government.

## References

- 1.Bazaraa MS, Jarvis JJ, Sherali HD (1990) Linear programming and network flows. Wiley, New YorkGoogle Scholar
- 2.Burden RL, Faires JD (1997) Numerical analysis, 6th edn. Brookes-Cole, New YorkGoogle Scholar
- 3.CDC Vaccine Price List (2006) Center for disease control and prevention, national immunization program website. Retrieved August 3, 2006, from www.cdc.gov/nip/vfc/cdc_vac_price_list.htm.
- 4.Centers for Disease Control and Prevention (2006) Recommended childhood and adolescent immunization schedule—United States. Morb Mort Wkly Rep 54(52):Q1–Q4Google Scholar
- 5.Centers for Disease Control and Prevention (2007) Recommended childhood and adolescent immunization schedule—United States. Morb Mort Wkly Rep 55(51):Q1–Q4Google Scholar
- 6.Centers for Disease Control and Prevention (2003) Prevention and control of infections with hepatitis viruses in correctional settings. Appendix hepatitis A and B vaccines. Morb Mort Wkly Rep 52(RR-1):34–36Google Scholar
- 7.Centers for Disease Control and Prevention (2002) General recommendations on immunization. Morb Mort Wkly Rep 51(RR-2)Google Scholar
- 8.Centers for Disease Control and Prevention (1999) Combination vaccines for childhood immunization. Morb Mort Wkly Rep 48(18)(RR-5)Google Scholar
- 9.Centers for Disease Control and Prevention (1995) Recommended childhood and adolescent immunization schedule—United States. Morb Mort Wkly Rep 44(RR-5):1–9Google Scholar
- 10.Chen RT, DeStefano F, Pless R, Mootrey G, Kramarz P, Hibbs B (2001) Challenges and controversies in immunization safety. Infect Dis Clin North Am 15(1):21–39CrossRefGoogle Scholar
- 11.Cochi SL (2005) 2005 National immunization survey. Presentation to National Press Club, July 27, 2005Google Scholar
- 12.Diekema DS (2005) Responding to parental refusals of immunization of children. Pediatrics 115(5):1428–1431CrossRefGoogle Scholar
- 13.Edwards KM, Decker MD (2001) Combination vaccines. Infect Dis Clin North Am 15(1):209–230CrossRefGoogle Scholar
- 14.Flanagan-Klygis EA, Sharp L, Frader JE (2005) Dismissing the family who refuses vaccines: A study of pediatrician attitudes. Arch Pediatr Adolesc Med 159(10):929–934CrossRefGoogle Scholar
- 15.Garey MR, Johnson DS (1979) Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman, New YorkGoogle Scholar
- 16.Hall SN (2006) The design and analysis of pediatric vaccine formularies: theory and practice. University of Illinois, Urbana, Illinois Ph.D. DissertationGoogle Scholar
- 17.Hochbaum DS (ed) (1997) Approximation algorithms for NP-hard problems. PWS, Boston, MassachusettsGoogle Scholar
- 18.Infectious Diseases in Children (2002) Almost 200 new drugs in development for use in children. 15(10):41–44Google Scholar
- 19.Jacobson RM, Targonski PV, Poland GA (2007) Meta-analyses in vaccinology. Vaccine 25:3153–3159CrossRefGoogle Scholar
- 20.Jacobson SH, Karnani T, Sewell EC (2003a) Analyzing the economic value of the hepatitis B-
*Haemophilus influenzae*type B combination vaccine by reverse engineering a formulary selection algorithm. Vaccine 21:2169–2177CrossRefGoogle Scholar - 21.Jacobson SH, Karnani T, Sewell EC (2004) Assessing the impact of wastage on pediatric vaccine immunization formulary costs using a vaccine selection algorithm. Vaccine 22:2307–2315CrossRefGoogle Scholar
- 22.Jacobson SH, Sewell EC (2002) Using Monte Carlo simulation to determine combination vaccine price distributions for childhood disease. Health Care Manage Sci 5:135–145CrossRefGoogle Scholar
- 23.Jacobson SH, Sewell EC, Allwine DA, Medina EA, Weniger BG (2003b) Designing pediatric vaccine formularies and pricing pediatric combination vaccines using operations research models and algorithms. Expert Rev Vaccines 2(1):15–19CrossRefGoogle Scholar
- 24.Jacobson SH, Sewell EC, Deuson R, Weniger BG (1999) An integer programming model for vaccine procurement and delivery for childhood immunization: a pilot study. Health Care Manage Sci 2:1–9CrossRefGoogle Scholar
- 25.Jacobson SH, Sewell EC, Karnani T (2005) Engineering the economic value of two pediatric combination vaccines. Health Care Manage Sci 8(1):29–40CrossRefGoogle Scholar
- 26.Jacobson SH, Sewell EC, Proano RA (2006) An analysis of the pediatric vaccine supply shortage problem. Health Care Manage Sci 9(4):371–389Google Scholar
- 27.Nemhauser GL, Wolsey LA (1999) Integer and combinatorial optimization. Wiley, New YorkGoogle Scholar
- 28.Srinivasan A (1999) Improved approximation guarantees for packing and covering integer programs. SIAM J Comput 29(2):648–670CrossRefGoogle Scholar
- 29.Sewell EC, Jacobson SH (2003) Using an integer programming model to determine the price of combination vaccines for childhood immunization. Ann Oper Res 119:261–284CrossRefGoogle Scholar
- 30.Sewell EC, Jacobson SH, Weniger BG (2001) Reverse engineering a formulary selection algorithm to determine the economic value of pentavalent and hexavalent combination vaccines. Pediatr Infect Dis J 20(11):S45–S56CrossRefGoogle Scholar
- 31.Weniger BG (1996) Economic analysis to meet the challenges of new combination vaccines. Presented at Vaccine Economics: Planning a Research Agenda for the Challenge of New and Improved Vaccines, Atlanta, GAGoogle Scholar
- 32.Weniger BG, Chen RT, Jacobson SH, Sewell EC, Deuson R, Livengood JR, Orenstein WA (1998) Addressing the challenges to immunization practice with an economic algorithm for vaccine selection. Vaccine 16(19):1885–1897CrossRefGoogle Scholar