1. IntroductionThe COVID-19 pandemic has had a devastating effect worldwide. In the U.S. alone, by the end of 2020 there were more than 20 million reported infections, greater than 1.1 million related hospitalizations, and nearly 364,000 related deaths [1]. As demand for beds, medical personnel, and equipment quickly outpaced supply, hospitals turned away the ill and suspended preventive and elective procedures [2,3]. While multiple COVID-19 vaccinations were developed at unprecedented speeds and made available to general adult U.S. populations by early 2021, challenges in the distribution of limited vaccine supplies quickly arose [4,5,6].In the U.S., states received the bulk of vaccine supplies from the federal government in an amount typically proportional to their population size [7]. Each state was tasked with downstream distribution to residents and local agencies. Although consumer vaccine distribution in most states was preliminarily undertaken in phases, based on infection susceptibility and likelihood [8,9], this, and subsequent distribution to the general adult population, was largely based on geographic population size. Though a local distribution method based on population size appealed to a sense of equality, it negated a typically more accepted needs-based approach. Currently, the possibility of optimal vaccine allocation during the early stages of a pandemic is not well understood.Due to the infectious mode of respiratory person-to-person transmission, location and population mobility continue to play a key role in the spread of COVID-19 [10]. Population mobility patterns inform risk of infectious disease exposure as well as highlight the varying and local non-pharmaceutical prevention mandates implemented during the pandemic. These mandates included limited business hours, reduced public location capacity, and stay-at-home orders. The time populations spend at locations where people are likely to interact—such as restaurants, health provider offices, grocery stores, and religious institutions or places of worship—are a dominant factor that has shaped this pandemic [11,12,13,14]. These factors need to be included in disease spread prediction models, and further, guide vaccination distribution methods. Integrating classical models of infectious disease transmission into a vaccine allocation optimization framework is challenging. In typical compartmental models, the equations governing transitions among classes are non-convex. Incorporating these transitions into an optimization model results in computationally challenging problems. In most cases, the allocations returned by state-of-the-art optimization techniques carry no guarantee of optimality [9]. Further, the complexity of such methods restricts their accessibility and use in practice, an important requirement as COVID-19 continues to spread, causing morbidity and mortality.

The purpose of this study was to identify optimal spatial allocation methods for COVID-19 vaccines with the objective of minimizing reported COVID-19 cases. Utilizing GPS data from smart devices, we incorporated mobility and location factors into a mixed-effect Poisson model predicting the spread of COVID-19 infections. When placed in an optimization framework, the resulting model is a convex math program, readily solvable through widely available software. By showing how to overcome major obstacles to better vaccine allocation, the methods we propose are both timely and practical.

3. ResultsAcross the 115 counties of Missouri, there are 6,154,913 residents. Of those, 22.4% (n = 1,378,701) are under 18 years of age, 54.2% (n = 3,335,963) are 19-64 years of age, and 17.6% (n = 1,083,264) are older than 65 years. The population size for each county is depicted in Figure 1A. The majority of the state’s residents are white, while 11.8% (n = 726,280) identify as Black/African American and 4.7% (n = 289,281) as Hispanic/Latino.

During the study period, a total of 173,656 COVID-19 cases were reported among Missouri counties for an average of 58.7 cases per week per county (SD 220.2). At the end of the study (July 2021), counties across the state had an average vaccination rate of 26.8% (SD 7.3%). Residents spent an average of 113.4 min (SD 64.2) when visiting senior living facilities, 99.4 min (SD 54.3) at healthcare facilities, 87.3 min (SD 26.2) at educational facilities, 43.1 min (SD 63.1) at grocery and food stores, 38.5 min (SD 16.1) at retail locations, and 37.2 min (SD 17.4) at restaurants and bars. Overall, residents traveled an average of 21.8 km (SD 13.3) to reach these locations during the study period.

Differences in county population sizes are depicted in the quantile map in Figure 1A. The quantile maps in Figure 1B and Figure 1C show differences in average time at the specified locations and average distance traveled, respectively. Figure 1D displays the number of vaccines distributed among all counties at each week of the study, peaking at 354,894 during week 14. Results of the mixed-effect regression, which was designed to predict number of COVID-19 cases, are detailed in Table 1 with the estimated variable coefficients expressed in a log link response. Each variable included in the model was shown to be significantly associated with the response variable. The cumulative percent of vaccinated individuals increased and the number of new COVID-19 cases decreased across the state significantly. COVID-19 case rates increased significantly as time spent at any commercial locations was documented. Each county’s population and their higher average distance traveled away from home was significantly associated with to higher case counts.For each of the nine mobility-supply scenarios, Figure 2 compares the performance of optimal allocation against a population-based allocation, where vaccines were distributed only according to population size. In the 100% mobility and 100% supply scenario (Figure 2, Scenario 5), the state of Missouri’s actual allocation policy was used as a second benchmark. In this scenario, we predict spatial optimization of vaccine allocation would have averted 72,781 COVID-19 cases, averted 1301 COVID-19 related deaths, and saved $54,893,389 in COVID-19 related hospital costs. The optimal vaccine allocation was 9 percentage points more effective, based on averted cases, than the population-based allocation and 8 percentage points more effective than Missouri’s actual allocation. The largest disparity between optimized allocation and population-based allocation was seen when resident mobility was doubled and vaccine supply was halved (Figure 2, Scenario 7). Under these parameters, optimized allocation averted twice as many cases as the population-based allocation method. Even under the most favorable parameters, with mobility halved and vaccine supply doubled (Figure 2, Scenario 3), the number of cases averted by the optimal allocation was 6 percentage points higher than the number averted by the population-based allocation. Finally, we examined the value of vaccines across time in an optimal allocation policy. For each of the 9 scenarios, Figure 3 displays the dual variables associated with weekly supply constraints (3). Due to large differences in dual values, we display the figures in three charts with identical horizontal scales but with different vertical scales for each mobility level (50%, 100%, 200%). A value in Figure 3 can roughly be interpreted as the decrease in case count that would have resulted from one additional vaccine available for allocation during a particular week. This information is a unique byproduct of mathematical optimization and cannot be obtained through other means. 4. Discussion

The purpose of this study was to understand the impact of spatially optimal COVID-19 vaccine allocation in Missouri. Results suggest optimal allocation would have markedly improved health outcomes, reducing the number of cases by 8% during a 6 month period of time. These findings suggest that including variables that increase risk of an infectious disease more accurately will reduce morbidity and mortality. While mobility data had not been widely used prior to the COVID-19 pandemic to inform public health and healthcare efforts, they have been found to be especially useful in predicting a respiratory infectious disease.

This study also found that across all scenarios in Figure 3, vaccines were generally more valuable when they were allocated earlier rather than later. For example, when mobility was 50% and supply was 200% (Figure 2, Scenario 3), an additional vaccine had more than 12 times the impact in early January 2021 than it would have had toward the end of June 2021. This difference increased to more than 73 times when mobility was 200% and supply was 50% (Figure 2, Scenario 7). Because COVID-19 infections grew at an exponential rate, a given number of vaccines was more effective at slowing disease spread in the early parts of a pandemic than the same amount would have been later. The kinks in each series are related to variations in the supply schedule. When the number of vaccines available for allocation during a particular week was lower than the supply the week before and the week after, additional vaccines during that week were more valuable.Second, across all time periods, the value of a vaccine increased substantially as mobility of the population increased. For instance, when supply was at 50%, the dual variable corresponding to the week-one supply constraint increased by more than 4 times when mobility moved from 50% to 100%, then by an additional 90 times when mobility increased to 200 percent. That is more than a 38,000% increase from low to high mobility. This enormous difference points to the importance of mobility in curbing a pandemic. It is not that a vaccine’s ability to inoculate somehow increases as individuals spend more time outside of their residences and venture further away from their homes. Rather, as Figure 2 shows, the number of infections to be averted is orders of magnitude higher, and thus the potential for a vaccine to decrease disease spread is also much higher.While other optimization models primarily utilized population age as a means to allocate vaccines [22,23,24], this study relies on mobility in rural, suburban, and urban communities. Including these factors in the vaccine optimization model allows for stronger predictive inputs that inform the output far more than the allocation method used in Missouri, and in many other states at the time of allocation. Though the literature on COVID-19 vaccine allocation is young, the same realism-tractability challenge faced by many fields is present here. Optimization models that integrate location, mobility, and disease dynamics are better representations of reality than those that do not, but they are significantly more difficult to solve [24,25].

Our work considered the roles of mobility and location in disease progression and also provided guidance for optimal vaccine allocation policies. We demonstrated how optimal policies could have averted infections, deaths, and hospital costs in the Missouri during the first half of 2021, a period when vaccine supplies were low, and COVID-19 infections continued to increase. Across a range of scenarios, we showed the potential for an optimal allocation of vaccines to improve upon policies based on population size. We found that the benefits of optimal allocation increased dramatically in scenarios with higher mobility and fewer vaccines. However, even when mobility was low, and supplies were more abundant, optimal allocation of vaccines still led to reductions in case rates, fatalities, and hospital costs.

To conceptualize findings and propel future research, several study limitations were identified. Due to data availability, this study worked under the assumption that distribution of vaccines equated to administration of vaccines. However, news sources revealed that at times vaccines go unused [26,27]. In addition to including geographic mobility, it may be beneficial for future studies to consider collective community beliefs and attitudes surrounding likelihood of vaccine uptake. While this study also gives an estimate of COVID-19 deaths and hospitalization costs, these values are based on national averages and, like infection rates, are likely a product of geographic variation [28,29]. Further, infections may have been unreported during the study period. Additional studies would benefit from deeper examination of these variables and the role they play in optimal vaccine allocation policies.

Our work provides an important public health tool for the future. In the face of new COVID-19 variants, our analysis can be used to guide the distribution of limited supplies of resources, as well as to prioritize communities that may be affected earlier than others due to mobility. Further, as we prepare for the possibility of other pandemics, this research lays a foundation for the integration of important environmental factors into predictive disease models and prescriptive optimization tools.