Using Monte Carlo Simulation to Determine ... - Springer Link

Health Care Management Science 5, 135–145, 2002  2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Using Monte Carlo Simulation to Determine Combination Vaccine Price Distributions for Childhood Diseases SHELDON H. JACOBSON Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2906, USA E-mail: [email protected]

EDWARD C. SEWELL Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA E-mail: [email protected] Received November 2000; Revised April 2001; accepted June 2001

Abstract. The Recommended Childhood Immunization Schedule provides guidelines that allow pediatricians to administer childhood vaccines in an efficient and effective manner. Research by vaccine manufacturers has resulted in the development of new vaccines that protect against a growing number of diseases. This has created a dilemma for how to insert such new vaccines into an already crowded immunization schedule, and prompted vaccine manufacturers to develop vaccine products that combine several individual vaccines into a single injection. Such combination vaccines permit new vaccines to be inserted into the immunization schedule without requiring children to be exposed to an unacceptable number of injections during a single clinic visit. Given this advantage, combination vaccines merit an economic premium. The purpose of this paper is to describe how Monte Carlo simulation can be used to assess and quantify this premium by studying four combination vaccines that may become available for distribution within the United States. Each combination vaccine is added to twelve licensed vaccine products for six childhood diseases (diphtheria, tetanus, pertussis, haemophilus influenzae type B, hepatitis B, and polio). Monte Carlo simulation with an integer programming model is used to determine the (maximal) inclusion price distribution of four combination vaccines, by randomizing the cost of an injection. The results of this study suggest that combination vaccines warrant price premiums based on the cost assigned to administering an injection, and that further developments and innovations in this area by vaccine manufacturers may provide significant economic and societal benefits. Keywords: economics, combination vaccines, pediatric immunization, integer programming, Monte Carlo simulation

1. Introduction and background The United States Recommended Childhood Immunization Schedule [5] has become increasingly crowded, requiring children to endure a large number of vaccine injections over several years [22]. Moreover, vaccine manufacturers have developed and are launching new vaccines for diseases not currently part of the schedule; such new products will exacerbate this crowding dilemma. Children and parents/guardians have limited tolerance for multiple injections during a single clinic visit [14]. Moreover, parents/guardians may not take the time (and bear the cost) to make additional clinic visits for deferred vaccination [7]. Such noncompliance with recommended vaccine scheduling places children at an increased risk to contract diseases that the vaccines are designed to prevent. These issues result in a significant cost to both the family unit, as well as the nation’s increasingly strained healthcare system. Several solutions have been proposed to overcome these complications (see [22] for a discussion on these issues). The ideal solution would be a single dose oral vaccine that immunizes children at birth from all childhood diseases [21]. A more realistic solution is to develop combination vaccines (i.e., vaccine products that combine several individual vac-

cines into a single injection [4]) to reduce the number of injections and clinic visits needed to comply with the Recommended Childhood Immunization Schedule [17]. If additional clinic visits can be avoided, combination vaccines may lead to a significantly lower economic burden on parents/guardians. Moreover, children are not exposed to a large number of injections during any single clinic visit. Lastly, as vaccine manufacturers develop new vaccines to protect against the large number of diseases to which children are exposed, combination vaccines provide a convenient avenue to make room for new vaccines that may need to be added to the already crowded Recommended Childhood Immunization Schedule. For example, in 2000, the United States Food and Drug Administration (FDA) approved the first pneumococcal conjugate vaccine for prevention of invasive pneumococcal disease in infants [1,6]. Moreover, the Centers for Disease Control and Prevention’s Advisory Committee on Immunization Practices (ACIP) recommends such a vaccine for child immunization, adding four injections during the first two years of life, hence resulting in further crowding of the Recommended Childhood Immunization Schedule. Vaccine manufacturers have recognized the advantages of developing combination vaccines. Aventis Pasteur (AVP), North American Vaccine (NAV), GlaxoSmithKline (GSK),

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and Wyeth Lederle (WAL) have worked to develop various combination vaccines that provide immunization against up to six diseases (diphtheria, tetanus, pertussis, haemophilus influenzae type B, hepatitis B, and polio). Given the increasing number of combination vaccines that may be available over the next decade, it will become increasingly more difficult for health care providers and immunization programs to assess the value of a particular combination vaccine product. For example, are the clinical and consumer savings incurred through vaccine combinations, hence reduced numbers of injections, sufficient to overcome the potential higher purchase prices for such products? Moreover, it is not clear how combination vaccines will impact vaccine formularies (i.e., the set of vaccines stocked by a healthcare provider) to ensure that the Recommended Childhood Immunization Schedule guidelines can be satisfied. Lastly, since combination vaccine products make room for additional vaccines in an immunization schedule, are the price premiums for such products justified by the resulting lower disease incidence (hence associated cost savings) for the diseases protected against with these additional vaccines? All these questions are difficult to address. The purpose of this paper is to provide insights into the economic value questions for four combination vaccines that may become available for distribution within the United States. This paper uses operations research models to determine the economic value of combination vaccines. Healthcare providers and parents/guardians each place a different value (hence cost) on each injection (or more significantly, on reducing the number of injections). For a given set of injection costs, there is an (maximal) inclusion price at which a combination vaccine provides a good economic value (i.e., the highest price at which the combination vaccine earns a place in the lowest cost vaccine formulary). The inclusion price can be determined by iteratively solving an integer programming model [8]. Monte Carlo simulation is used to sample the injection costs from a set of injection cost probability distributions, where each injection cost probability distribution corresponds to the values that a population of parents/ guardians place on avoiding an injection. The resulting set of inclusion prices for each combination vaccine is used to create (estimate) the probability distribution of inclusion prices for that combination vaccine, where this probability distribution can be used, for example, to determine the probability that the population of parents/guardians will deem the combination vaccine a good value (over existing vaccines) at a given inclusion price. Therefore, the inclusion price probability distribution provides valuable information for vaccine manufacturers, since it provides a tool for estimating the market share that can be secured from a particular population (based on the injection cost probability distribution). By using different injection cost probability distributions, the sensitivity of the form of this probability distribution on the inclusion price distribution can be assessed. The study uses Monte Carlo simulation in conjunction with an integer programming model to study the relationship between injection costs and combination vaccine inclu-

S.H. JACOBSON, E.C. SEWELL

sion prices for immunization against six childhood diseases (diphtheria, tetanus, pertussis, haemophilus influenzae type B, hepatitis B, and polio) over the first five years of the Recommended Childhood Immunization Schedule. Note that this study is a natural extension of the study reported in [19] that uses an integer programming model to reverse engineer the inclusion price of these four combination vaccines based on specified injection costs. The results from the study reported here focus on building probability distributions for the inclusion price of combination vaccines, hence provides a systematic approach to determining the economic value of combination vaccines in the marketplace. This in turn allows one to assess whether price premiums for such vaccine products are economically justified. The paper is organized as follows: section 2 describes the assumptions for the study, including the key issues addressed and the limitations of the methodologies being used. Section 3 discusses the integer programming model developed and used in the study. Section 4 discusses the results obtained from the integer programming model, including combination vaccine product pricing information and formulary compositions. Section 5 presents a discussion on the limitations of the results, as well as concluding comments and directions for future research.

2. Model 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. Note that the vaccine for diptheria, tetanus, and pertussis is packaged as a single injection, hence by definition is a combination vaccine. However, it was not analyzed in this study, since vaccines for the individual components are not under Federal contract with the CDC. 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 -HIB-HBV, and DTPa -HIB-HBV-IPV, respectively) are analyzed by adding them, one at a time, to the list of twelve vaccine products that were licensed and under contract for distribution by the Centers for Disease Control and Prevention (CDC) as of March 2000 (table 1). To provide boundaries for the scope of the results presented, certain assumptions are needed. Wherever possible, the assumptions used in the study reported in [19] are also used in this study.

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Table 1 List of existing vaccine products under contract with the CDC (March 2000) that are used in the integer programming model, including vaccine formulation and packaging features, manufacturer, Federal contract vaccine price (as of March 2000), and assumed nursing time preparation cost to prepare and administer, and assumed total cost per dose. Vaccinea DTPa DTPa DTPa DTPa DTPa -HIB HIB HIB HIB HIB-HBV HBV HBV HBV HBV IPV IPV

Formulation/ packagingb

Manufacturerc

March 2000 Federal price/dose

Preparation cost/dose

Total cost/dosed

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

a Vaccine abbreviations (from www.cdc.gov/nip/visi/prototypes/vaxabbrev.htm): DTP – diphtheria toxoid, a

tetanus toxoid, and acellular pertussis vaccine; HIB – Haemophilus influenzae type B conjugate vaccine; DTPa -HIB – diphtheria toxoid, tetanus toxoid, acellular pertussis, and Haemophilus influenzae type B conjugate vaccine; HBV – hepatitis B virus vaccine; HIB-HBV – Haemophilus influenzae type B conjugate vaccine and hepatitis B virus vaccine; IPV – polivirus vaccine, inactivated. b [s] = liquid vaccine packaged in prefilled syringe; [v] = liquid vaccine packaged in vial; [p] = powdered (lyophilized) vaccine requiring reconstitution step. c Manufacturer abbreviations (from www.cdc.gov/nip/visi/prototypes/mfgnames.htm): AVP – Aventis Pasteur; NAV – North American Vaccine (absorbed in 2000 into Baxter Healthcare Corporation); GSK – GlaxoSmithKline; MRK – Merck and Co.; WAL – Wyeth-Ayerst Laboratories (includes Wyeth Laboratories, Lederle Laboratories Division American Cyanamid Company, and Wyeth Lederle Vaccines marketing arm). d Excludes clinic visit cost and injection cost.

The Federally negotiated vaccine price list includes five 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 (as of March 2000). These five manufacturers produce twelve vaccine products that protect against six diseases (vaccine for diphtheria, tetanus, pertussis = DTPa , vaccine for haemophilus influenzae type B = HIB, vaccine for hepatitis B = HBV, and vaccine for polio = IPV). The cost (objective) function components that are used to determine the inclusion prices for the four combination vaccines, as well determine the overall cost of the resulting vaccine formularies, include: • the purchase price of all licensed vaccines; • the cost of each clinic visit; • the cost of vaccine preparation by medical staff; • the cost of administering each injection. There are several other factors that impact the cost of immunization. These include cold chain costs (i.e., the cost of providing and maintaining the cold chain for vaccines that require such storage, as well as the costs associated with cold chain failure), product shelf life and vaccine expiration costs

(i.e., the cost of vaccine wastage due to inadequate or poor inventory management), adverse reaction costs (i.e., the cost of treating undesirable side effects associated with vaccination), and the costs associated with vaccine-preventable disease incidence (i.e., the cost of treating the diseases that were not prevented by the vaccines being administered). Unfortunately, reliable data is difficult to secure to support these costs, hence they are not included in the cost function for the integer programming model developed here. Of these four cost factors, the costs associated with vaccine-preventable disease incidence are the most significant. These costs are influenced by several factors, including delays in the earliest achievable age of immunity [2], the vaccine response distribution (i.e., the manner in which vaccine failure occurs) [13,20], the relative efficacies of vaccines [2,15], vaccination coverage for each vaccine in the target populations [2,15], and the basic reproduction numbers for each vaccine-preventable disease [2]. Moreover, the relationship between each of these factors may be highly nonlinear [9]. However, if such costs increase the cost of immunization, then not including them in this study may result in the inclusion price distributions obtained here possibly underestimating the actual inclusion price distributions at which the combination vaccines provide a good economic value; further research is needed to fully assess this effect.

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The vaccine purchase prices used are the Federally negotiated prices as of March 2000; these prices are displayed (in US dollars) in table 1. The cost of a clinic visit is set at $40. This value was used in the CDC pilot study [22], and includes the direct and indirect costs associated with a clinic being able to deliver vaccines. The cost of vaccine preparation by the medical staff is classified into three categories (with associated preparation times):

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 three perspectives are used to set this mean cost of an injection:

• liquid vaccine packaged in pre-filled syringe (0.5 min) [s];

(ii) $20.00 = cost from perspective (i) plus the direct medical costs associated with repeat clinic visits for deferred injections, 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”;

• liquid vaccine packaged in pre-filled vial (1.5 min) [v]; • powdered (lyophilized) vaccine requiring reconstitution step (3.0 min) [p]. The times for these three categories, labeled [s] for pre-filled syringe, [v] for pre-filled vial, and [p] for powder, were distributed around a reported mean time of 1.6 min to administer an injection [11,18], with modifications and rounding to account for the three categories of vaccine formulations and packaging. A medical staff compensation rate of $0.50/min was used in the CDC pilot study [22], 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, HBV manufactured by GlaxoSmithKline and Merck, and IPV manufacturer by Aventis Pasteur, are available in both pre-filled syringes and liquid vial formulations. Though their purchase prices are the same, the pre-filled syringes requires 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. The cost associated with administering an injection can be broken down into several telescoping components [22]. The first component is the actual direct cost of administering the vaccine, which is estimated to be $5.00 for each injection [12]. The second component is the direct cost for repeat clinic visits if injections are refused by the parent/guardian (e.g., when four or more injections are required at a particular clinic visit). This cost is estimated to be approximately $3.00 for each injection. The third component is the indirect cost of parental/guardian lost time from work for repeat clinic visits if injections are refused by the parent/guardian. This cost is estimated to be approximately $12.00 for each injection. The fourth component is the indirect cost of “pain and emotional distress to the child” (hence indirectly to the parent/guardian) 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 $25.00 per injection [12] or more conservatively, as $8.00 per injection [16]. The results in [10] independently support this range of values. It is difficult to assess a single value/cost 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,

(i) $5.00 = the marginal direct medical costs. Note that this cost reflects the perspective of the payer (e.g., an HMO or health insurer), but not the parents/guardian and society;

(iii) $45.00 = cost from perspective (ii) plus the indirect cost of $25.00 per injection for “pain and emotional distress.” The following assumptions are used in the analysis. These assumptions were all used in the study reported in [19]: (i) The 2000 Recommended Childhood Immunization Schedule was followed for immunization against six diseases: diphtheria, tetanus, pertussis, haemophilus influenzae type B, hepatitis B, and polio; (ii) injections can be administered in months 0–1 (within one month of birth), 2, 4, 6, 12–18, and 60, providing six opportunities (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 vaccines under Federal contract as of March 2000 (with the CDC) 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 is administered in month 0–1; (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 HBV, HIB, and IPV vaccines; (viii) extravaccination is permitted for HBV, HIB, and IPV vaccines, but not for DTPa vaccines; (ix) the DTPa -HIB vaccine by Aventis Pasteur can only be administered in the 12–18 or 60 month periods; (x) the vaccine prices are the Federally negotiated discount prices effective March 2000. Note that these assumptions are based on the guidelines as set forth in the Recommended Childhood Immunization Schedule [5]. Individual situations that deviate from this schedule are not considered, and are beyond the scope of this study.

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

Vaccine

Manufacturer

DTP1,. DTP3,. DTP4,. DTP5,. 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 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

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

DTPa -HIB-HBV-IPV4,.

DTPa -HIB-HBV-IPV

GSK

3. Model description An integer programming model was developed to determine the inclusion price for the four different combination vaccines. Monte Carlo simulation was then applied in conjunction with this model to estimate the distribution for these inclusion prices for a given population of parents/guardians (as defined by different injection cost distributions). The decision variables for the integer programming model are all either non-negative 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 combination 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 and ordered alphabetically by the manufacturer’s name. The resulting integer programming model contains 96 integer variables, of which 90 are binary variables, and 51 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 550 MHz IBM-compatible personal computer. 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. Also, to capture a skipped month 6 injection for the HIB vaccine manufactured by MRK (see constraint (vi)), the binary decision variable HIBskip2,6 was defined, where HIBskip2,6 = 1 if 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 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. The inclusion price distributions were obtained using Monte Carlo simulation in conjunction with the integer programming model. Three different probability distributions for the injection cost were considered, each with three different mean injection costs corresponding to the three perspectives ($5, $20, $45) described in section 2. To provide a breadth of probability distribution classes, a uniform distribution, a normal distribution, and an exponential distribution for the cost of administering an injection were used. For the uniform distribution case, the distributions used were U($0, $10), U($0, $40), and U($0, $90). For the normal distribution, the distributions used were N(µ = $5, σ = $1), N(µ = $20, σ = $4), and N(µ = $45, σ = $9). For the exponential distribution, the distributions used were exponential with means $5, $20, and $45. It would be very difficult to determine the exact distribution for the cost of administering an injection within a population of parents/guardians. Therefore, the three distributions were chosen to provide different levels for the coefficients of variation (σ/µ). In particular, the uniform distributions all have coefficients of variation 2/(12)1/2 ≈ 0.577 (which is moderate), the normal distributions all have coefficients of variation 0.2 (which is small), and the exponential distributions all have coefficients of variation 1 (which is large). Therefore, a total of nine Monte Carlo simulation experiments were run for each of the four combination vaccines. For a given combination vaccine, each Monte Carlo simulation experiment generated a total of 500 vaccine injection costs, where each such cost was used as an input to the integer programming model. The resulting integer programming model was then solved to determine the inclusion price for the combination vaccine such that the resulting vaccine formulary contained this combination vaccine one time and two or three times (i.e., the one/two or three dose inclusion price for a combination vaccine occurs when the vaccine is administered no less than one/two or three time(s) during the entire immunization schedule). Note that when the two dose vaccine formulary is not reported, this means that the inclusion price at which the combination vaccine enters the resulting vaccine formulary for two doses is the same as the inclusion price that it enters for three doses. Each set of Monte Carlo experiments for each of the combination vaccines took approximately five hours to execute. 4. Results This section presents the results of using Monte Carlo simulation in conjunction with the integer programming model to estimate the distributions for the inclusion prices for the four combination vaccines, given a particular population of parents/guardians (as defined by different injection cost distributions).

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Table 3 Hypothetical pentavalent and hexavalent combination vaccines (in planned or current development, or sold outside the U.S. market). Combination vaccines

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]

Formulation/ Manufacturer(s)a Assumed packaging preparation cost/dose 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

a Manufacturers listed are those with either a licensed product of the vac-

cine type sold anywhere outside the United States, or which have ever conducted clinical trials of the vaccine type, or which indicated plans to do so, even if subsequently abandoned. Sources of information are personal communications with vaccine manufacturers.

Table 3 provides a list of the formulations and packaging for the four combination vaccines, as well as the manufacturers who are either selling these products (either within the United States or in other countries) or have conducted clinical trials. Since DTPa manufacturer brand matching is required (assumption (vii)), then a total of six different cases for the four combination vaccines had to be studied. For DTPa -HIBHBV, 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. Four manufacturers are either manufacturing or conducting clinical trials for DTPa HIB-IPV. 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 constraint (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 cases in the tables of results (see tables 6 and 9). The results of the Monte Carlo simulation experiments provide the inclusion price distributions for nine different parent/guardian populations, for the six different cases described above. This information not only provides the manufacturers of these vaccines with market share information for these populations, but also allows public health officials and health care providers to assess whether such combination vaccines should be stocked in their vaccine formularies, based on the fraction of their particular population that value the convenience that such vaccines provide. This provides a broad perspective on both the economic and the societal benefits of the results presented here. For each of the six combination vaccine cases, and for each of the nine Monte Carlo simulation experiments, the integer programming model was used to solve for 500 inclusion prices for the one dose vaccine formulary and 500 inclusion prices for the two or three dose optimal formulary. Figure 1

Inputs: Probability distribution of the cost of administering an injection for a given population. Desired number of times to administer the combination vaccine. Goal: Approximate the distribution of the inclusion prices of the combination vaccine. For j = 1 to 500 Cj = random number generated from the probability distribution of the cost of administering an injection. Use the Bisection method (see figure 2) with the cost of administering an injection equal to Cj . Pj = inclusion price of the combination vaccine found by the Bisection method. End Output: (P1 , P2 , . . . , P500 ), a sample of inclusion prices from the given population. Figure 1. Monte Carlo simulation. Inputs: Cost of administering an injection. Desired number of times to administer the combination vaccine. Goal: Determine the inclusion price at which the combination vaccine will be used the desired number of times in the lowest cost formulary. HI = $999 (The combination vaccine will be used zero times at this price.) LO = $0 (The combination vaccine will be used at least three times at this prices.) Repeat MID = (HI + LO)/2. Set the price of the combination vaccine equal to MID. Solve the integer program for the lowest cost formulary. If the combination vaccine is used fewer times in the lowest cost formulary than the desired number of times, then set HI = MID. Else Set LO = MID. Until (HI − LO) < 0.01 Output: LO is the desired inclusion price of the vaccine. Figure 2. Bisection method to find the inclusion price of a combination vaccine.

provides an overview of the Monte Carlo simulation. Each of the 500 inclusion prices obtained from the Monte Carlo simulation experiments required the integer programming model to be solved several times. This was done using a bisection search algorithm [3], where an upper and lower bound for the inclusion 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 middle point between the upper and lower bound replaced either the upper or the lower bound (see figure 2). Note that bisection search is effective in reaching an optimal formulary provided the cost function is well behaved, such as being convex over the feasible region of possible formularies. Each set of 500 inclusion prices was used to compute a mean (X) and a standard deviation (s) for the these prices, as well as the 20th percentile inclusion price, the 50th (median) percentile inclusion price, and the 80th percentile inclusion price. The 20th percentile inclusion price corresponds to the price of the combination vaccine at which 20% of the population will not purchase the vaccine, for the designated number of doses. Similarly, the 80th percentile inclusion price corresponds to the price of the combination vaccine at which 80% of the population will not purchase the vaccine, for the de-

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Table 4 DTPa -HIB-HBV inclusion price results. Cost of an injection probability distribution

X

s

U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

35 51 76 35 52 77 34 51 77

5 12 26 2 4 9 7 22 49

One dose 20% 50% 29 39 49 34 48 69 27 34 42

35 52 77 35 52 77 32 45 62

Two doses 20% 50%

80%

X

s

40 64 104 37 55 84 40 65 106

27 34 50 28 33 50 27 35 52

1.3 6 17 0.3 3 6 2 13 31

26 28 31 28 31 45 25 28 28

80%

28 34 50 28 34 50 28 29 41

28 41 68 28 36 55 28 42 70

Three doses 20% 50%

80%

Table 5 DTPa -HIB-IPV inclusion price results. Cost of an injection probability distribution

X

s

U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

33 53 90 34 52 90 33 55 92

5 16 38 2 6 14 8 31 71

One dose 20% 50% 28 38 48 32 47 78 26 33 40

34 53 91 34 52 90 31 44 68

80%

X

s

39 70 130 36 57 101 39 72 134

30 49 74 30 50 75 31 49 75

4 14 27 1 5 9 7 23 49

26 34 47 30 47 68 25 30 37

31 51 76 31 51 76 29 43 61

34 62 102 31 54 93 35 64 105

Table 6 DTPa -HIB-IPV inclusion price results (AVP case). Cost of an injection probability distribution U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

X

s

33 56 103 34 54 103 33 59 107

5 19 49 2 7 19 8 39 94

One dose 20% 50% 28 38 48 32 47 87 26 33 40

34 53 104 34 53 103 31 44 74

80%

X

s

39 77 157 36 60 118 39 79 162

30 49 74 30 50 75 31 49 75

4 14 27 1 5 9 7 23 49

Three doses 20% 50% 26 34 47 30 47 68 25 30 37

80%

31 51 76 31 51 76 29 43 61

34 62 102 31 54 83 35 64 105

Two doses 20% 50%

80%

Table 7 DTPa -HBV-IPV inclusion price results. Cost of an injection probability distribution

X

s

U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

37 53 78 37 54 79 36 53 79

5 12 26 2 4 9 7 22 49

One dose 20% 50% 31 41 51 36 50 71 29 36 44

37 54 79 37 54 79 34 47 64

signated number of doses. All these values are reported in tables 4–9. Note that all the inclusion prices listed are rounded to the nearest dollar.

80%

X

s

42 66 106 39 57 86 42 67 108

35 51 76 36 51 76 35 51 77

4 12 26 1 4 9 6 22 48

31 39 48 35 48 68 29 36 41

36 52 77 36 51 76 34 45 62

39 63 103 37 55 84 40 64 106

This analysis suggests that the combination vaccines provide good economic value at such prices, based on nine different cost probability distributions associated with adminis-

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S.H. JACOBSON, E.C. SEWELL

Table 8 DTPa -HIB-HBV-IPV inclusion price results. Cost of an injection probability distribution U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

X

s

48 79 128 48 80 129 47 80 131

8 24 52 3 8 19 13 44 97

One dose 20% 50% 39 55 74 46 73 114 37 46 59

48 80 131 48 80 130 44 67 101

80%

X

s

56 104 183 51 86 145 56 106 189

40 61 97 41 60 97 40 62 100

4 16 38 1 6 14 7 31 72

Three doses 20% 50% 36 44 56 40 55 86 34 41 46

41 61 98 41 60 98 39 51 76

80% 44 78 137 42 65 109 45 80 142

Table 9 DTPa -HIB-HBV-IPV inclusion price results (AVP case). Cost of an injection probability distribution U($0, $10) U($0, $40) U($0, $90) N(µ = $5, σ = $1) N(µ = $20, σ = $4) N(µ = $45, σ = $9) Exp(µ = $5) Exp(µ = $20) Exp(µ = $45)

X

s

48 79 128 48 80 129 47 80 131

8 24 52 3 8 19 13 44 97

One dose 20% 50% 39 55 74 46 73 114 37 46 59

48 80 131 48 80 130 44 67 101

tering each injection. For example, if a health care provider believes that for their patient population, the cost of an injection follows a normal distribution with mean $20 and standard deviation $4, and this provider wishes to stock combination vaccine DTPa -HIB-HBV for administration of two doses per child during the first five years of the immunization schedule (see table 4), with a child already having been administered a 0–1 month dose of HBV, then if the price of this combination vaccine is $36, approximately 20% of this provider’s patient population will deem this combination vaccine a good economic value over the currently licensed vaccines. However, if the price is dropped to $31, then approximately 80% of the provider’s patient will deem this combination vaccine a good economic value over the currently licensed vaccines. On the other hand, if the health care provider believes that for their patient population, the cost of an injection follows an exponential distribution with mean $20 (hence the population has a larger coefficient of variation for their injection cost distribution), then the 20% and 80% prices are $42 and $28, respectively. Therefore, higher coefficients of variation tend to result in a wider variation in combination vaccine prices. This information provides health care providers with valuable practical information on how many doses of this combination should be stocked, since if too many doses are purchased, and the price is set too high, then the provider, for example, may have to bear the cost of vaccine spoilage from expiration. This also provides useful formulary information for health care providers, as well as information that can benefit suppliers and insurance companies in assessing a priori the impact of combination vaccines on ordering and reimburse-

80%

X

s

56 104 183 51 86 145 56 106 189

41 69 97 41 69 118 41 70 121

4 22 38 1 8 19 10 42 96

Two doses 20% 50% 36 44 56 40 62 103 34 41 48

41 69 98 41 69 119 39 56 90

80% 45 93 137 42 75 134 45 95 178

ment processing, respectively. Similar results and analysis can be obtained from each of the other tables. The standard deviations of the inclusion prices are positively related to the standard deviations of the injection cost distributions. For example, in table 7, for the one dose formulary, the values for s corresponding to the three injection cost distributions (uniform, normal and exponential distributions with mean $20, hence their standard deviations are $5.77, $4, and $20, respectively) are $12, $4, and $22, respectively. This means that for a uniform($0,$40) injection cost distribution, the standard deviation for the inclusion prices of DTPa -HBV-IPV is $5.77, while for an exponential($20) injection cost distribution, the standard deviation for the inclusion prices of DTPa -HBV-IPV is $22. Therefore, as the injection cost distribution standard deviation increases, the inclusion price distribution standard deviation also increases, though this relationship may be nonlinear (particularly for the uniform distribution) and depends on the form of the injection cost distribution. One consequence of this observation is that health care providers are more easily able to predict the volume of various combination vaccines to stock in their formularies based on the homogeneity of their patient population (as measured by the variability in cost that they associate with avoiding an injection). Moreover, vaccine manufacturers must be more sensitive to how they set the price of their combination vaccines for such populations, since small changes in their price can lead to significant changes in the volume of vaccines that they are able to administer. The results presented in tables 4–9 are based on the Federally negotiated prices for the twelve vaccines under contract

MONTE CARLO SIMULATION FOR CHILDHOOD DISEASES

Figure 3. Inclusion price distribution for DTPa-HBV-IPV with U($0, $40) injection cost distribution.

Figure 4. Inclusion price distribution for DTPa-HBV-IPV with N($20, $4) injection cost distribution.

with the CDC, hence represent the maximum Federal prices at which the four combination vaccines enter the vaccine formulary one and two or three times. Note that for DTPa-HIBIPV, three doses rather than two doses are reported since three doses of this vaccine can be used without any extravaccination occurring. The other combination vaccines result in some extravaccination occurring if administered three times. The nine distributions for the cost of an injection analyzed for each of the 6 cases resulted in a total of 54 Monte Carlo experiments. For each Monte Carlo experiment, a histogram plot of the inclusion prices was obtained. For the normal and exponential distributions, the histogram plot shape for the inclusion prices preserves these distributions, with shifted means and variances as given in tables 4–9 (see figures 4 and 5). However, for the uniform distributions, the histogram plots for the inclusion prices do not appear to preserve the form of the uniform distribution (see figure 3). Figures 3–5 provide these histogram plots for the DTPa -HBV-IPV vaccine product reported in table 7, with injection cost distributions U($0, $40), N($20, $4), and exponential with mean

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Figure 5. Inclusion price distribution for DTPa-HBV-IPV with Exp(20) injection cost distribution.

$20, respectively. Given that the distributions are preserved for the normal and the exponential injection cost distributions suggest a possible linear relationship between the cost of an injection and the inclusion prices; this is consistent with the relationship between these injection cost distribution standard deviations and the inclusion price standard deviations. However, the result in figure 3 suggests that this is not the case for the uniform distribution, which may be due in part to the fact that for the uniform injection cost distribution, many of the injection costs may be close to zero. Therefore, each injection cost distribution must be treated individually to determine the correct inclusion price distribution for a given combination vaccine for a particular population. The inclusion prices in tables 4–9 also provide valuable marketing information for vaccine manufacturers. For example, if the manufacturer of a combination vaccine would like to target their product at a particular sector of a community that places a high value on avoiding an injection (e.g., exponential with mean µ = $45), then the manufacturer can price their product accordingly and know what fraction of that sector will deem this combination vaccine a good economic value over the currently licensed vaccines. 5. Conclusions and limitations of results This paper uses Monte Carlo simulation and integer programming modeling to determine inclusion price distributions for four combination vaccines that are being developed for immunization against six childhood diseases. The results of this study suggest that combination vaccines warrant cost premiums based on the value and the distribution assigned to administering an injection, and that further developments and innovations in this area by vaccine manufacturers can provide significant economic and societal benefits. The inclusion price distributions given for the four combination vaccines listed in tables 4–9 provide useful information for determining what fraction of a population are willing to purchase a given combination vaccine at a specified

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price. Note that these distributions may be highly sensitive to the data described in section 2, including the price of each vaccine product listed in table 1, the removal or addition of any vaccine product listed in table 1, and the formulation and packaging (hence the preparation costs) for each vaccine product listed in table 1. Note that this study did not attempt to match each of the combination vaccines against each other, but rather looks at the effect of adding each combination vaccine individually to the available and Federally licensed vaccine products. As these combination vaccines are priced (once they enter the marketplace), such a study would be both appropriate and useful to help health care providers assess whether any or all such products have a place within their vaccine formulary. The factors stated in section 2 that impact the cost of immunization would add an additional level of realism to the results reported here. As data is collected to determine such costs, the resulting cost components can be incorporated into the integer programming model, hence provide a more accurate inclusion price distribution for combination vaccines. However, the costs associated with vaccine-preventable disease incidences may lead to a non-linear cost function, resulting in a non-linear integer programming model. The nature of this nonlinearity (e.g., convexity, multi-modal) will determine the extent to which the approach used in this paper can be extended to handle such cost factors, or whether a new modeling approach is required. The inclusion price distributions for the four combination vaccines are obtained based on the assumptions stated in section 2. Note that the effect of issues such as brand loyalty and other behavioral factors were not included in the study hence the results reported do not capture such factors. In addition, the data used to obtain these distributions, such as the injection cost distributions and the cost of vaccine preparation, are highly dependent on specific factors germane to each individual health care provider or clinic, hence any changes in these data may result in changes to the inclusion price distributions for the combination vaccines. Therefore, given these limitations, the prices given in tables 4–9 should serve as general guidelines, rather than precise values as to how the combination vaccines should be priced in the marketplace, since any or all of these other factors may serve to either increase or decrease such values. As new vaccine combination products enter the market and become licensed for administration, the combinatorial explosion of choices available to health care providers will make it even more challenging to make sound economic decisions. The operations research modeling approach presented is this paper provides a systematic methodology to address such issues, hence encourages intelligent and cost effective decisionmaking in the rapidly expanding combination vaccine development arena. Moreover, once these combination vaccines are priced in the marketplace, healthcare providers and insurance companies can use the operations research modeling approach used in this study to assess whether such products provide a good value for their particular patient population and circumstances.

S.H. JACOBSON, E.C. SEWELL

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