Using an Integer Programming Model to Determine the ... - Springer Link

Annals of Operations Research 119, 261–284, 2003  2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

Using an Integer Programming Model to Determine the Price of Combination Vaccines for Childhood Immunization EDWARD C. SEWELL [email protected] Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA SHELDON H. JACOBSON [email protected] Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2906, USA

Abstract. The Recommended Childhood Immunization Schedule has become sufficiently crowded that the prospect of adding additional vaccines to this schedule may not be well received by either health-care providers or parents/guardians. This has encouraged vaccine manufacturers to develop combination vaccines that can permit new vaccines to be added to the schedule without requiring children to be exposed to an unacceptable number of injections during a single clinic visit. This paper develops an integer programming model to assess the economic premium that exists in having combination vaccines available. The results of this study suggest that combination vaccines provide a cost effective alternative to individual vaccines and that further developments and innovations in this area by vaccine manufacturers can provide significant economic and societal benefits. Keywords: integer programming, combination vaccines, pediatric immunization, reverse engineering, economic analysis

1.

Introduction

As vaccine manufacturers develop an increasing number of new vaccines to protect children from childhood diseases, it is becoming a challenge to incorporate them into an already crowded immunization schedule in the United States [4,18]. Health-care providers, children and parents, are all hesitant to increase the number of injections (per clinic visit) required to take advantage of such new products [12]. In 2000, the four recommended doses of oral polio vaccine (OPV) [2] were replaced with four injections of inactivated polio vaccine (IPV) [4]. Also in 2000, four doses of a new conjugate vaccine for pneumococcal disease were inserted into the immunization schedule [5]. These changes resulted in eight additional injections that are needed to satisfy immunization requirements for a child from birth through six years of age, bringing the total number of injections to nineteen (as of July 2000). This means that a child may require four or five injections at each of three recommended immunization visits (2, 4, and 6 months) in the first year of life.

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Combining antigens protective against multiple diseases into a single injection can ease this burden [14]. Licensed combination vaccines are officially preferred over their separate component vaccines in order to reduce the pain and indirect consequences of multiple injections [3]. Such consequences include the emotional discomfort to child and parents from being exposed to multiple injections, possibly resulting in deferred or missed vaccinations, resulting in an economic toll for the direct medical costs and indirect work-loss of extra visits, since if these visits do not occur, then the child may be at risk of the disease [6]. Research has been conducted in conjunction with personnel from the Centers for Disease Control and Prevention (CDC) to assess the costs of extra injections by determining the amount of money at which parents would be willing to pay to avoid them for their children [8,11,13]. These studies found this amount clustering around $25 for each injection avoided. In addition, direct medical costs of each injection were estimated at $10 each, while the overall cost of deferring excessive injections was approximately $23 for each scheduled vaccination visit [11]. Note that such data can be incorporated into a publicly available web site (funded through the CDC), www.vaccineselection.com, for selecting optimal vaccine formularies. In spite of the initial optimism for combination vaccines as a solution to the problem of excessive injections, the development of such vaccines has been very challenging and costly. This has been due in part to immunologic interference (i.e., antigens for one vaccine causing another vaccine to be less effective; see [16,17]). Therefore, whenever a new combination vaccine is put forward for Food and Drug Administration (FDA) approval, the CDC, vaccine manufacturers, health-care providers, and insurance companies are all faced with the daunting task of assessing its value and worth [9]. The two questions posed are mirror images of the same issue. Vaccine manufacturers ask, “How much should we charge for the combination vaccine?” Health-care providers and insurance companies ask, “How much should we pay for the combination vaccine?” This paper presents an integer programming model that can be used to help address these questions, hence allow the CDC, health-care providers, insurance companies, and parents/guardians make well informed vaccine formulary decisions [7,19]. Note that this integer programming model is an extension of the model presented in [7]. In particular, the model used in this paper includes combination vaccines that are not included in the model contained in [7]. The resulting analysis is predicated on the principle that decisions based on purchase price alone can be more costly in the long run by ignoring the economic value of distinguishing features among competing vaccine products. Optimization models are developed such that vaccine formulary decisions can be made based on how combination vaccines fit into the market place. More specifically, these models assemble from among all available vaccine products at their market prices the vaccine formulary that provides the best value within the constraints of the immunization schedule, achieving the lowest overall cost to society or any other desired perspective. Therefore, the model selects from among a set of monovalent (i.e., single antigen) and combination vaccines those products that should be used at which scheduled visits within the Recommended Childhood Immunization Schedule [4].

USING AN INTEGER PROGRAMMING MODEL

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Users of these optimization models have the choice to assign any cost for each economic variable included, such as the cost of each injection. Note that although the model was designed principally for use by major vaccine purchasers, such as public health agencies, health-care organizations, and large private clinics, it can also be used to reverse engineer the price of new combination vaccine products for vaccine manufacturers. This price information provides such companies with a tool for determining the price at which a proposed new vaccine product would win a place in the best value vaccine formulary solution, hence perhaps guide investment decisions and development priorities towards products with the most beneficial economic impact on the immunization system. At present, there are four new pentavalent and hexavalent combination vaccine products that are in various stages of development. These four vaccine products contain five or all six of the following vaccines: Diphtheria, Tetanus, Pertussis (acellular) (DTPa), Haemophilus influenzae type B (HIB), Hepatitis B (HBV), and Polio (IPV). More specifically, these four vaccine products are: DTPa-HBV-IPV, DTPa-HIB-IPV, DTPa-HIB-HBV, DTPa-HIB-HBV-IPV. To illustrate the technique of solving the model in reverse, the maximum price at which each of these four combination vaccines listed above would win a place in a lowest-cost vaccine formulary was computed. Each product competed against the preexisting vaccines licensed in the United States at current federally negotiated public sector prices. Cost assumptions used in the 1998 Centers for Disease Control and Prevention (CDC) pilot study model [7,19] were updated with more detailed data that has become available over the past two years. Background and assumptions The integer programming model developed for this analysis captures the first five years of the Recommended Childhood Immunization Schedule for immunization against six childhood diseases (Hepatitis B, Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, and Polio). The vaccine for Diptheria, Tetanus, and Pertussis is packaged as a single injection, and is not considered a combination vaccine. To determine the economic impact of combination vaccines, four different vaccine combinations are considered: • • • •

Diphtheria, Tetanus, Pertussis, Hepatitis B, Polio; Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, Polio; Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, Hepatitis B; Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, Hepatitis B, Polio.

These combination vaccines (labeled DTPa-HBV-IPV, DTPa-HIB-IPV, DTPa-HIBHBV, and DTPa-HIB-HBV-IPV, respectively) are analyzed by adding them, one at a time, to the list of twelve vaccine products currently licensed and under contract for distribution by the CDC. Note that until these combination vaccines are actually priced in

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the market by their manufacturers, one cannot determine their relative values. Therefore, adding two or more combination vaccines to the list at the same time is not considered. To provide boundaries for the scope of the results presented, certain assumptions are needed. To maintain the scientific rigor and validity of the analysis, wherever possible, the assumptions used in the study described in [15] are also used in this study. Note that this paper provides the technical details of this study, while [15] focuses on the medical issues and implications of the study. The federally negotiated vaccine price list effective March 2000 includes five drug companies (labeled AVP = Aventis Pasteur, MRK = Merck, NAV = North American Vaccine, GSK = GlaxoSmithKline, WAL = Wyeth–Lederle) that manufacture all the vaccines that were licensed and under contract with the CDC for childhood immunization. These five manufacturers produce twelve vaccine products that protect against six diseases (vaccine for Hepatitis B = HBV, vaccine for Diphtheria, Tetanus, Pertussis = DTPa, vaccine for Haemophilus influenzae type B = HIB, and vaccine for Polio = IPV). The cost function is used to price the four combination vaccines, as well as determine the overall cost of the resulting vaccine formularies. The cost function includes • • • •

the purchase price of all licensed vaccines, the cost of each clinic visit, the cost of vaccine preparation by medical staff, the cost of each injection administered.

The vaccine purchase prices used are the federally negotiated prices as of March 2000; these prices in US dollars are displayed in table 1. The cost of a clinic visit is set at $40. This value was used in the CDC pilot study [7,19], and includes the direct and indirect costs associated with a clinic being able to deliver vaccines (see [19]). The cost of vaccine preparation by the medical staff is classified into three categories with associated preparation times: • liquid vaccine packaged in pre-filled syringe • liquid vaccine packaged in pre-filled vial • powdered (lyophilized) vaccine requiring reconstitution

(0.5 minutes) [s], (1.5 minutes) [v], (3.0 minutes) [p].

These categories are based on the packaging of each vaccine and the associated time required for medical staff (such as nurses) to prepare the vaccine for administration to a child. The times for these three categories, labeled [s] for pre-filled syringe, [v] for pre-filled vial, and [p] for powder, are based on a mean time of 1.6 minutes to administer an injection [10], with modifications and rounding to account for the three categories of vaccine formulations and packagings. A medical staff compensation rate of 0.50/minute was used in the CDC pilot study [7,19], hence is also used here. Table 1 contains the preparation costs for the twelve vaccine products licensed and under contract with the CDC. Note that three of the vaccine products, two brands of HBV and one brand of IPV,


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Table 1 Vaccine purchase prices and preparation costs. Vaccine

Formulation/ packaging

Manufacturer ‡

March 2000 Federal price/dose

Preparation cost/dose

Total cost/dose

DTPa DTPa DTPa DTPa DTPa-HIB HIB HIB HIB HIB-HBV HBV HBV HBV HBV IPV IPV

liquid/vial [v] liquid/vial [v] liquid/vial [v] liquid/vial [v] powder/vial [p] powder/vial [p] liquid/vial [v] liquid/vial [v] liquid/vial [v] liquid/syringe [s] liquid/vial [v] liquid/syringe [s] liquid/vial [v] liquid/syringe [s] liquid/vial [v]

AVP NAV GSK WAL AVP AVP MRK WAL MRK MRK MRK GSK GSK AVP AVP

$9.25 $9.25 $9.25 $9.25 $22.01 $5.20 $7.75 $5.25 $20.99 $9.25 $9.25 $9.00 $9.00 $7.75 $7.75

$0.75 $0.75 $0.75 $0.75 $1.50 $1.50 $0.75 $0.75 $0.75 $0.25 $0.75 $0.25 $0.75 $0.25 $0.75

$10.00 $10.00 $10.00 $10.00 $23.51 $6.70 $8.50 $6.00 $21.74 $9.50 $10.00 $9.25 $9.75 $8.00 $8.50

are available in both pre-filled syringes and liquid vial formulations. Though their purchase prices are the same, the pre-filled syringes require one fewer minute of preparation time, hence from an economic standpoint, would always be chosen over the pre-filled vials. Therefore, for these three vaccine products, only the pre-filled syringes are considered in the analysis. Note that the total cost per dose listed in table 1 does not include the cost of administering each injection. Weniger et al. [19] observe that the cost associated with administering an injection can be broken down into several components. The first component is the actual direct cost of administering the vaccine. Lieu et al. [11] suggest this cost to be approximately $5 for each injection. The second component is the direct cost for repeat clinic visits if injections are refused by the parents/guardians (e.g., when four or more injections are required at a particular clinic visit). This cost is estimated to be approximately $3 for each injection. The third component is the indirect cost of lost time from work by parents/guardians for repeat clinic visits if they refuse injections. This cost is estimated to be approximately $12 for each injection. The fourth component is the indirect cost of “pain and emotional distress” associated with each injection, as measured by a parent’s/guardian’s “willingness-to-pay” to avoid such pain. This mean cost has been estimated to be as high as $25 per injection by [11] or more conservatively, as $8 per injection by [13]. The results reported in [8] independently support these values. It is difficult to assess a single value associated with administering an injection, since each parent/guardian and health-care provider may place widely disparate values on each of the four components described above. Therefore, for a given population of parents/guardians and/or health-care providers, one can assign a mean value for the cost associated with administering an injection. The following five telescoping perspectives

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are used to set this mean cost of an injection: Perspective (i). $5 = the marginal direct medical costs. Note that this cost reflects the perspective of the payer (e.g., an HMO or health insurer), but not parents/guardians and society. Perspective (ii). $8 = cost from perspective (i) plus the direct medical costs associated with repeat clinic visits for deferred injections. As in perspective (i), this cost also reflects the perspective of the payer. Perspective (iii). $20 = cost from perspective (ii) plus the indirect cost of parental/guardian lost time from work for repeat clinic visits if injections are refused by the parent/guardian. Note that this cost captures the perspective of both the payer and society as a whole, since it includes the indirect costs of lost work time by parents/guardians. Perspective (iv). $30 = cost from perspective (iii) plus the indirect conservative $10 per injection cost of “pain and emotional distress”. Perspective (v). $45 = cost from perspective (iii) plus the indirect $25 per injection cost of “pain and emotional distress”. The following assumptions are used in the analysis. These assumptions were all used in the CDC pilot study reported in [7,19]: (i) the Recommended Childhood Immunization Schedule was followed for immunization against six diseases: Hepatitis B, Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, and Polio (note that Polio was not included in the CDC pilot study, but is included in this study), (ii) injections can be administered in months 0–1, 2, 4, 6, 12–18, and 60, providing six months/periods to administer vaccines, only one clinic visit can occur in each of these months/periods, and all injections in a given month/period are administered in a single clinic visit, (iii) only the twelve currently available and under CDC contract vaccines are included in the model, with the exception of the four combination vaccines, (iv) HIB vaccines can only be administered in month 2 or later, (v) the first HBV injection must be administered in either month 0–1 (referred to as the HBV birth dose) or month 2 (referred to as the HBV nonbirth dose), (vi) if HIB vaccine products by Merck are administered in both months 2 and 4, then no HIB vaccine is required in month 6, (vii) manufacturer brand matching is required for DTPa vaccines, but not for HIB and HBV vaccines. In addition to the assumptions listed above which were used in the CDC pilot study, the following lists the assumptions used for this study: (a) the 2000 Recommended Childhood Immunization Schedule was used instead of the 1997 schedule (see [4]),


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(b) an IPV vaccine is included in the model, (c) extravaccination of IPV is permitted, (d) manufacturer brand matching is not required for the IPV vaccines, (e) the vaccine prices are the CDC prices as of March 2000, (f) a DTPa vaccine by North American Vaccine is included, (g) HBV can be administered in month 0–1 and month 2, (h) the vaccine preparation costs are computed using an expanded breakdown, (i) the DTPa-HIB vaccine by Aventis Pasteur can only be administered in the 12–18 or 60 month periods, (j) the DTPa-HIB vaccine by Wyeth–Lederle is not under contract with the CDC, hence is excluded from the model. Note that these assumptions are based on the guidelines as set forth in the 2000 Recommended Childhood Immunization Schedule (see [4]). Individual situations that deviate from this schedule are not considered, and are beyond the scope of this study. 1.1. Integer programming model An integer programming model was developed to determine the price at which four different combination vaccines provide a good economic value in the marketplace under different scenarios and perspectives. This model was also used to identify the optimal vaccine formularies over the first five years of immunization that include each of these combination vaccines individually, hence demonstrate the impact on the number of vaccines and the overall cost when different combination vaccines become available. The decision variables for the integer programming model are all either nonnegative integers or binary (0–1). A set of binary decision variables is defined for each month in which a particular vaccine (by manufacturer) can be administered (months 0–1, 2, 4, 6, 12–18, and 60), where a value of one (zero) indicates that the vaccine should (not) be administered in that month. The decision variables are also indexed by the particular manufacturer of each vaccine. These indices are numbered from one to five, ordered as follows: 1 ≡ Aventis Pasteur, 2 ≡ Merck, 3 ≡ North American Vaccine, 4 ≡ GlaxoSmithKline, and 5 ≡ Wyeth–Lederle. Each of the binary decision variables denotes whether a vaccine is scheduled for a particular month’s visit. For example, if vaccine IPV1 manufactured by AVP is (not) scheduled for month j ∈ M = {0−1, 2, 4, 6, 12−18, 60}, then IPV1,j = 1 (0), where M represents the set of months/periods when vaccines can be scheduled and administered. Also, to capture a skipped month 6 injection for the HIB vaccine manufactured by MRK (see assumption vi), the binary decision variable HIBskip2,6 was defined, where HIBskip2,6 = 1 if manufacturer MRK HIB vaccine is used in months 2 and 4, hence the month 6 vaccine can be skipped, or 0 otherwise. All these binary decision variables are

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Table 2 Integer programming model: vaccine binary decision variables. Decision variable

Total cost/dose coefficient

Vaccine

DTPa1,. DTPa3,. DTPa4,. DTPa5,. HIB1,. HIB2,. HIBskip2,. HIB5,. DTPa-HIB1,. HBV2,. HBV4,. HIB-HBV2,. IPV1,. DTPa-HBV-IPV4,. DTPa-HIB-IPV1,. DTPa-HIB-IPV3,. DTPa-HIB-IPV4,. DTPa-HIB-IPV5,. DTPa-HIB-HBV4,. DTPa-HIB-HBV-IPV1,. DTPa-HIB-HBV-IPV4,.

c-dtpa1 c-dtpa3 c-dtpa4 c-dtpa5 c-hib1 c-hib2 – c-hib5 c-dtpa-hib1 c-hbv2 c-hbv4 c-hib-hbv2 c-ipv1 c-dtpa-hbv-ipv4 c-dtpa-hib-ipv1 c-dtpa-hib-ipv3 c-dtpa-hib-ipv4 c-dtpa-hib-ipv5 c-dtpa-hib-hbv 4 c-dtpa-hib-hbv-ipv1 c-dtpa-hib-hbv-ipv4

DTPa DTPa DTPa DTPa HIB HIB HIB HIB DTPa-HIB HBV HBV HBV-HIB IPV DTPa-HBV-IPV DTPa-HIB-IPV DTPa-HIB-IPV DTPa-HIB-IPV DTPa-HIB-IPV DTPa-HIB-HBV DTPa-HIB-HBV-IPV DTPa-HIB-HBV-IPV

Manufacturer AVP NAV GSK WAL AVP MRK MRK WAL AVP MRK GSK MRK AVP GSK AVP NAV GSK WAL GSK AVP GSK

summarized in table 2. Lastly, the binary decision variables mj are defined to capture whether there is a clinic visit scheduled for month j , j ∈ M, where mj = 1 (0) if there is (not) a clinic visit in month j . The integer decision variables sj are defined to capture the number of injections given at the clinic visit in month j , j ∈ M. Note that if a particular manufacturer does not provide a particular vaccine, then the associated total cost/dose coefficient is set to infinity, or for practical purposes, to an arbitrarily large positive value to ensure that it does not enter the optimal solution. For example, since MRK does not provide DTPa, then c-dtpa2 = +∞. The objective function for the integer programming model is 5 [c-dptai DTPai,j + c-hibi HIBi,j + c-dtpa-hibi DTPa-HIBi,j 40mj + 15sj + j ∈M

i=1

+ c-hbvi HBVi,j + c-hib-hbvi HIB-HBVi,j + c-ipvi IPVi,j + c-dtpa-hbv-ipv i DTPa-HBV-IPVi,j + c-dtpa-hib-ipv i DTPa-HIB-IPVi,j + c-dtpa-hib-hbv i DTPa-HIB-HBVi,j

+ c-dtpa-hib-hbv-ipv i DTPa-HIB-HBV-IPVi,j ] .


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The constraints for the model are given below, based on the assumptions defined in section 1, (i)–(vii) and (a)–(j): • Recommended Childhood Immunization Schedule (March 2000) constraints: – Case 1: HBV is administered in the month 0–1 visit HBV2,0−1 + HBV4,0−1 = 1; Case 2: HBV is not administered in the month 0–1 visit HBVi,2 + HIB-HBV2,2 + DTPa-HBV-IPV4,2 + DTPa-HIB-HBV4,2 i∈{2,4}

+

DTPa-HIB-HBV-IPVi,2 = 1.

i∈{1,4}

– Vaccine containing HBV must be scheduled in the month 2 or 4 visits HBVi,j + HIB-HBV2,j + DTPa-HBV-IPV4,j + DTPa-HIB-HBV4,j j ∈{2,4}

i∈{2,4}

+

DTPa-HIB-HBV-IPVi,j 1.

i∈{1,4}

– Vaccine containing HBV must be scheduled in the month 6 or 12–18 visits HBVi,j + HIB-HBV2,j + DTPa-HBV-IPV4,j j ∈{6,12−18}

i∈{2,4}

+ DTPa-HIB-HBV4,j +


i∈{1,4}

– Must have two vaccines containing HBV in the first 4 months of visits HBVi,j + HIB-HBV2,j + DTPa-HBV-IPV4,j j ∈{0−1,2,4}

i∈{2,4}

j ∈{2,4}

+ DTPa-HIB-HBV4,j +


i∈{1,4}

– Vaccine containing DTPa must be scheduled in the month 2, 4 and 6 visits DTPai,j + DTPa-HBV-IPV4,j + DTPa-HIB-HBV4,j i∈{1,3,4,5}

+

[DTPa-HIB-IPVi,j + DTPa-HIB-HBV-IPVi,j ] 1,

i∈{1,4}

j = 2, 4, 6.

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– Vaccine containing DTPa must be scheduled in the month 12–18 and 60 visits DTPai,j + DTPa-HIB1,j + DTPa-HBV-IPV4,j + DTPa-HIB-HBV4,j i∈{1,3,4,5}

+

[DTPa-HIB-IPVi,j + DTPa-HIB-HBV-IPVi,j ] 1,

j = 12−18, 60.

i∈{1,4}

– Extravaccination is not permitted for DTPa DTPai,j + DTPa-HIB-HBV4,j + DTPa-HBV-IPV4,j j ∈M

i∈{1,3,4,5}

+

[DTPa-HIB-IPVi,j

+ DTPa-HIB-HBV-IPVi,j ]

i∈{1,4}

+

DTPa-HIB1,j = 5.

j ∈{12−18,60}

– Vaccine containing HIB must be scheduled in the month 2 and 4 visits HIBi,2 + HIB-HBV2,2 + DTPa-HIB-HBV4,2 i∈{1,2,5}

+

[DTPa-HIB-IPVi,2 + DTPa-HIB-HBV-IPVi,2 ] 1,

j = 2, 4.

i∈{1,4}

– Vaccine containing HIB must be scheduled in the month 6 visit if a Merck brand HIB vaccine was not used in the months 2 and 4 visits HIBi,6 + HIB-HBV2,6 + DTPa-HIB-HBV4,6 i∈{1,2,5}

+

[DTPa-HIB-IPVi,6 + DTPa-HIB-HBV-IPVi,6 ] + HIBskip2,6 1,

i∈{1,4}

HIB2,j + HIB-HBV2,j − HIBskip2,6 0, j = 2, 4, [HIB2j + HIB-HBV2,j ] − HIBskip2,6 1, j ∈{2,4}

HIBi,6 + HIBskip2,6 1.

i∈{1,2,5}

– Vaccine containing HIB must be scheduled in the month 12–18 visit HIBi,12−18 + DTPa-HIB1,12−18 + HIB-HBV2,12−18 i∈{1,2,5}

+ DTPa-HIB-HBV4,12−18 +

[DTPa-HIB-IPVi,12−18

i∈{1,4}

+ DTPa-HIB-HBV-IPVi,12−18 ] 1.


271

– Vaccine containing IPV must be scheduled in the month 2, 4 and 60 visits [DTPa-HIB-IPVi,j IPV1,j + DTPa-HBV-IPV4,j + i∈{1,4}

+DTPa-HIB-HBV-IPVi,j ] 1,

j = 2, 4, 60.

– Vaccine containing IPV must be scheduled in the month 6 or 12–18 visits IPV1j + DTPa-HBV-IPV4,j j ∈{6,12−18}

+

[DTPa-HIB-IPVi,j

+ DTPa-HIB-HBV-IPVi,j ] 1.

i∈{1,4}

• Manufacture Matching is required for DTPa: – Matching for Aventis Pasteur brand DTPa vaccines DTPa1,j + I {j = 12−18}DTPa-HIB1,j + DTPa-HIB-IPV1,j + DTPa-HIB-HBV-IPV1,j DTPa1,k + I {k = 12−18 or k = 60}DTPa-HIB1,k + DTPa-HIB-IPV1,k + DTPa-HIB-HBV-IPV1,k , (j, k) ∈ (2, 4), (4, 6), (6, 12−18), (12−18, 60) , where I {j = j1 } = 1 if j = j1 , and 0 otherwise. – Matching for North American brand and Wyeth–Lederle brand DTPa vaccines DTPai,j + DTPa-HIB-IPVi,j DTPai,k + DTPa-HIB-IPVi,j , i = 3, 5, (j, k) ∈ (2, 4), (4, 6), (6, 12−18), (12−18, 60) . – Matching for GlaxoSmithKline brand DTPa vaccines DTPa4,j + DTPa-HBV-IPV4,j + DTPa-HIB-IPV4,j + DTPa-HIB-HBV4,j + DTPa-HIB-HBV-IPV4,j DTPa4,k + DTPa-HBV-IPV4,k + DTPa-HIB-IPV4,k + DTPa-HIB-HBV4,k + DTPa-HIB-HBV-IPV4,k , (j, k) ∈ (2, 4), (4, 6), (6, 12−18), (12−18, 60) . – Compute the Number of Injections per Clinic Visit HBV2,0−1 + HBV4,0−1 = s0−1 , DTPai,j + HIBi,j + HBVi,j + IPV1,j + HIB-HBV2,j i∈{1,3,4,5}

i∈{1,2,5}

i∈{2,4}

+ DTPa-HBV-IPV4,j + DTPa-HIB-HBV4,j + I {j = 12−18}DTPa-HIB1,12−18

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+

[DTPa-HIB-IPVi,j + DTPa-HIB-HBV-IPVi,j ] = sj ,

i∈{1,4}

j = 2, 4, 6, 12−18, DTPa1,60 + DTPa3,60 + DTPa4,60 + DTPa5,60 + IPV1,60 + DTPa-HIB1,60 + DTPa-HBV-IPV4,60 + DTPa-HIB-IPV1,60 + DTPa-HIB-IPV4,60 + DTPa-HIB-HBV4,60 + DTPa-HIB-HBV-IPV1,60 + DTPa-HIB-HBV-IPV4,60 = s60 . – Ensure that a visit occurs whenever an injection is given s0−1 m0−1 , sj 10 mj , 2.

j = 2, 4, 6, 12−18, 60.

Results

The integer programming model was designed such that all the experiments could be conducted using this model, with the appropriate decision variables set to zero to obtain each of the desired scenarios (HBV birth dose versus HBV nonbirth dose) and perspectives (five different injection costs) across the four combination vaccines (six cases), resulting in sixty different integer programming model experiments and solutions. Each integer programming model contains 94 integer variables, of which 88 are binary variables, and 76 constraints. The integer programming model was created using AMPL Plus 1.6 and solved using the CPLEX 6.5 LP and MIP Solver on a Pentium 233 MHz IBM-compatible personal computer. The reason that there are six cases needed to analyze the four combination vaccines is a result of manufacturer brand matching for DTPa (assumption (vii)). Table 3 provides a list of the formulations and packaging for the four combination vaccines, as well as the manufacturers who either have a licensed product of the vaccine type sold anywhere outside the United States, have ever conducted clinical trials of the vaccine Table 3 Combination vaccine information. Combination vaccines

Formulation/ packaging

Manufacturer(s)

Assumed preparation cost/dose

DTPa-HIB-HBV [p] DTPa-HIB-IPV [p] DTPa-HIB-IPV [v] DTPa-HBV-IPV [v] DTPa-HBV-IPV [s] DTPa-HIB-HBV-IPV [v] DTPa-HIB-HBV-IPV [s]

powder powder liquid/vial liquid/vial liquid/syringe liquid/vial liquid/syringe

GSK AVP, GSK NAV, WAL GSK GSK AVP, GSK AVP, GSK

$1.50 $1.50 $0.75 $0.75 $0.25 $0.75 $0.25


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type, or have indicated plans to do so, even if subsequently abandoned. Using this information, since DTPa matching is required, then a total of six different cases for the four combination vaccines had to be studied. For DTPa-HIB-HBV, there is just one case, since only GlaxoSmithKline is manufacturing or conducting clinical trials for this vaccine. This is also true for DTPa-HBV-IPV. For DTPa-HIB-IPV, four manufacturers are either manufacturing or conducting clinical trials for this vaccine. However, since the purchase price and packaging for all the DTPa products are the same across all the manufacturers, then due to DTPa manufacturer brand matching (see assumption (vii)), the results for any one manufacturer will be the same for any of the other manufacturers. The only exception to this is for Aventis Pasteur, which markets a DTPa-HIB combination vaccine. Therefore, the analysis must be done separately for this combination vaccine when manufactured by Aventis Pasteur. The same reasoning explains why two cases must be considered for DTPa-HIB-HBV-IPV. These two additional cases will be referred to as the AVP variations in the tables of results (see tables 6 and 8). Each of the sixty different integer programming model formulations had to be solved several times to determine the maximum price at which the specified combination vaccine provided a good economic value. This was done using a bisection search [1], where an upper and lower bound for the maximum price of the combination vaccine was set, and based on whether the upper or lower bound resulted in the vaccine entering the optimal formulary, the mid-point between the upper and lower bound replaced one of these values. Note that the resulting maximum prices obtained depend on whether these combination vaccines are part of the optimal vaccine formulary one, two, three or four times (depending on which combination vaccine is considered). All the prices listed in tables 4–9 are rounded to the nearest dollar. The price of each combination vaccine was initially set at a large value ($500) such that it would not appear in the optimal solution. These prices were then changed, one at a time, to determine the maximum prices as reported in tables 4–9. For each combination vaccine, the prices listed in tables 4–9 are for the vaccine entering the formulary one, two, three, or four times. Note that the pricing structure is such that for the HBV nonbirth dose scenario, the price for the two dose formulary is the same as the three dose formulary. This is also the case for the HBV birth dose scenario for perspectives (i) and (ii) between the two dose and three dose formularies in tables 5 and 6 and between the three and four dose formularies in tables 4 Table 4 Maximum DTPa-HIB-HBV prices (GSK variation, powder [p]).

Cost of an injection

One dose

Perspective (i) Perspective (ii) Perspective (iii) Perspective (iv) Perspective (v)

$34 $38 $50 $60 $75

$5 $8 $20 $30 $45

HBV birth dose scenario Two Three Four doses doses doses $27 $27 $32 $39 $49

$20 $23 $32 $39 $49

$20 $23 $32 $39 $49

HBV nonbirth dose scenario One Three Four dose doses doses $34 $38 $50 $60 $75

$30 $31 $37 $43 $53

$20 $23 $35 $43 $53

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Table 5 Maximum DTPa-HIB-IPV prices (NAV and WAL variation, liquid in vial [v])a .


One dose


$33 $38 $52 $67 $89

$5 $8 $20 $30 $45


$30 $34 $50 $60 $75

$22 $25 $37 $47 $62


$29 $32 $44 $53 $69

$22 $25 $37 $47 $62

a The price for the GSK product would be $0.75 less than the prices shown in this table because it is

packaged as a powder, not as a liquid in a vial. Table 6 Maximum DTPa-HIB-IPV prices (AVP variation, powder [p]).


One dose


$33 $37 $51 $71 $101

$5 $8 $20 $30 $45


$29 $33 $49 $59 $74

$22 $25 $37 $47 $62


$28 $31 $43 $53 $68

$22 $25 $37 $47 $62

Table 7 Maximum DTPa-HBV-IPV prices (GSK variation, liquid in syringe [s]).


HBV birth dose scenario One Two Three Four dose doses doses doses

HBV nonbirth dose scenario One Three Four dose doses doses


$37 $41 $53 $63 $78

$37 $41 $53 $63 $78

$5 $8 $20 $30 $45

$36 $39 $51 $61 $76

$23 $26 $38 $48 $63

$23 $26 $38 $48 $63

$36 $39 $51 $61 $76

$23 $26 $38 $48 $63

and 7, as well as for perspectives (iii)–(v) between the two dose, three dose, and four dose formularies in table 4, between the three dose and four dose formularies in table 7, and between the two dose and three dose formularies in table 9. The results presented in tables 4–9 are the maximum CDC prices that can be set for the four combination vaccines such that they enter the optimal formulary a specified number of times. Therefore, at such prices, the combination vaccines provide good economic value, based on the five different cost perspectives associated with administering each injection. For example, if a health-care provider values the cost of an injection as


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Table 8 Maximum DTPa-HIB-HBV-IPV prices (AVP variation, liquid in syringe [s]).


One dose


$48 $55 $79 $99 $129

$5 $8 $20 $30 $45


$34 $40 $64 $84 $114

$23 $26 $38 $48 $63


$44 $49 $73 $93 $123

$23 $26 $38 $48 $63

Table 9 Maximum DTPa-HIB-HBV-IPV prices (GSK variation, liquid in syringe [s]).


One dose


$48 $55 $79 $99 $129

$5 $8 $20 $30 $45


$34 $40 $60 $75 $97

$23 $26 $38 $48 $63


$44 $48 $66 $81 $104

$23 $26 $38 $48 $63

Table 10 Optimal formulary with CDC licensed vaccines. Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

HIB-HBV2 , DTPa1 , IPV1 HIB-HBV2 , DTPa1 , IPV1 DTPa1 HIB-HBV2 , DTPa1 , IPV1 DTPa1 , IPV1

$37.99 $37.99 $9.25 $37.99 $17.00

$1.75 $1.75 $0.75 $1.75 $1.00

$60 $60 $20 $60 $40

$40 $40 $40 $40 $40

$139.74 $139.74 $70.00 $139.74 $98.00

$140.22

$7.00

$240

$200

$587.22

Total costs

$20 (perspective (iii)), and wishes to stock combination vaccine DTPa-HIB-HBV for two doses per child for administration during the first five years of the immunization schedule, with a child already having been administered a birth dose of HBV, then this combination should only be stocked if its price is at or below $32. On the other hand, if a second health-care provider values the cost of an injection as $8 (perspective (ii)), and wishes to stock combination vaccine DTPa-HIB-HBV for three doses per child for administration during the first five years of the immunization schedule, with a birth dose of HBV not already administered, then this combination should only be stocked if its price is at or below $31.

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Table 11 Optimal formulary with combination vaccine DTPa-HIB-HBV (GSK variation, powder [p]). Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

DTPa-HIB-HBV4 , IPV1 DTPa-HIB-HBV4 , IPV1 DTPa4 , HIB5 , IPV1 DTPa-HIB-HBV4 DTPa4 , IPV1

$44.99 $44.99 $22.25 $37.24 $17.00

$1.75 $1.75 $1.75 $1.50 $1.00

$40 $40 $60 $20 $40

$40 $40 $40 $40 $40

$126.74 $126.74 $124.00 $98.74 $98.00

$166.47

$7.75

$200

$200

$574.22

Total costs

Table 12 Optimal formulary with combination vaccine DTPa-HIB-IPV (NAV and WAL variation, liquid in vial [v])a . Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

HBV4 , DTPa-HIB-IPV∗ HBV4 , DTPa-HIB-IPV∗ DTPa-HIB-IPV∗ HIB-HBV2 , DTPa* DTPa∗ , IPV1

$52.49 $52.49 $43.49 $30.24 $17.00

$1.00 $1.00 $0.75 $1.50 $1.00

$40 $40 $20 $40 $40

$40 $40 $40 $40 $40

$133.49 $133.49 $104.24 $111.74 $98.00

$195.71

$5.25

$180

$200

$580.96

Total costs

a The costs shown in this table are for the NAV and WAL combination vaccines, which are packaged as a

liquid in a vial. The GSK combination is packaged as a powder. This causes the per dose cost of the GSK vaccine to be $0.75 less than the cost shown in this table, and the preparation cost to be $0.75 more. Table 13 Optimal formulary with combination vaccine DTPa-HIB-IPV (AVP variation, powder [p]). Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

HBV4 , DTPa-HIB-IPV1 HBV4 , DTPa-HIB-IPV1 DTPa-HIB-IPV1 HIB-HBV2 , DTPa1 DTPa1 , IPV1

$51.74 $51.74 $42.74 $30.24 $17.00

$1.75 $1.75 $1.50 $1.50 $1.00

$40 $40 $20 $40 $40

$40 $40 $40 $40 $40

$133.49 $133.49 $104.24 $111.74 $98.00

$193.46

$7.50

$180

$200

$580.96

Total costs

The reverse engineering procedure used to obtain the prices for the four combination vaccines listed in tables 4–9 also provides an optimal vaccine formulary (hence indicates not only the number of combination vaccines included in this formulary but the actual administration months/periods in the immunization schedule). These formularies are presented for the HBV nonbirth dose scenario in perspective (iii) for the cost of each injection, with three doses of the combination vaccine administered. Tables 11–16 present the details of these formularies, including their total cost, the specific vaccines


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Table 14 Optimal formulary with combination vaccine DTPa-HBV-IPV (GSK variation, liquid in syringe [s]). Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

DTPa-HBV-IPV4, HIB2 DTPa-HBV-IPV4, HIB2 DTPa-HBV-IPV4 DTPa4 , HIB5 DTPa4 , IPV1

$58.74 $58.74 $50.99 $14.50 $17.00

$1.00 $1.00 $0.25 $1.50 $1.00

$40 $40 $20 $40 $40

$40 $40 $40 $40 $40

$139.74 $139.74 $111.24 $96.00 $98.00

$199.97

$4.75

$180

$200

$584.72

Total costs

Table 15 Optimal formulary with combination vaccine DTPa-HIB-HBV-IPV (AVP variation, liquid in syringe [s]). Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

DTPa-HIB-HBV-IPV1 DTPa-HIB-HBV-IPV1 DTPa-HIB-HBV-IPV1 DTPa-HIB1 DTPa1 , IPV1

$72.73 $72.73 $72.73 $22.01 $17.00

$0.25 $0.25 $0.25 $1.50 $1.00

$20 $20 $20 $20 $40

$40 $40 $40 $40 $40

$132.98 $132.98 $132.98 $83.51 $98.00

$257.20

$3.25

$120

$200

$580.45

Total costs

Table 16 Optimal formulary with combination vaccine DTPa-HIB-HBV-IPV (GSK variation, liquid in syringe [s]). Month

Scheduled vaccines

Vaccine cost

Preparation cost

Injection cost

Clinic visit cost

Total cost

2 4 6 12–18 60

DTPa-HIB-HBV-IPV4 DTPa-HIB-HBV-IPV4 DTPa-HIB-HBV-IPV4 DTPa4 , HIB5 DTPa4 , IPV1

$66.49 $66.49 $66.49 $14.50 $17.00

$0.25 $0.25 $0.25 $1.50 $1.00

$20 $20 $20 $40 $40

$40 $40 $40 $40 $40

$126.74 $126.74 $126.74 $96.00 $98.00

$230.97

$3.25

$140

$200

$574.22

Total costs

(including the number of vaccine products required), and the number of manufacturers represented in the solution. As a base case, the optimal vaccine formulary that includes just the twelve vaccine products currently licensed and under contract for distribution by the CDC was also obtained and is depicted in table 10. Comparing the total cost from table 10 with the total costs from tables 11–16 suggest that there is no significant cost advantage in stocking and administering combination vaccines to help satisfy the Recommended Childhood Immunization Schedule guidelines for Hepatitis B, Diphtheria, Tetanus, Pertussis, Haemophilus influenzae type B, and Polio, if all the combination vaccines products are optimally priced. However, the

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solution in table 10 uses twelve injections, whereas the solutions in tables 11–16 use between six and ten injections. Note that the total injection cost in table 10 is $240, while the total injection costs in tables 11–16 range from $120 to $200. Therefore, the integer programming model solution shifts the full cost savings from fewer injections into the economic premium associated with administering fewer total numbers of vaccines using combination vaccine products. The economic advantage to the health-care system of one combination vaccine over another can only be assessed once these products are actually priced in the marketplace. Therefore, as such combination vaccine products become licensed and available, the results depicted in tables 4–9 provide health-care providers with practical guidelines to evaluate whether such products are economically viable for their particular immunization environments. To further assess the relationship between the cost of an injection and the maximum combination prices, a sensitivity analysis of the prices for the four combination vaccines with respect to the injection cost was obtained. In particular, by considering the range of injection costs from $1 to $45, in increments of $1, one can determine how the combination prices change. Figures 1–6 provide graphs for the four combination vaccines, at the two dose and three dose levels for the birth dose case, and for three doses for the nonbirth dose case. These figures are ordered by the types of combination corresponding to the results in tables 4–9. These figures suggest several observations. First, all the curves are piece-wise linear and nondecreasing. Moreover, only in figure 1 is there a flat region where an increase in the injection cost does not result in an increase in the maximum price of a combination vaccine. Also, for combination vaccines that contain HBV, the maximum prices

Figure 1. Maximum DTPa-HIB-HBV prices (GSK variation, powder [p]).


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Figure 2. Maximum DTPa-HIB-IPV prices (NAV and WAL variation, liquid in vial [v]).

Figure 3. Maximum DTPa-HIB-IPV prices (AVP variation, powder [p]).

for the HBV nonbirth dose scenario are always larger than the maximum prices for the HBV birth dose scenario. This is not surprising, since the HBV birth dose scenario results in one fewer possible dose of a HBV containing combination vaccine to satisfy the childhood immunization schedule, hence puts downward pressure on its price.

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Figure 4. Maximum DTPa-HBV-IPV prices (GSK variation, liquid in syringe [s]).

Figure 5. Maximum DTPa-HIB-HBV-IPV prices (AVP variation, liquid in syringe [s]).

Moreover, for the combination vaccines that do not contain HBV, this pricing result is reversed. Each of the figures provides unique insights into how the different combination vaccines fit into the recommended childhood immunization schedule. For example, in


281

Figure 6. Maximum DTPa-HIB-HBV-IPV prices (GSK variation, liquid in syringe [s]).

figure 1, the two dose HBV birth dose scenario has a flat region where increases in the cost of an injection do not translate into an increase in the maximum vaccine price. Moreover, the two dose and three dose maximum prices remain the same once the injection cost reaches $12. In figure 2, for injection costs up to $4, the two dose and three dose HBV birth dose scenarios and the three dose HBV nonbirth dose scenario all have the same maximum combination price. However, beyond this injection cost, the maximum prices diverge. A similar result is observed in figure 3. In figure 4, the maximum prices for the two dose HBV birth dose scenario and the three dose HBV nonbirth dose scenario are identical. In both figures 5 and 6, the maximum prices for the two dose HBV birth dose scenario is sandwiched between the maximum prices for the three dose HBV birth dose and nonbirth dose scenarios. The key conclusions to draw from these observations is that each combination vaccine results in a unique sensitivity with respect to the cost of an injection. 3.

Conclusions

This paper uses an integer programming model to reverse engineer the price of four combination vaccines products for childhood immunization. The prices for the four combination vaccines listed in tables 4–9 are the maximum prices at which these products provide good economic value under different cost of injection perspectives and HBV birth dose versus HBV nonbirth dose scenarios. Note that these prices may be highly sensitive to the data described in section 1, including the price of each vaccine product listed in table 1, the removal of any vaccine product listed in table 1, the addition of any new vaccine or vaccine combination into the list of vaccines in table 1 (recall that

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this study does not match each of the combination vaccines against each other, but rather looks at the effect of adding each combination vaccine individually to the currently available and licensed vaccine products), and the preparation costs for each of the vaccines listed in table 1. The primary practical advantage of using combination vaccine products is that they require fewer injections. Therefore, combination vaccines create an opportunity for additional vaccine products to be added to the Recommended Childhood Immunization Schedule, without the additional burden of children enduring excessive numbers of injections at each clinic visit. The cost savings to society and the health-care system in having fewer children contracting the diseases prevented by such additional vaccines is a factor that is not captured in this analysis. Assuming that vaccination for disease prevention is cost-effective, this would only serve to elevate the value of combination vaccines, hence justify possibly higher prices than those indicated in this study. This also provides incentive for the development and introduction of new vaccines for existing diseases and new combination vaccine products for childhood immunization. The maximum prices for the four combination vaccines are obtained based on the assumptions stated in section 1. Note that the effect of issues such as brand loyalty and other behavioral factors were not included in this study, hence the results reported do not capture such factors. In addition, the data used to obtain the maximum prices, such as the cost of administering an injection and the cost of vaccine preparation, are highly dependent on specific factors germane to individual health-care providers, hence any changes in these data may result in changes to the maximum prices for the combination vaccines. Note that under assumption (ii), changes in the cost of a clinic visit will change the cost of the optimal vaccine formulary, but has no effect on the vaccine products in this optimal formulary. Therefore, given these limitations, the maximum prices given in tables 4–9 should serve as general guidelines, rather than precise values as to how the combination vaccines should be valued in the marketplace, since any or all of these other factors may serve to either increase or decrease such prices. One concern of using combination vaccines is that extravaccination may occur. With the exception of DTPa (for which extravaccination is not permitted in the integer programming model developed), there are no recognized negative side effects of extravaccination for HIB, HBV, or IPV [5]. However, for the optimal formularies listed in tables 11–16, there was no extravaccination. Since no extravaccination also occurred in the base case (see table 10), these results suggest that using combination vaccines should result in fewer vaccine products being needed (hence fewer injections to be administered) to satisfy the guidelines outlined in the Recommended Childhood Immunization Schedule. Moreover, by carefully stocking both combination vaccines and certain monovalent vaccine products, the need to stock a large number of different vaccine products may be reduced. As new vaccine combination products enter the market and become licensed for distribution and administration, the combinatorial explosion of choices available to health-care providers will make it even more challenging to make sound economic decisions [9]. The operations research modeling approach presented in this paper pro-


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vides a systematic methodology to address such issues, hence encourages intelligent, cost-effective decision-making in the rapidly expanding combination vaccine development arena. Moreover, once these combination vaccines are priced in the marketplace, health-care providers and insurance companies can use the operations research modeling approach introduced in this study to assess whether such products provide a good value for their particular health-care environments. The publicly available web site, www.vaccineselection.com, provides one such avenue for such analysis.

Acknowledgments The authors would like to thank the special issue editor, Dr. Eva K. Lee, the Associate Editor, and two anonymous referees for their insightful comments and suggestions. The authors would also like to express their thanks and gratitude to Dr. Bruce G. Weniger of the Centers for Disease Control and Prevention for his helpful comments and feedback on this paper, for providing access to data, as well as for his overall long-term support for this line of research. The second author is supported in part by the Air Force Office of Scientific Research (F49620-01-1-0007). This research is supported by the National Science Foundation (OMI-0222554, OMI-0222597).

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