key: cord-0110537-y1t9sahd
authors: Zhong, Zhen; Sengupta, Ribhu; Paynabar, Kamran; Waller, Lance A.
title: Multi-Objective Allocation of COVID-19 Testing Centers: Improving Coverage and Equity in Access
date: 2021-09-21
journal: nan
DOI: nan
sha: 24204cfb70cb3b97417504776db94bc76ee73a12
doc_id: 110537
cord_uid: y1t9sahd

At the time of this article, COVID-19 has been transmitted to more than 42 million people and resulted in more than 673,000 deaths across the United States. Throughout this pandemic, public health authorities have monitored the results of diagnostic testing to identify hotspots of transmission. Such information can help reduce or block transmission paths of COVID-19 and help infected patients receive early treatment. However, most current schemes of test site allocation have been based on experience or convenience, often resulting in low efficiency and non-optimal allocation. In addition, the historical sociodemographic patterns of populations within cities can result in measurable inequities in access to testing between various racial and income groups. To address these pressing issues, we propose a novel test site allocation scheme to (a) maximize population coverage, (b) minimize prediction uncertainties associated with projections of outbreak trajectories, and (c) reduce inequities in access. We illustrate our approach with case studies comparing our allocation scheme with recorded allocation of testing sites in Georgia, revealing increases in both population coverage and improvements in equity of access over current practice.

and Σ % ( ) = 45 (7) 47 # , the derivative of the covariance matrix of . The inverse Fisher information matrix is straightforward to compute and will asymptotically converge to the real covariance matrix. Therefore, instead of minimizing the determinant of the covariance matrix of , we instead minimize the determinant of inverse Fisher information matrix, an approach known as meeting the local optimal design criterion [4] . Suppose due to limited resources, only k facilities can be operational and $ is a binary variable indicating facility $ is operational or not. We define 9 ( , ) = − log ( , ), which is an evaluation of the uncertainty of under an allocation scheme indicated by = { $ ; = 1, . . , }. As a result, when seeking to minimize prediction error, the best test site allocation scheme should result in the lowest 9 . Therefore, the local optimal design criterion should be:

: ( 9 ) = ; 9 ( ; 9 )

We note this criterion requires the availability of a preliminary estimate of , denoted by 9 . In the beginning of the pandemic, such an estimate was not available due to limited prevalence data.

Thus, we consider minimizing the worst-case scenario through a minimax optimal design criterion [4] defined by

Since, for different values, we will have different ; 9 ( , ) , we adopt a criterion that removes the influence of different value of ; 9 ( , ). Specifically, we define " = ! ( , ) = ; max 7∈= ( 9 ( , ) − 9 ( ( ), )

where ( ) represent the local optimal design over . The function " is used as the D-optimal criterion in our multi-objective optimization.

In this section, we introduce the equity measure used in the optimization model. Following [9], we claim equity is achieved when the probability of being tested given social and demographic factors including race and sex is equal to the probability of being tested unconditional of any socioeconomic factor. Mathematically, if is the binary variable indicating being tested or not and is the set of observed social and demographic factors, then our definition of equity can be measured by comparing the expectation of the marginal distribution of to that of the conditional distribution | . Since is an indicator function, the expectation of marginal distribution ( ) is equal to ( ) and the expectation of conditional distribution ( | ) is equal to ( | ). To move toward an equitable allocation of testing resources, we minimize the quadratic loss between ( | ) and ( ). Therefore, we define the following equity criteria as:

Here ! , … B represent labels for sociodemographic subpopulations, while ! , … B represent the total number of different labels in each group, j

k is the coverage probability for a specific sociodemographic group, while j

k yields the probability to be covered among all different sociodemographic groups. Ideally, these two quantities should be the same, and we define or criteria for optimizing equity in test site allocation as the mean square loss of the difference between these two probabilities.

As discussed in the problem formulation, we consider two types of constraints. cover area % (i.e., $% = 1), or this facility is not selected, then the area % is not covered (i.e., % = 0).

In practice, we need to determine the matrix = W $% X. This matrix reflects the location and coverage (by the site-specific testing capacity, number of tests available each day) of each test site and can be obtained by using following algorithm.

Step 1

Based on the testing capacity of each test site, we define the total population that each test site can cover, denoted as . Also, we denote the population of area by % .

While (Not all candidate test sites are analyzed):

Step 2

The location of potential test site is denoted by ( $! , $" ), which is the centroid of an area. We calculate the distance between test site and area , and define it as $% .

Step 3

Rank the $% from ascendingly, with the corresponding subscript (%,!) , … , (%,,) .

Step 4 ,) . Mark $ (+,!) , … , $ (+,,) = 1, and any other $ is set to zero. $ (+,,) is a weight for the population which is always set to 1.

For the above algorithm, we assume people always choose the closest test site to their home to be tested and we claim one area is covered if the population size of the area is below the capacity of the site.

To handle various test sites with different capacities or types of test that they provide, we

(.) denotes the ( , ) th element of the matrix of type test sites and 9 represents the total number of test site types. This holds since a person is covered if covered by at least one type of test site.

Taken together the objective and constraints, our problem formulation becomes:

The proposed optimization model is a non-convex optimization problem constrained to integer solutions. As the size of the problem can become very large for some states, applying exact integer programming algorithms may prove computationally infeasible, and we explore the use of a genetic algorithm approximation approach to solve the optimization problem. To solve this model, we use the Genetic Algorithm function from the gramEvol package in R. We define the number of chromosomes as the total number of test centers to be allocated and the best genome result to be the solution. The fitness function is our objective function.

To illustrate and validate the proposed allocation approach, we apply it to data from multiple 

. Note that larger coverage scores are preferred, while for the D-optimality and equity scores, the lower the better. Setting our weights as ! = 10 1" and # = 1, we obtain the results shown in Table 1 . For our target population percentage, we set it to 10% meaning a census tract will be covered if the testing center can cover 10% of its population. Further discussion about the choice of these weights is discussed below. From the above table, we find our algorithm always generates better coverage and equity scores than the observed distribution of test sites for all counties considered. However, for the Doptimality score, our approach does not always perform as well as the current testing distribution, suggesting competition between our optimality criteria. That is, simply improving coverage can make equity worse. Our solution considers the weights assigned to the equity and coverage criteria.

In the case study, both weights are set to 10 -2 . However, if we wish to improve the D-optimality criteria, we can simply increase or decrease the values of our weights ! and # .

Overall, the higher the value given to each weight, the more importance it takes in the final solution. For choosing weights, our initial analyses for the state of Georgia suggest that the coverage score's magnitude ranges from 10 0 to 10 1 , the D-optimality score's magnitude ranges from 10 -5 to 10 -4 , and the equity score's magnitude is typically 10 -1 . With these magnitudes in mind, we suggest the user changes the weight of ! to increase/decrease the weightage of coverage. For decision makers, we assign four general levels of importance (in the order of Less Important, Somewhat Important, Important, and Very Important) for both weights, and suggest the following illustrative values for each in Table 2 . 

These results merit a closer look. As noted above, our model is focused on public testing sites, which means common private testing sites such as CVS/Walgreens and doctors' offices are not considered. Keeping in mind the well-established inequities in regular health care access between non-white urban areas and predominantly white suburbs, our data (and hence, our model) does not consider preexisting availability of private testing in suburban areas [15] . Our crowdsourced data (which is incomplete) show us that in these areas where our model is proposing to move some centers to, there are already many private options available. A map of an incomplete list of private test centers in Fulton County can be seen in Figure 2 illustrating many preexisting private test centers in the areas the model suggests moving test centers to for equity purposes, when only based on existing public testing sites.

The Fulton County example illustrates a need for comprehensive data regarding all available sources of testing for the proposed optimization to have greatest impact, and a potential for the optimization results to compound existing inequities, if based on incomplete data.

We conduct a similar case study on Cobb County. A comparison of the current allocation and the suggested allocation of public testing sights for Cobb County can be seen below in Figure 3 .

Compared to Fulton County, Cobb County is comprised of more suburban areas and not many urban centers. The model decides to spread out testing sites across the county and is choosing census tracts which have close to 50% White vs 50% non-White racial distribution. This allocation lowers the equity score compared to the current allocation. The model is also choosing some census tracts strategically to help increase the number of census tracts covered due to the high weight given to coverage in our case study.

Similar to Fulton County, the incomplete crowd-sourced data available on existing test sites shows that there is a large number of private test sites allocated in Cobb County. The allocation of these private test sites is visible in Figure 4 . The high number of private test sites compared to public test sites is presumably due to the high number of suburban areas such as Smyrna, Marietta, and

Kennesaw. In fact, most of the private test centers on the map fall within and around these suburbs.

This large difference also highlights the difference in approaches a county could take in terms of reliance on public or private testing sites to fulfill testing demands.

To expand our scope beyond the Greater Atlanta Area, we also consider Chatham County, home to the city of Savannah, with results visible in Figure 5 . The Chatham County results show that the current allocation of public testing sites are clustered in the main urban area of Savannah and there are no public centers allocated to the suburbs of Savannah. Based on the large allocation of private centers in Fulton and Cobb County for Atlanta suburbs, upon checking the allocation of private centers in Chatham County, we found that there were only three total private test centers in Chatham County according to the crowdsourced data. The placement of these private sites can be seen in Figure 6 . 

Overall, there seems to be a small number of test sites in Chatham County which raises important limitations in using our model. When there are a small number of test sites to be allocated, the equity portion of the optimization model will try and place test sites into areas which are more evenly split along socioeconomic strata..

Similar to the Fulton County results, the model decides to moves test centers from predominatly non-white areas and put them in areas with more sociodemographically balanced populations (for a map with racial breakdowns of Chatham County, please reference Appendix C). The proposed allocation provides important insight in the application of a criteria seeking to minimize an equity score: If the allocation of public sites already favors non-white populations, perhaps to fill in gaps in coverage by public test sites, minimizing our equity score based only on public sites will move sites to more a racially balanced allocation. However, if the white population has access to additional private testing opportunities not included in the data, our optimization model can (and, often, does) compound existing inequities by proposing adjustments of public site locations away from non-white populations. Future work will extend the proposed approach to include both public and private test sites, but will require more comprehensive data on test sites to avoid the biases seen in public-only results.

The public-only allocations illustrated above seem to be hindering access for historically underserved communities of color, especially in urban centers in cities like Atlanta and Savannah.

However, the combination of reliance on private testing centers in suburban areas (often predominatnly White) and the overall low number of test centers available require public-only model results to be interpreted with caution, especially when used with incomplete data. For example, in Chatham County, given the low total number of public or private test sites listed in the crowdsourced data, our model results should give public officials an indication that there is a need for more test centers. As more test centers are added, the model would place more test centers in areas where the testing need is high in urban areas, and would do so in a manner to still minimize the inequity in access.

These results above raise an important point for such multicriteria optimization models used for test center allocation. This allocation happens within a context of the current modus operandi for allocating public resources. Some states may pursue a public-first philosophy where the main objective is to give all residents easy access to public testing sites. From there the state can forge partnerships with private firms to fill in the gaps in availability. On the other hand, states could use public sites as a second resort after all private testing centers are established. In this case, the public allocation would be used to fill in the gaps in availability provided by private sites, often resulting in allocation of public sites to urban areas with higher proportions of non-white residents.

Based on the data in our case study, this seems to be the case for many counties in Georgia. Such allocations compensate for inequities in location of private testing facilities, but, as we see above, by applying our optimization procedure to only the public test sites and requiring equity in access, the algorithm undoes the preferential assignment and proposes reallocation of public sites away from the very neighborhoods where they were placed to help. Future considerations will consider extensions to better include original goals as well as criteria for future allocation of both private and public test sites.

Expanding this discussion, decision makers must also consider many socioeconomic and geographic factors when allocating resources. For example, suburban residents usually own a motor vehicle and can travel longer distances to get access to testing and utilize larger testing sites with capabilities such as drive-through testing. In contrast, urban residents may be more reliant on public transportation and thus must have testing centers which are within a walkable distance or are located in higher density neighborhoods that do not have the space for larger drive-through testing centers. This context is critical for understanding the examples above and for further refining the approach. Eventually, extra constraints can be added to this model to allow for certain areas to not receive testing sites (as they could already be serviced by private sites), to allow for differences in transportation access, and to allow for multiple test-center types to be allocated.

Note that currently the web-based app allows for two test center types, regular test centers and mega test centers.

Another key aspect of our model is data support, i.e., the availability and type of information where all data on all active test sites, public and private, are available readily for users.

This paper proposes a novel test site allocation scheme by formulating a multi-objective optimization problem. The objectives are (i) increase in coverage; (ii) reduction of prediction uncertainties; and (iii) improvement of equity among different social demographic groups. With this framework we have built a model and an interactive tool which can take these objectives into consideration and deliver a Covid-19 Test Center allocation for a given number of test centers and their capacities to public health officials. Our tool allows the user to choose the percentage of the population they want to cover and the relative importance of each component in the optimization problem. Most importantly, our methodology allows for optimal allocations within the context of minimizing inequity in a given allocation. Using our model compared to existing location allocation algorithms in practice, our allocation scheme can outperform current test site allocation schemes. However, for our models to be used accurately, consolidated data sources containing full knowledge of existing public and private test centers are crucial for achieving best results. As the examples illustrate, ignoring existing differences in access and placement of public and private test centers can result in increasing overall inequity while attempting to minimize inequities for public sites alone. Moving forward, our model can serve as a baseline for many public health decisioning frameworks in the future to not only deliver optimal allocations, but allocations which help tackle historic inequities within society.

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