Optimizing Distribution of Influenza Antiviral Drugs
Optimizing Distribution of Influenza Antiviral Drugs
Texas has 1,939 US postal code (ZIP code) areas in 254 counties; 1,023 of these ZIP code areas contain ≥1 pharmacy (Table 1). We obtained the addresses of all community and clinic pharmacies with active licenses listed by the Texas Pharmacy Board. The largest chains (present in the most ZIP code areas) in Texas are Brookshire, Costco, CVS, HEB, Kmart, Kroger, Randalls, Sam's Club, Target, Tom Thumb, United, Walgreens, and Walmart. Other pharmacies, independent or small chain, are listed as independents. The Texas DSHS provided the list of pharmacies selected to dispense antiviral drugs to underinsured populations during the 2009 influenza pandemic; these pharmacies were in 723 ZIP code areas. To approximate the size of the uninsured and underinsured population in each ZIP code area (direct statistics were not available), we used the number of persons in households with an annual income <$20,000 (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
Our optimization model uses a geographic resolution of ZIP code areas based on ZIP code tabulation areas (ZCTAs). ZCTAs differ slightly from US Postal Service ZIP code areas and may include ≥1 US Postal Service ZIP code area. We mapped each pharmacy and residential ZIP code area to its corresponding ZCTA, and, for simplicity, we refer to these as ZIP code areas.
We used National Household Travel Survey (NHTS) data for 2009 to estimate the distances persons are willing to travel in Texas to obtain antiviral drugs sufficient for a course of treatment during an influenza pandemic (model described below). We created a willingness-to-travel model, which follows an exponentially decaying distribution, by fitting the model to national-scale NHTS data for privately operated vehicle travel (Figure 1). This included ≈330,000 person trips (83% of all person trips in the database), totaling 3.3 million miles, including ≈30,000 person trips originating in Texas. We made the simplifying assumption that health care–seeking behavior in Texas during an influenza pandemic will resemble national willingness to travel by privately operated vehicle for work, school, family, and social reasons. Although there are probably major differences in these estimates, we believe that this model conservatively underestimates actual accessibility of pharmacies during a pandemic.
(Enlarge Image)
Figure 1.
Willingness-to-travel curve for receiving antiviral drugs during the 2009 influenza pandemic given by equation (2) (in Methods section) fit to National Household Travel Survey (NHTS) data on privately operated vehicle travel for the entire US underinsured population.
Using a least-squares fit, we obtained the following model (Equation 1):
in which the
term is the fraction of the target population willing to travel at least d miles. As the required travel distance increases, the fraction of the population willing to travel distance d decreases. We used a piecewise model that allows for different coefficients below and above a distance threshold of 5 miles to enable urban and rural populations to exhibit different willingness-to-travel patterns ( http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
To estimate travel patterns for the underinsured population, we considered NHTS data for households with incomes <$20,000 (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf) and found that the travel patterns for this group are given by Equation 2:
The estimated willingness-to-travel for the underinsured population is slightly greater (<1%) than that for the entire population. The adjusted R values for each model exceed 0.99.
The optimization model we used identifies ZIP code areas for pharmacy-based distribution of SNS and state-cache antiviral drugs to maximize access in the target population (either underinsured or entire population). It is a facility-location type model with an objective function defined in terms of the expected number of persons willing to obtain antiviral drugs from the nearest dispensing point. We estimated this quantity by using our willingness-to-travel model for the distance between the home ZIP code centroid and pharmacy ZIP code centroid. For the distance to a pharmacy within the home ZIP code area, we used a correction factor based on the size of the ZIP code area (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
The optimization model takes as input the total number of ZIP code areas to be included in the distribution network (b). The model does not account for the number of available antiviral drug doses, the number to be shipped to each pharmacy, or the capacity of individual pharmacies. Additional details on methods are available at http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf.
The Web-based decision-support tool based on this model provides solutions for a range of values of b (Figure 2) and displays the trade-off between the expected access for the target population and the number of dispensing points. This tool also enables the user to select specific solutions for further analysis.
(Enlarge Image)
Figure 2.
Antiviral drug access in underinsured populations achieved by the Texas antiviral drug distribution network during the 2009 influenza A pandemic and by optimized antiviral drug distribution networks, for A) small ZIP code (US postal code) areas (i.e., ZIP code areas with <1,000 underinsured persons) and B) statewide. Access is the expected fraction of the underinsured population willing to travel to the nearest dispensing pharmacy to obtain antiviral drugs. The black vertical and horizontal lines indicate the number of ZIP code areas that participated in the Texas 2009 distribution network and the estimated access achieved, respectively. For each network size (number of dispensing ZIP code areas), a hybrid optimization was performed to maximize coverage in small ZIP code areas and overall (see Methods for details). Color indicates which combination of 13 major pharmacy chains plus independents were considered in the optimization. For a distribution network of size 723 (comparable to the Texas 2009 H1N1 antiviral drug distribution), the best performing single-chain (Walgreens), 2-chain combination (Walgreens and Walmart), and 3-chain combination (Walgreens, Walmart, and CVS) provided near optimal coverage statewide, but critically underserved the smallest ZIP code areas.
We considered 3 types of objective functions, all of which focus exclusively on the underinsured population in Texas: maximizing statewide access, maximizing access in small ZIP code areas (i.e., ZIP code areas with <1,000 underinsured persons), and a hybrid that combines the first 2 objectives. For our hybrid optimization model, we first specified a percentage of all dispensing points to focus on small ZIP code areas (P). Second, we optimized P of all dispensing points solely for access in small ZIP code areas and recorded the access achieved in small ZIP code areas (As). Third, we started over and optimized all dispensing points by using the statewide objective function with the added constraint that the solution must achieve a minimum of 0.95As access in small ZIP code areas. This method simultaneously achieves near maximal coverage statewide and in small ZIP code areas.
Methods
Data
Texas has 1,939 US postal code (ZIP code) areas in 254 counties; 1,023 of these ZIP code areas contain ≥1 pharmacy (Table 1). We obtained the addresses of all community and clinic pharmacies with active licenses listed by the Texas Pharmacy Board. The largest chains (present in the most ZIP code areas) in Texas are Brookshire, Costco, CVS, HEB, Kmart, Kroger, Randalls, Sam's Club, Target, Tom Thumb, United, Walgreens, and Walmart. Other pharmacies, independent or small chain, are listed as independents. The Texas DSHS provided the list of pharmacies selected to dispense antiviral drugs to underinsured populations during the 2009 influenza pandemic; these pharmacies were in 723 ZIP code areas. To approximate the size of the uninsured and underinsured population in each ZIP code area (direct statistics were not available), we used the number of persons in households with an annual income <$20,000 (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
Our optimization model uses a geographic resolution of ZIP code areas based on ZIP code tabulation areas (ZCTAs). ZCTAs differ slightly from US Postal Service ZIP code areas and may include ≥1 US Postal Service ZIP code area. We mapped each pharmacy and residential ZIP code area to its corresponding ZCTA, and, for simplicity, we refer to these as ZIP code areas.
Willingness-to-travel Model
We used National Household Travel Survey (NHTS) data for 2009 to estimate the distances persons are willing to travel in Texas to obtain antiviral drugs sufficient for a course of treatment during an influenza pandemic (model described below). We created a willingness-to-travel model, which follows an exponentially decaying distribution, by fitting the model to national-scale NHTS data for privately operated vehicle travel (Figure 1). This included ≈330,000 person trips (83% of all person trips in the database), totaling 3.3 million miles, including ≈30,000 person trips originating in Texas. We made the simplifying assumption that health care–seeking behavior in Texas during an influenza pandemic will resemble national willingness to travel by privately operated vehicle for work, school, family, and social reasons. Although there are probably major differences in these estimates, we believe that this model conservatively underestimates actual accessibility of pharmacies during a pandemic.
(Enlarge Image)
Figure 1.
Willingness-to-travel curve for receiving antiviral drugs during the 2009 influenza pandemic given by equation (2) (in Methods section) fit to National Household Travel Survey (NHTS) data on privately operated vehicle travel for the entire US underinsured population.
Using a least-squares fit, we obtained the following model (Equation 1):
in which the
term is the fraction of the target population willing to travel at least d miles. As the required travel distance increases, the fraction of the population willing to travel distance d decreases. We used a piecewise model that allows for different coefficients below and above a distance threshold of 5 miles to enable urban and rural populations to exhibit different willingness-to-travel patterns ( http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
To estimate travel patterns for the underinsured population, we considered NHTS data for households with incomes <$20,000 (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf) and found that the travel patterns for this group are given by Equation 2:
The estimated willingness-to-travel for the underinsured population is slightly greater (<1%) than that for the entire population. The adjusted R values for each model exceed 0.99.
Optimization Model
The optimization model we used identifies ZIP code areas for pharmacy-based distribution of SNS and state-cache antiviral drugs to maximize access in the target population (either underinsured or entire population). It is a facility-location type model with an objective function defined in terms of the expected number of persons willing to obtain antiviral drugs from the nearest dispensing point. We estimated this quantity by using our willingness-to-travel model for the distance between the home ZIP code centroid and pharmacy ZIP code centroid. For the distance to a pharmacy within the home ZIP code area, we used a correction factor based on the size of the ZIP code area (http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf).
The optimization model takes as input the total number of ZIP code areas to be included in the distribution network (b). The model does not account for the number of available antiviral drug doses, the number to be shipped to each pharmacy, or the capacity of individual pharmacies. Additional details on methods are available at http://www.bio.utexas.edu/research/meyers/_docs/publications/SinghEID14Supplement.pdf.
The Web-based decision-support tool based on this model provides solutions for a range of values of b (Figure 2) and displays the trade-off between the expected access for the target population and the number of dispensing points. This tool also enables the user to select specific solutions for further analysis.
(Enlarge Image)
Figure 2.
Antiviral drug access in underinsured populations achieved by the Texas antiviral drug distribution network during the 2009 influenza A pandemic and by optimized antiviral drug distribution networks, for A) small ZIP code (US postal code) areas (i.e., ZIP code areas with <1,000 underinsured persons) and B) statewide. Access is the expected fraction of the underinsured population willing to travel to the nearest dispensing pharmacy to obtain antiviral drugs. The black vertical and horizontal lines indicate the number of ZIP code areas that participated in the Texas 2009 distribution network and the estimated access achieved, respectively. For each network size (number of dispensing ZIP code areas), a hybrid optimization was performed to maximize coverage in small ZIP code areas and overall (see Methods for details). Color indicates which combination of 13 major pharmacy chains plus independents were considered in the optimization. For a distribution network of size 723 (comparable to the Texas 2009 H1N1 antiviral drug distribution), the best performing single-chain (Walgreens), 2-chain combination (Walgreens and Walmart), and 3-chain combination (Walgreens, Walmart, and CVS) provided near optimal coverage statewide, but critically underserved the smallest ZIP code areas.
We considered 3 types of objective functions, all of which focus exclusively on the underinsured population in Texas: maximizing statewide access, maximizing access in small ZIP code areas (i.e., ZIP code areas with <1,000 underinsured persons), and a hybrid that combines the first 2 objectives. For our hybrid optimization model, we first specified a percentage of all dispensing points to focus on small ZIP code areas (P). Second, we optimized P of all dispensing points solely for access in small ZIP code areas and recorded the access achieved in small ZIP code areas (As). Third, we started over and optimized all dispensing points by using the statewide objective function with the added constraint that the solution must achieve a minimum of 0.95As access in small ZIP code areas. This method simultaneously achieves near maximal coverage statewide and in small ZIP code areas.
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