## 2 Abstract

Vaccine allocation decisions during the ongoing COVID-19 pandemic have proven to be challenging due to competing ethical, practical, and political considerations. Complicating decision making, policy makers need to consider vaccine allocation strategies that balance needs both within and between populations. Due to limited vaccine stockpiles, vaccine doses should be allocated in locations where their impact will be maximized. Using a susceptible-exposed-infectious-recovered (SEIR) model we examine optimal SARS-CoV-2 vaccine allocation decisions across two populations considering the impact of population size, underlying immunity, continuous vaccine roll-out, and heterogeneous population risk structure. We find that in the context of an emerging pathogen, where many epidemiologic characteristics might not be known, equal vaccine allocation between populations performs optimally in most scenarios. In the specific case considering heterogeneous population risk structure, first targeting individuals at higher risk of transmission or death due to infection leads to equitable resource allocation across populations.

## 3 Introduction

Since its emergence, SARS-CoV-2 has caused considerable suffering and disruption of societies, resulting in over 173 million infections and 3.7 million deaths worldwide as of June 2021 [1]. Vaccines are currently the most effective public health intervention available against SARS-CoV-2, with eight vaccines currently approved for full use and an additional seven authorized for early or limited use globally [2, 3].

Even with the approval of multiple vaccines, distributed across different regions globally, roll-out has been slow. While some countries are falling far behind their anticipated timelines, vaccination rates are highly unequal across countries [4]. Indeed, as some countries have already vaccinated over 60% of their population, and are on track to vaccinate all willing adults by the end of the summer of 2021, others still do not have access to vaccines [4]. Even as COVAX– a Coalition for Epidemic Preparedness Innovations (CEPI), Gavi the Vaccine Alliance (GAVI) and World Health Organization (WHO) collaboration – aims to deliver approximately 1.8 billion doses to 92 low- and middle-income economies by the end of 2021 [5], it could take up to 2023 [6, 7] for all countries to have access to a sufficient number of doses. Overall, vaccination allocation decisions are currently being made under the constraint of a limited vaccine stockpile and multiple factors need to be considered in order to maximize the effect of each dose across populations.

As previous work [8, 9, 10] has shown, targeting specific subgroups within a given population, including older individuals, results in decreased COVID-19 morbidity and mortality. Further, previous theoretical work [11] has shown that unequal vaccine allocations might be favorable, but are less optimal when incorporating realistic assumptions about population heterogeneity and contact structure. This leaves a potentially conflicting message for policy makers when considering optimal allocation decisions. We build upon this work, by not only considering the optimal decision, but also how the decision compares to all possible allocations across two populations. Our results show that the efficiency gains for unequal allocations that are found in models with highly-simplified epidemics are typically small; moreover, they vanish and can even reverse under settings more relevant to the current pandemic. Similar to previous findings, we show in more realistic scenarios, incorporating population heterogeneity and interaction between populations, equitable distributions are not only optimal, but vastly outperform unequal distributions.

## 4 Methods

We use a deterministic, two-population, susceptible-exposed-infectious-recovered (SEIR) compartmental model. We assume people are initially susceptible (S). Susceptible individuals move to the exposed state (E) after an effective contact with an infectious individual. After a latent period, exposed individuals become infectious (I). After the infectious period has elapsed, infectious individuals move to a recovered state (R). We do not account for waning immunity and assume once individuals have recovered, they stay immune to SARS-CoV-2 for the duration of our simulation, here modeled as one year, which is longer than the epidemic duration in the vast majority of simulations. We start by assuming that there is no interaction between the two populations, so all disease transitions happen in parallel between the two populations.

We extend the traditional SEIR model to allow for underlying immunity and vaccination (Figure S3). At the start of the epidemic, in each population, individuals can be in the susceptible (S), infectious (I), or recovered (R) compartments. When there is underlying immunity, a set proportion of individuals are placed in R. Individuals in R, whether through underlying immunity or infection through the course of the simulation, can never be re-infected. When vaccine doses are distributed to the population, vaccinated individuals are placed in R if the vaccination is successful. We assume that the vaccine is all-or-nothing with 90% efficacy, meaning 90% of those who are vaccinated are placed in R and the remainder stay in S. When there is underlying immunity and vaccination, immune individuals may be vaccinated; vaccination has no effect on them and they remain in R. Finally, we initialize each simulation by placing one individual in I and allow the epidemic to run, unmitigated except by vaccination, through each population. Full model parameters and equations are shown in SI Appendix A.2.

To recreate the results of Keeling and Shattock (2012) [11], we model two homogeneous populations with identical characteristics apart from population size. In this scenario we assume population 2 is double the size of population 1 (Figure 2). For later scenarios, which consider the impact of heterogeneity within populations, we simulate two populations that are identical in size, but vary in their population characteristics (e.g. fraction high risk) (Figure 5 and 6).

Next, we extend the model by allowing for heterogeneous risk groups. Within each population, we first model efficient transmitters of the SARS-CoV-2 virus (e.g. young adults) [12]. In this scenario, we assume high-transmitters are four times more likely to transmit compared to other groups. We fix the within-population structure to allow the global *R*_{0} to equal 2. The full derivation is shown in SI Appendix A.2.4.

In addition, we also model individuals at elevated risk of death from severe COVID-19 disease, which can represent elderly individuals or other individuals with co-morbidities known to exacerbate COVID-19 disease [13, 14]. For simplicity of the model, we assume these individuals are five times more likely to die than other individuals infected with SARS-CoV-2. Note that death rates are assumed to be constant throughout the epidemic, which may not be realistic as health care resources are strained by large case loads or case management improves over time.

Similarly to the homogeneous two population scenario described above, we initialize the model by placing individuals from each population in the susceptible or recovered state based on the pre-existing immunity level and the vaccine allocation scenario. The total number of vaccine doses are split amongst the two populations based on the scenario. Within each population, high-risk individuals are vaccinated first, with leftover doses then allocated to the low-risk population, as described in SI Appendix A.2.5.

Finally, we model the scenario where vaccines are unavailable at the start of the epidemic, but are slowly rolled out over the course of the epidemic. For this simulation we vary both the timing of roll-out, and the fraction of the population vaccinated each day. We allow vaccine roll-out to start 1, 10, 30, 50, or 100 days after the epidemic has begun and vary the proportion of the population vaccinated from 1% to 3% per day. For these simulations, the vaccine is allocated within and across populations identically to the scenarios described above for the homogeneous and heterogeneous scenarios.

For each simulation we calculate the cumulative number of infections and deaths from the deterministic SEIR model at the end of the epidemic. Across each scenario we define the optimal allocation strategy as the one that minimizes the total epidemic size (cumulative number of infections) across both populations. Within the high morbidity scenario, we define the optimal allocation strategy as the one that minimizes the total number of deaths across both populations. This is equivalent to maximizing the total number of people across both populations that escape infection (or death) [15].

We conduct sensitivity analyses to assess the robustness of our results. First, we model a leaky vaccine scenario where we assume the vaccine reduces susceptibility to SARS-CoV-2 by 90%. As a result, all vaccinated individuals (except those previously immune through natural infection) can become infected with the virus, although the probability of infection for each contact with an infected individual is lower than for an unvaccinated individual. We further extend the model by relaxing the assumption that the two populations do not interact. We allow a fraction *i* of infected individuals in both populations to contribute to the force of infection in the other population instead of their own population. An *i* value of 0 corresponds to no interaction, and an *i* value of 0.5 corresponds to complete interaction between the two populations (i.e. is equivalent to one large population). Full model equations are shown in SI Appendix A.2.3.

All analyses were conducted in R version 4.0.3.

## 5 Results

### 5.1 Literature review

We reviewed the literature on optimal vaccine allocation across populations that was published prior to the emergence of SARS-CoV-2 (see SI Table A.3.1). Multiple papers [11, 16, 17, 18, 19] have shown that allocation proportional to population size is rarely optimal. Further, previous studies have highlighted that the timing of vaccine allocation [20, 21, 22], heterogeneity in population composition, as well as the stochasticity in infection dynamics affect the optimal distribution [23, 24]. Duijzer et al. (2018) [15] provide important contributions by showing that the optimal vaccination threshold is often not the herd immunity threshold.

Additionally, we summarize some of the recent literature modeling optimal SARS-CoV-2 vaccine allocation in one population in SI Table A.3.2.

### 5.2 Optimal allocation in two populations of equal size

We build upon the existing literature by first examining allocation decisions in the simple scenario of two identical, non-interacting populations with no underlying SARS-CoV-2 immunity (see Figure 1). In the simplest case, with a small number of vaccine doses available, pro-rata allocation performs comparably to highly unequal allocation strategies. As the number of vaccine doses increases, highly unequal strategies gain advantage over pro-rata allocation. This occurs because one population can be vaccinated close to, but lower than, its herd immunity threshold, maximizing the indirect effect of the vaccine doses [15]. When sufficient vaccines are available for both populations to reach that threshold, more unequal strategies use the doses less efficiently, as indicated by the increasing arms of the “W” shapes in Figure 1. Allocating doses to the population that has reached its threshold provides limited benefit in that population and withholds doses from the other. When there are nearly enough doses to reach the thresholds in both populations, the optimal strategy becomes equal allocation between the two populations.

### 5.3 Optimal allocation in two populations of unequal size

Extending the simple case of non-interacting populations of equal size, previous studies have shown how optimal allocation across populations of different sizes is not linear, but varies with the number of doses available in a characteristic, and often counter-intuitive, “switching” pattern [11, 15, 17].

As shown in Figure 2 (top), when the number of doses available is very limited, optimal allocation concentrates all vaccine doses to the smallest population, not assigning any to the largest population (regime 1). As the number of doses allocated to the smaller population reaches its threshold, additional doses are gradually allocated to the larger population (regime 2). Strikingly, for 777,500 doses available, a drastic switch happens and all doses are allocated to the larger population and none to the smaller one (regime 3). Then, as the largest population itself reaches its threshold, supplementary doses are assigned to the smaller population (regime 4). When the number of vaccines available allows both populations to attain their respective thresholds, vaccines are allocated proportionally to the population sizes (regime 5).

For most values of vaccine available, the optimal allocation is highly unequal, as previously shown [11, 15]. This counterintuitive result is caused by the non-linearity of the indirect effect from each additional vaccine dose. Additional doses are allocated to the population where they have the largest benefit. For example, in regime 1 of Figure 2, additional doses bring a larger benefit in the smaller population then they would in the larger population.

Importantly, while prior literature [11] demonstrates that unequal allocations can be optimal, these results show that the benefit of such unequal, optimal allocations over more nearly equal ones is often small. As shown in Figure 2 (bottom), for low numbers of vaccine doses (regimes 1 and 2), although concentrating all doses to the smallest population is optimal, other strategies do not perform much worse. Each regime is characterized by a different allocation profile that gives rise to a different optimum, indicated by black points. In regime 4, the characteristic W shape appears where a fully unequal allocation is sub-optimal, regardless of which population is vaccinated.

### 5.4 Impact of underlying immunity

As vaccines continue to get rolled out at varying speeds, populations will have varying degrees of underlying immunity to the virus due to prior infections. Serological surveys estimate that around a fifth of the population had already been infected in areas hardest hit during the spring of 2020 (23% in NYC [25], 18% in London [26] and 11% in Madrid and Paris [27, 28]). Select high-risk groups, including health care workers and nursing home residents, have been shown to have an even higher prevalence of SARS-CoV-2 antibodies [29]. To account for underlying immunity, we further simulate optimal allocation decisions with varying levels of underlying immunity in each population to mirror the fact that allocation decisions are made during an ongoing pandemic.

Comparing two populations with varying amounts of underlying immunity, the optimal strategy favors prioritizing the population that is closer to their herd immunity threshold (Figure 3). Figure 3 shows optimal allocation decisions across two homogeneous populations of equal size with no immunity (top left, repeating Figure 1) or increasing degrees of immunity in population 1. With increasing immunity in population 1, the characteristic V- or W-shape becomes more lopsided as fewer doses are required in population 1 to reach the threshold at which doses should be split between populations. Extremely unequal allocation strategies either waste doses or fail to minimize the cumulative number of infections in both populations if given completely to population 1 or 2, respectively. In addition, allocating vaccines to population 1 beyond the amount needed to reach its threshold results in the highest cumulative number of COVID-19 cases because it confers little additional benefit in population 1, and deprives population 2 of vaccines needed to mitigate cases. As before, once the number of doses is large enough to approach or reach the threshold in both populations, optimal strategies move to a more equitable approach.

### 5.5 Impact of delayed vaccine roll-out in a homogeneous population

Next, we examine the impact of vaccine roll-out over the course of the epidemic. We find both the timing and speed of roll-out play an important role in minimizing the final size of the epidemic. As shown in Figure 4, the cumulative number of cases across both populations is minimized when vaccine roll-out occurs as soon as possible after the start of the epidemic. Further, the final size is minimized when roll-out speed is increased, vaccinating a larger proportion of the population each day.

For the early and efficient roll-out (10 days and speed of 2 or 3% of the population/day), the vaccination performance profile looks similar to that of prophylactic vaccine deployment strategy shown in Figure 1. However for slower or later roll-out we see highly unequal approaches per-form poorly across almost all doses and more equitable approaches result in the smallest final size. This is because a larger fraction of the population is naturally infected, minimizing the gains from concentrating vaccine doses in one population.

### 5.6 Impact of heterogeneous population structure

Looking within a population, many studies have shown optimal strategies favor prioritizing older individuals (e.g., those aged 60 or over) when the goal is minimizing mortality. If the goal instead is minimizing final size, targeting adults 20-49 with an effective transmission-blocking vaccine minimizes cumulative incidence [8, 9]. Here we model the impact of heterogeneous population structure to examine the impact of strategies across populations. These simulations consider populations with heterogeneous transmission or with heterogeneous risk of death.

Targeting high transmission or high mortality groups first within a population shifts the optimal allocation across the two populations in the direction of greater equity (Figure 5 and 6). In Figure 5 we first model the impact of prophylactic vaccination in a heterogeneous population structure with 25% of each population at either high risk of transmission (top) or death (bottom). In the high-transmission scenario, the behavior looks similar to that in Figure 1 for a low number of doses, representing the trade-off between vaccinating the high-transmitters in both populations. Once there are enough doses available to vaccinate enough high-transmitters to reduce transmission dramatically, the optimal strategy favors more equitable allocations across the two populations as high-transmitters are driving the bulk of transmission. This shift to more equitable allocations occurs at a lower number of vaccine doses compared to Figure 1. In the high-mortality scenario, we see the optimal allocation rapidly shift to equitable strategies, starting at a very low number of vaccine doses. Interestingly, the sequence of profiles from Figure 1 is repeated twice. First, for a low number of vaccine doses there is a trade-off between vaccinating the high-mortality individuals in both populations. Then for higher vaccine counts the trade-off is repeated, this time between all individuals of both populations. While this trade-off exists, equitable allocation is heavily favored across almost all levels of available vaccine doses.

Next we considered the impact of continuous roll-out for both the high transmission and high mortality scenarios. We find that across both high-risk scenarios and all vaccine roll-out times and speeds, unequal allocation is highly sub-optimal (Figure 6). Similarly to Figure 4, we vary both the start date of vaccination roll-out (1, 10, 30, 50, or 100 days) and the daily vaccination rate (1, 2 or 3% per day). We find that both the speed and timing of vaccine roll-out are important factors in minimizing the cumulative number of cases or deaths across the two populations, and see the greatest reduction in cumulative deaths and final size with the earliest and fastest roll-out. Specifically, for vaccine stockpiles larger than 500,000 doses, the achievable impact of vaccination is more dependent on the timing (solid vs. dashed curves) and speed (different panels) of vaccine roll-out rather than on the total number of doses available.

### 5.7 Sensitivity Analyses

We assessed the robustness of our results by varying the characteristics of the vaccine and connection between populations to be more representative of the current pandemic. As expected, the leaky and all-or-nothing vaccine perform identically because both have the same critical vaccination threshold [30] (Figure S1).

In the previous situations we have only considered the scenario of non-interacting populations. As we relax this strict assumption, we find that as the amount of interaction between the two populations increases, equitable strategies are most favorable (Figure S2). When the force of infection in each population depends on epidemic dynamics in both populations, accounting for interaction drastically changes the optimal allocation profiles and favors equal allocation between populations, as seen in previous work [11, 15]. Even for low values of the interaction parameter *i*, equal allocation rapidly becomes optimal. Indeed, for *i* values higher than 0.01 — which corresponds to one out of every hundred infected individuals contributing to infection in the other population — equal allocation between the two population always performs best. As *i* further increases, unequal strategies progressively approach the optimal (equal) allocation as indicated by the flattening of the curves. For *i* equal to 0.5, when the two populations concretely behave like one large population, all allocation strategies perform almost identically. Compared to the non-interacting case, allowing for interaction between the two populations leads to a higher cumulative number of infections for all possible vaccine allocation strategies, and the “W”-shaped allocation curve no longer appears.

## 6 Discussion

As countries continue to roll-out vaccines, challenging allocation decisions will need to be made due to resource constraints. Previous studies [11] have shown simple scenarios favor unequal allocation. We recreated those findings, and further apply the theory to the current COVID-19 pandemic.

In the simple case of two non-interacting populations of identical size we show that for very high quantities of vaccine, relative to population size, equal allocation strategies are optimal. For very few doses, all strategies provide comparable results. This supports the European Commission’s decision to allocate vaccine doses proportional to population size among the 27 European Nations [31]. In this simplest model, until there is enough vaccine for both populations to approach their critical herd immunity threshold, optimal strategies favor a highly unequal approach, allocating doses to either population 1 or 2 until the population has reached its threshold. If the populations vary in size, allocation decisions vary, and as the number of vaccines increases, focus switches from the smaller population to the larger one, as supported by Keeling and Shattock (2012) [11].

We consider more realistic scenarios that better mirror the current COVID-19 pandemic including underlying immunity, population interaction, continuous vaccine roll-out, and heterogeneous population structure. While many of these parameters are either unknown or changing throughout the course of the epidemic, we find that, across a range of scenarios, optimal allocation decisions favor equal allocation across populations. Since these strategies are often optimal or nearly-optimal across a range of parameters, while unequal allocations are only generally optimal for narrow parameter ranges, more equitable strategies might be the best option under uncertainty in an ongoing epidemic.

Specifically, for scenarios considering heterogeneous population risk, we find that first targeting high risk individuals, either high-transmitters or those at higher risk of death after infection, results in more equal allocations between populations being optimal. Targeting high-risk individuals first, then shifting priority to lower-risk individuals is supported by previous modeling work, looking at SARS-CoV-2 vaccine allocations within a single population [9, 8, 32, 33, 10], and is in concordance with the ongoing COVAX strategy, targeting early doses to high-risk individuals, and the USA’s plan to vaccinate health care workers and elderly individuals first [34, 35].

Our modeling analyses are subject to many simplifying assumptions on population dynamics and vaccine characteristics that may not be applicable to the current pandemic. We consider a vaccine that prevents both SARS-CoV-2 disease and infection, thus providing indirect protection to a fraction of the population. Vaccines appear to be able to reduce infectiousness but this effect still needs to be precisely assessed. We do not model vaccine refusal, and assume that all individuals given doses accept them. Recent studies [36] show vaccine hesitancy as a threat to successful COVID-19 response. Next, we do not consider delays between doses, but model the epidemic from a final dose which confers 90% efficacy. Due to vaccine shortages, the delay between the first and second dose could impact our findings as individuals may be able to get infected in the interim. Further, we only model one available vaccine. The current SARS-CoV-2 vaccine landscape is complex, and multiple vaccine candidates are rapidly undergoing testing. Considerations for optimal allocation in this context are more complicated, especially if the vaccines have different immunogenicity profiles, and the quantity of doses and timing of roll-out varies across candidates. Further, we only consider one strain of SARS-CoV-2, not taking into account variants with varying transmissibility and mortality. Finally, we do not consider the impact of non-pharmaceutical interventions (NPIs) in conjunction with vaccination.

Furthermore, we only model allocation strategies within two symmetric populations. It is likely that policy makers will face allocation decisions across multiple countries, or across multiple regions within a country. While our analyses do not extend to more than two locations, general principles should remain the same, as illustrated for three populations in Duijzer et al. (2018) [15] and Keeling and Shattock (2012) [11].

Future modeling work on SARS-CoV-2 vaccination strategies is needed, as multiple vaccine candidates continue to be rolled out. These studies should also consider the effects of vaccines on reducing hospitalizations and preserving hospital capacity, which may have indirect benefits for mortality rates for COVID-19 and other diseases beyond the direct prevention of infection in high-risk populations. In addition, other work should also consider populations with varying epidemic dynamics, and distribution capacity. Indeed, it has been argued that populations at higher immediate risk of SARS-CoV-2 spread and populations where vaccine roll-out is most efficient should be prioritized for SARS-CoV-2 vaccine allocation [37].

With vaccine supplies still severely constrained, rapid allocation decisions will need to be made. Due to the magnitude of SARS-CoV-2 spread and impact globally, further political and economic constraints will likely play a large role in allocation decisions. Mathematical modelling can provide insight into optimal allocation strategies that maximize the benefit from each dose. Conclusions from such models should be balanced with ethical considerations on the fairness of allocation that also minimize disparities in access. We show key principles that should be considered in the design of realistic and implementable allocation strategies.

## Data Availability

All relevant data and code will be available on GitHub.

## 7 Acknowledgments

We thank Dr. Kendall Hoyt for helpful discussion. ER, LKS and ML were supported by the Morris-Singer Fund. This work was supported in part by Award Number U01CA261277 from the US National Cancer Institute of the National Institutes of Health. The content of this article is solely the responsibility of the authors and does not necessarily represent the official views of the Morris-Singer Fund or the National Institutes of Health.