Outreach Strategies for Vaccine Distribution: A Multi-period Stochastic Modeling Approach

Abstract

Vaccination has been proven to be the most effective method to prevent infectious diseases. However, in many low- and middle-income countries with geographically dispersed and nomadic populations, last-mile vaccine delivery can be extremely complex. Because newborns in remote population centers often do not have direct access to clinics and hospitals, they face significant risk from diseases and infections. An approach known as outreach is typically utilized to raise immunization rates in these situations. A set of these remote locations is chosen, and over an appropriate planning period, teams of clinicians and support personnel are sent from a depot to set up mobile clinics at these locations to vaccinate people there and in the immediate surrounding area. In this paper, we model the problem of optimally designing outreach efforts as a mixed integer program that is a combination of a set covering problem and a vehicle routing problem. In addition, because elements relevant to outreach (such as populations and road conditions) are often unstable and unpredictable, we address uncertainty and determine the worst-case solutions. This is done using a multi-period stochastic modeling approach that considers updated model parameter estimates and revised plans for subsequent planning periods. We also conduct numerical experiments to provide insights on how demographic characteristics affect outreach planning and where outreach planners should focus their attention when gathering data.

Appendix 1. Data Description

The data used in our simulated examples was derived from four countries in sub-Saharan Africa. Table 5 overviews some of their broad demographic characteristics. Because of confidentiality issues, the countries are not identified and data has been scaled to make the smallest value correspond to a value of 1.0. Country C is relatively small but with a very high population density. Conversely, A and B are large but very sparsely populated (although they both have some pockets of denser population interspersed with regions—such as areas in the Saharan desert—that are much more sparsely populated). Country D is in between in terms of area and overall population density.

Table 5 Characteristics (scaled for confidentiality)

Table 6 lists the vaccine regimen in each country and presentations and names used there; these were what we were told each country uses based on WHO recommended guidelines [31], and differ somewhat by country. Table 7 is derived from Table 6: we first find the volume of each vaccine required per child by computing $(dose\ volume\times dosage)/(1-OVW)$ and then add the across all vaccines to find the total volume of vaccines demanded per child (last row). Multiplying this by the number of annual births for location i (based on its population and the birth rate for the country) yields the total annual volume demanded. We then distribute this evenly over each planning period to obtain ${b}_{i}$. Note that a child is not given the entire regimen at the same time; inoculations are based on the child’s age. However, we assume that ages are uniformly distributed in the population, so that total volume is representative of overall annual demand.

Table 6 Vaccine details

Table 7 Total vaccine volumes (cc) by country

Table 8 Model inputs and parameters for instances with smaller, densely populated regions

Table 9 Model inputs and parameters for instances with moderately populated regions

Table 10 Model inputs and parameters for instances with in larger, sparsely populated regions

We now describe how we generate the problems presented in Tables 2, 3, and 4 in Sect. 6. We decided to study cases that represented larger but more sparsely populated areas; smaller, more densely populated areas; and areas that are in between. We did not have access to detailed population data across all locations in the four countries we looked at. However, we had approximate estimates of populations across parts of the different countries, as well as some information on what might be a typical area that is covered for outreach. Based upon this information, we first decided that moderate to densely populated regions would have an annual demand corresponding to between about 120 and 150 children with a few outliers, while the sparser regions would have annual demand between 80 and 100. Similarly, we define a “small” outreach area to be a square grid with an area between 6² and 12² sq. km, a “large” one to be between 12² and 22² sq. km, and a “moderate” one to be between 9² and 12² sq. km. The depot was then placed at the center of the grid and the number and locations of population centers on the grid were randomly generated. In general, the total targeted population and number of population centers that we assigned in dense regions (usually 12 to 14) tend to be larger than in the sparser regions (usually 6 to 10), and over a smaller grid. Cost and demand parameters were subjectively assigned from one of the four countries based on how these areas resembled parts of these countries. The facility cost tends to be more important than the total transportation cost in the dense regions, while in the sparse regions the facility cost tends to play a smaller role.

Tables 8, 9 and 10 correspond to each of the problems generated as above and listed in Tables 2, 3 and 4 respectively, and describe the underlying problem characteristics. The letter of the entry in the first column identifies the country on which the problem is based, while the number indexes the problems from each country. Vehicle capacities and speeds, as well the average fixed (${{\varvec{f}}}_{{\varvec{i}}}$) and transportation costs (c) are computed based on information provided by the local public health officials as what is typical in the country from which the problem is derived.

Table 11 summarizes the same problem characteristics, but from the perspective of each of the four countries.

Table 11 Summary of inputs and parameters across all counties

Lastly, the distances between locations $i$ and $j$ is assumed to be the Euclidean distance on the graph that we generate and the travel times are computed using this distance and the vehicle travel speed shown in Tables 8, 9 and 10. We assume that the MCD is set at a value of $D$ = 5 km, which is consistent with the standard WHO guidelines, and we use a value of 14 h for the MTD along with a service time of 3 h at each location.