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Michael and , b eateto ouainHat cecs colo ulcHat,GogaState Georgia Health, Public of School Sciences, Health Population of Department doi:10.1073/pnas.2025786118/-/DCSupplemental ulse pi ,2021. 2, April Published at online information supporting contains article This 1 under distributed BY).y is (CC article access open This Submission.y Direct PNAS a is article This interest.y competing no con- declare authors and help The with paper, data the collected wrote M.R.S. J.B. and G.C.y J.H.B. M.R.S. from project; and and visualization; the J.B. and investigation administered research; ducted conceptualized and M.R.S. methodology and developed G.C., J.H.B., contributions: Author lower with against associated policies is vaccination COVID-19 mirror example, simply For pan- ongoing not influenza. the should against strategies demic Sharp vaccination that (8). 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ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES susceptibility to infection among children and adolescents (9, 10) chance they transmit the disease if infected. Policies that account and has a substantially higher infection fatality rate overall that for the greater risk essential workers are exposed to may be more also increases markedly with age (11). Toner et al. (ref. (5), p. just and highlight a group of individuals “who have been over- 24) provide a detailed overview of the 2018 pandemic influenza looked in previous allocation schemes” (5). Furthermore, these vaccination plan and conclude that “the priority scheme envi- policies may be more effective at mitigating morbidity and mor- sioned ... does not comport with the realities of the COVID- tality, as they can account for a key factor driving transmission of 19 pandemic and new guidance is needed.” Fitzpatrick and the disease. Galvani (12) concur, detailing how the unique “epidemiological, To account for the gradual rollout of vaccines, we employ clinical, behavioral, and vaccine-related relationships” of SARS- stochastic nonlinear programming techniques to solve for vac- CoV-2 motivate the need for “pathogen-specific transmission cine prioritization policies that distribute vaccine to susceptible modeling.” individuals and change on a monthly time step responding to We develop and apply a mathematical model to assess the changes in the epidemiological status of the population (shares optimal allocation of limited COVID-19 vaccine supply in the of the population in different disease states). These dynamic United States across sociodemographic groups differentiated by policies account for a key feature of the policy-making process, age and essential worker status (see Methods). The transmission since the supply of vaccine is likely to be constrained, with avail- dynamics are modeled using a compartmental model tracking able doses administered as they become available over a period eight demographic groups through the nine disease states as of several months. shown in Fig. 1. The parameters are calibrated to capture our The transmission of COVID-19 is a complex process con- current understanding of the epidemiology of COVID-19, and tingent on the characteristics of the disease and ever-changing our analysis is designed to capture two key features of COVID- social behavior. Furthermore, many of the key dynamics can 19 prioritization: essential workers and the gradual availability of change depending on the spatial scale considered, with differ- vaccines over time. A large number of workers are constrained ences in the transmission process within and between commu- in their ability to work from home (essential workers), expos- nities. We seek to summarize the features of the complex and ing them to a higher level of risk of infection, and increasing the evolving processes that are most relevant to the spread of the

Fig. 1. Schematic of the modeled movement of individuals between epidemiological states defined in Methods (A), the portion of individuals from the US population in each demographic group determined by essential worker status (∗) and age (B), and the contact rates between demographic groups, given by average daily number of contacts a group on the horizontal axis makes with a group on the vertical axis (C).

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ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES Table 1. Descriptions of alternative scenarios relative to the base model (see SI Appendix, section A for specific levels) Scenario Change from base scenario parameters Source Base scenario None (Base parameter values are provided in SI Appendix, section A) High initial infections Increased number of initial symptomatic Assumed: pandemic state will infections (300% increase) vary between localities when vaccine first available to the general public Strong NPI NSD NPI are strong, resulting in Consistent with R < 1 a declining infection rate Weak NPI NSD NPI are weak, resulting in a sharply Consistent with R  1 increasing burden of infection Weak vaccine Lower vaccine effectiveness (success rate) for Minimum value required by all age groups relative to the base scenario FDA guidelines Weak vaccine 60+ Lower vaccine effectiveness for Informed by influenza vaccine ages 60+ y effectiveness Even susceptibility All ages are equally susceptible to infection; Assumed: tests sensitivity increase in susceptibility for ages < 20y to age-dependent susceptibility relative to Base described by refs. 9 and 20 Low supply Sufficient supply for 5% of the population Assumed: vaccine supply is monthly (50% of supply relative to base uncertain and known to impact scenario; prioritization changes every 10% of the optimal allocations (21) population vaccinated, such that decision period is 2 mo) Ramp up Vaccine supply is 5% per month for Informed by comments from the first 2 mo and 10% per month thereafter the scientific head of the US (first decision period is 2 mo, so increments vaccine development of 10% of the population are vaccinated each program (22) decision period) Open schools Rate of social contact in schools increased Assumed: tests sensitivity of from 30% in base model to 70% optimal allocations to school closure intensity High contacts Increased number of contacts outside Assumed: tests sensitivity to the home, school and workplace (50% increase relaxed distancing relative to base)

older essential workers and school-age children (5 y to 19 y), not for deaths. However, even while the age-only and static since these groups have higher contacts and thus higher risk of policies do not substantially impede performance in minimiz- contraction and spread. ing deaths, these constrained approaches still suffer substantial In Fig. 3A, we show the dynamic path of infections, start- performance loss (9 to 18 percentage points) in the other two ing from the period in which vaccines become available, under outcomes not optimized (YLL and infections) but clearly still of various policies. As expected, infections are highest given no interest.¶ In other words, accounting for both essential work- vaccines. Results for allocating vaccines in a manner propor- ers and a dynamic prioritization strategy provides substantial tional to each group’s size shows the substantial value of even improvements in the metric being optimized and/or the other two “untargeted” vaccines. As expected, the policy for minimizing metrics of interest. infections leads to the lowest level of infections. In general, we find that, no matter the policy objective In Fig. 3B, we show the performance of various policies for pursued in targeted vaccine allocation, some improvement resulting outcome metrics (infections, YLL, and deaths) in terms is made on all three metrics. However, there are trade-offs of the percentage improvement relative to an untargeted vac- in what can be achieved between the objectives. For exam- cine allocation. We consider the optimal policies presented in ple, policies that minimize infections result in substantially Fig. 2 where the objective is minimizing infections (green), more deaths than a policy that minimizes deaths. We also YLL (purple), or deaths (orange) with no constraints (“none”). find that differentiating essential workers substantially reduces We also consider two constrained alternatives: an “age-only” these trade-offs between objectives relative to age-only or static dynamic policy that does not differentiate by essential worker policies. status, and a “static” policy where the fractional allocation across groups does not change over decision periods.§ We find that Sensitivity of Vaccine Prioritization. To assess how robust our base the unconstrained policy—that is dynamic and differentiated scenario findings are to key uncertainties in the model, we con- by essential workers—outperforms the untargeted approach by duct three different sensitivity analyses. First, we consider a set of approximately 31 to 40% depending on the objective. Rela- 10 alternative plausible scenarios involving a broad set of model tive to the unconstrained policy, the age-only and static policies inputs; then, we focus on a narrower set of four parameters, perform substantially worse for infections and YLL, although each explored in richer gradient detail; finally, we examine a few fundamental changes to model structures.

§Excess vaccine is allocated without targeting if all of the susceptible individuals in a ¶See resulting YLL and resulting infections for the deaths objective with age-only and given group have already been vaccinated. static constraints.

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ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES infections, stronger or weaker nonsocial distancing (NSD) non- We find that performance costs, in percentage terms, from pharmaceutical interventions (NPI) like mask wearing, weaker applying the wrong policy from this set are typically modest vaccine effectiveness overall or for seniors (60+ y), lower vac- (low single digits), albeit with notable exceptions. For exam- cine supply or supply that starts low and ramps up, more open ple, when the “truth” is that we have a weak vaccine for ages schools, or higher contact rates overall. 60+ y, several policies applied perform very poorly relative To compare and contrast optimal early vaccination alloca- to the true optimal policy, since they substitute vaccine away tion for each scenario and objective, in Fig. 4A, we show the from younger essential workers to ages 75+ y. A few of the percentage of each group vaccinated after 30% of the overall policies were generally less robust across various true models, population is covered (typically in 3 mo, except for alternative specifically, those for high initial infections, strong NPI, and supply scenarios). We find that high priority groups—by percent weak vaccine 60+. The base scenario policy performed rea- of group vaccinated—are typically but not always robust to the sonably well across true alternative models, with the largest alternative scenarios. For example, when deaths are considered loss arising (7%) when children are not less susceptible (even (Fig. 4 A, Top), we see substitution between younger essential susceptibility). workers (29 y to 39 y∗) and ages 60 y to 74 y, and, when YLL Equivalent versions of Fig. 4B for minimizing deaths or infec- are considered, there is substitution between younger essential tions are provided in SI Appendix, section C. When the focus is workers and ages 75+ y. To illustrate differences in the relative minimizing deaths, the pattern of performance between scenar- order of these high-priority groups, in SI Appendix, Fig. S5, we ios is very consistent with YLL in Fig. 4. However, the scope for show optimal prioritization of vaccination in the very first deci- performance loss is larger overall—up from a maximum of 26% sion period across objectives and scenarios. We find that, when for YLL to 46% for deaths. When the focus is infections, the YLL are considered, essential workers ages 40 y to 59 y are the range of performance loss is much less intense, at 7%. For infec- highest-priority group in all scenarios. However, when deaths tions, this relatively robust performance arises because optimal are considered, ages 75+ y are the highest-priority group under policies are much more similar across scenarios when minimiz- several alternative scenarios. ing infections (compared to the other objectives). Given greater For insight into the cost of error in specifying the correct scenario-driven heterogeneity in policies for minimizing YLL or scenario, we assessed the performance of the policy identified deaths, there is greater opportunity for performance loss from for each of the 11 alternative scenarios, depending on which specification error. of these 11 is the “true” scenario. In Fig. 4B, we show these A gradient over four key parameters. For further sensitivity results for the YLL objective. For example, the first column analysis, as shown in Fig. 5, we assessed how optimal vaccine allo- shows the performance loss (in percentage of additional YLL cation policy changed along a gradient for four key model inputs: above the optimum) when the true scenario is the base model NSD NPI effectiveness (e.g., mask wearing) which determines but the decision maker applies a policy matched to any of the initial reproductive number (when the vaccine first becomes the alternative scenarios (rows). By construction, when the pol- available), initial infections, monthly rate of vaccine supply, and icy applied matches the true scenario, the performance loss is vaccine effectiveness. zero. When YLL is the focus and the base specification is the Echoing sensitivity results reported above, variation in these “true” scenario, the greatest performance loss (9%) comes from parameters had little effect on the optimal policy for minimizing mistakenly applying the high initial infections policy. infections. But we found systematic differences in the policies

Fig. 4. The cumulative percent of each demographic group (horizontal axis) vaccinated after the first 30% of the population is vaccinated under the alternative scenarios (vertical axis) and each objective to be minimized including deaths (Top), YLL (Middle), and infections (Bottom) (A); init. infect., initial infections. The percentage of additional YLL in excess of the optimum when applying a policy for a given alternative scenario (row) when a particular scenario is the “truth” (column) (B). *, indicating essential worker groups.

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ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES concentrated essential workers, where, relative to the baseline Static policies identify a set of high-priority groups but not how scenario, the portion of the working age population deemed to order them when phased deployment is necessary. More strik- “essential” is half (20%), and they have approximately double ingly, targeting essential workers (or other adults with a large the contact rate; and 3) leaky vaccine, where, rather than work- number of work contacts) reduces not just the adverse out- ing perfectly for 90% of individuals, vaccinated individuals have come of focus but also trade-offs for remaining outcomes. For reduced susceptibility to infection, infectiousness, and risk of example, when minimizing deaths, allocation that differentiates death if infected. A more detailed discussion of these models is essential workers substantially lessens the degree to which infec- included in SI Appendix, section D. tions and YLL climb from the minimum achieved when each is We found that the qualitative nature of the solutions remained optimized on its own. constant across each of these alternative models, with some Existing published analysis of optimal COVID-19 vaccination minor differences. Treating the essential worker group as a targeting includes Matrajt et al. (16) and Bubar et al. (17). Before cluster increased the proportion of vaccine allocated to ages comparing and contrasting results, some key modeling differ- 60+ y when deaths and YLL were considered. This shows ences should be noted. These two analyses consider a wider that, when essential worker contacts are clustered within group, range of vaccine availability than considered here. Our analy- this reduces the indirect protection that vaccinating these sis uniquely incorporates NPI, including social distancing and individuals provided to others. Conversely, concentrating the NSD (e.g., mask wearing). Doing so allows us to account for essential worker group (to a more select group with higher differences between groups like essential workers constrained in contact rates) increased the fraction of these individuals vac- distancing versus others who are much less so. All three preprints cinated. This shows that select essential workers with espe- implement static optimization where vaccines are allocated and cially high contact rates (e.g., medical professionals and essen- administered in a one-shot process. Our allocation is dynamic, tial retail workers) are particularly strong candidates for early responding to changing epidemiological conditions over a 6-mo vaccination. period. Finally, all three model vaccines as “leaky,” that is, reduc- ing the probability that a susceptible individual will be infected Discussion [and the probability of severe disease (19)]. Bubar et al. also Key insights and results from our analysis are summarized in consider an “all-or-nothing” vaccine that is 100% effective for Box 1. Together, these lessons show the strong implications of a fraction of the population. In our base model, the vaccine is considering dynamic solutions, social distancing, and essential “all-or-nothing,” although we also consider a leaky vaccine, as workers (given their limitations in social distancing) for vaccine discussed at the end of Results. prioritization. Matrajt et al. (16) found that optimal strategies to minimize Our analysis of COVID-19 vaccine prioritization uniquely deaths and YLL will either exclusively target groups with high accounts for two critical needs: 1) dynamic prioritization given infection fatality rates, maximizing the direct benefit of vaccines, gradual rollout of vaccine during an active pandemic, and 2) or will target groups with high rates of infection, maximizing attending to substantial heterogeneities in work contacts among the indirect benefits of the vaccine. In contrast, our results indi- the adult population due to the ability of many to work from cate that optimal policies initially target groups with high risk home. These two novel features demonstrably change opti- of infection and then switch to targeting groups with high infec- mal vaccine prioritization. Given gradual vaccine deployment, tion fatality. This difference most likely follows from our dynamic static policies are out-performed by dynamic policies, which nar- versus static allocation. The switching behavior we identify is rowly target a small number of demographic groups and (after consistent with past work on pandemic influenza vaccine pri- substantial coverage of them) switch to lower-priority groups. oritization, which suggests that, early in an outbreak when the

Box 1. Key insights and results

1) Benefits: Prioritization can reduce a particular undesirable outcome (deaths, YLL, or infections), by 32 to 40% in the base scenario (or 17 to 44%, depending on the alternative scenario). 2) Objectives: Moving from minimizing infections to YLL to deaths boosts each of the following: benefits from vaccina- tion targeting, prioritization differences between scenarios, and (therefore) the sensitivity of optimal prioritization to scenario. 3) Dynamic prioritization: Dynamic prioritization 1) is responsive to the initial and evolving disease status and 2) generates substantial improvement in outcomes relative to a static prioritization, indicating that a phased approach to vaccine distribution is well justified. However, diminishing marginal returns to additional vaccination within a group drives a shift to other groups before 100% vaccination of the first group is achieved. 4) Widening prioritization: As vaccination rates rise, precise prioritization becomes less critical, and targeting widens to a larger set of groups. 5) Trade-offs: Policies that target one objective forgo opportunities to reduce alternative metrics. For example, policies that minimize deaths do not reduce infections nearly to the same degree as policies that minimize infections. These trade-offs are typically stronger when policies do not allow for targeting based on essential worker status. 6) Essential workers: Relative to an age-only model, policies that allow targeting of essential workers provide the greatest improvements when minimizing infections and YLL are the focus. In the base scenario, essential workers are a high priority group under all three objectives (i.e., they are among the first 30% of the population to receive vaccines). However, their priority relative to ages 60+ y is affected by key model parameters (see Sensitivity next). 7) Sensitivity: The high-priority groups remain consistent across the range of parameters considered. However, when min- imizing deaths or YLL, the fraction of vaccine allocated to essential workers and ages 60+ y depends on the number of infections and reproductive number when the vaccine became available, the supply of vaccines, and vaccine effective- ness. In effect, when the vaccine has a limited ability to quickly reduce the transmission of the virus, optimal policies more heavily prioritize older individuals.

8 of 12 | PNAS Buckner et al. https://doi.org/10.1073/pnas.2025786118 Dynamic prioritization of COVID-19 vaccines when social distancing is limited for essential workers Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 yai roiiaino OI-9vcie hnsca itnigi iie o seta workers essential for limited is distancing social when vaccines COVID-19 of prioritization Dynamic al. et Buckner fet,pirtzto ih hf wyfo hs rus(26). var- through groups occur will these vaccination from perspective, logistical away a shift From side might vaccination prioritization significant effects, experience seniors) groups or certain if children example, (e.g., For work. future for remain extensions allocation vaccine should strategies. occupation-differentiated makers policy consider that the strongly indicating to relative models, population impor- age-only adult the with the standard lower demonstrate in group heterogeneity to with this us group a of allows tance a This over contacts. and work contacts workers) of of rates (essential contacts number work total high simplified the is averaging differences these by of representation nonessen- the is and workers, model essential tial reality, between our differences In capturing While in even order. occupations. unique individuals, in “essential” across different are continuously varies between caveats infection of of risk number the a educational process, essential making and broadly. poverty) more workers in essential to those groups turning before (e.g., sociodemographic sub- workers risk various However, high adults. on at older at workers focuses by care priority followed health infection, sequent frontline of of risk prioritization high the audience— here. on global considered a agree for not broadly underly- distinction more consider guidelines—written salient WHO also a guidelines conditions, CDC health ing The workers, from age. mortality essential increasing with by older motivated infection latter and distinc- the younger on the will priority between with in which here differ prioritization conditions, made recommendations that specific tion Our to in location. sensitive findings, over was our vary pair with this consistent within work- set is essential not sub- frontline This and is over are priority ers. and y clear B 75 a clear aged and persons with Notably, between although A priority. here, top Categories high- considered supporting than with remaining. logic scale y finer over 64 4) a everyone to at and workers; 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ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES recovered (R), and deceased (D). In Fig. 1, we display the compartmental Table 2. Weights on contact rates for a given disease and diagram and directions of transitions between epidemiological states. symptom type (m) and location/activity (x) under social We modeled the COVID-19 transmission dynamics using a system of cou- distancing pled ordinary differential equations for each demographic group, indexed by i and j. The transmission rate was given by the product of the trans- Contact rate weights, αm,x mission probability (q), the age-specific susceptibility (s ), strength of NPIs i Disease and symptom type Home Work School Other (θ), the relative infectiousness of each symptom type (τm)—where m ∈ M ≡ {asym, pre, sym}—and the rate of contact (rm,i,j) between infected individ- Symptomatic 1.0 0.036 0.036 0.075 uals with symptom type m from group j and susceptible individuals from Susceptible or asymptomatic 1.0 0.4*, 0.1 0.3 0.4 group i. The exogenously given population vaccination rate at time t is given by v(t), where units of time are days.# In our base model, we assume that, When essential and nonessential worker weights are both needed, the for each individual, the vaccine either works or it does not (although we also former is marked with an asterisk. consider vaccines that are partially effective for all vaccinated in our sensi- tivity analysis). Individuals in group are vaccinated at a rate of µ ( ), and a i iv t when this is not the case, we may overestimate or underestimate the impor- fraction of the those ( ) are protected, while a fraction remain susceptible i tance of school contacts, depending on the time of year when vaccines are and move to the failed vaccination category ( ).k Once infected, individ- F distributed. uals move from exposed to presymptomatic at rate γ−1. Presymptomatic exp The proportion of essential workers in the population was set to be con- individuals become symptomatic or asymptomatic at rates σ /γ and asym pre sistent with estimates of the portion of jobs that can be done from home (1 − σ )/γ , respectively. Asymptomatic individuals recover at an uni- asym pre (31) and estimates from the US Cyber-security and Infrastructure Security form rate γ−1 , and symptomatic individuals either recover or die at a asym Agency, which indicate that 70% of the workforce is involved in these essen- rate of (1 − δ )/γ or δ /γ , respectively, where δ is the age-specific a sym a sym a tial activities (e.g., healthcare, telecommunications, information technology infection fatality rate. These assumptions yield the system of differential systems, defense, food and agriculture, transportation and logistics, energy, equations described in , section A, with parameter values also SI Appendix water, public works, and public safety) (32). However, essential workers are given in , section A. SI Appendix not a cleanly defined group of individuals, and there is heterogeneity in the level of contact rates within this group. As a robustness check on this base Contact Rates. Contact rates indicating the level of direct interaction of indi- scenario approach, we also tested a model with a smaller number of essen- viduals within and between groups drive the transmission dynamics in the tial workers with higher contact rates. Results from this model are discussed model. We built the contact matrices used in this model from the contact in SI Appendix, section D. matrices estimated for the United States in ref. 14. These estimates are given for age groups with 5-y age increments from 0 y to 80 y. These estimates Transmission Rate and Vaccine Effectiveness. The model was calibrated to were aggregated to provide estimates for the six-level age structure used match the predicted R0 for COVID-19 in the United States (see SI Appendix, in our model. We also extended these data to estimate the contact rates Table S1) by solving for probability of transmission q, assuming a naive of essential workers. A detailed derivation of these contact rates can be (prepandemic) population. Details of this procedure are provided in SI found in SI Appendix, section A. In short, we assumed that essential work- Appendix, section A. ers have, on average, the same pattern of contacts as an average worker In our base model, we considered vaccine effectiveness of 90%. This in the population in the absence of social distancing. We then scaled the level is at the low end of the range of estimates reported (90 to 95%) contact rates for essential and nonessential workers to represent the effects for reduction in symptomatic infections in the fall of 2020 from phase of social distancing, and calculated the resulting mixing patterns assuming three clinical trials (33). We selected the low end, since real-world per- homogeneity between these groups. formance is typically somewhat lower than clinical trial effectiveness, for We constructed contact matrices for each of four locations, ∈ x example, due to imperfect implementation of dual-dose timetables and/or { , , , }, following ref. 14. The total contact rate for an home work school other cold storage requirements. We also assume this effectiveness is the same asymptomatic individual before the onset of the pandemic is given by the across age groups, since initial evidence does not show substantial differ- sum of these location-specific matrices. However, it is clear that populations ences between subgroups (34). As an alternative, lower-bound scenario, we are exhibiting social distancing in response to the pandemic (29). We further considered vaccine effectiveness of 50%, since this is the minimum expec- expect symptomatic individuals to change their behavior in response to the tation of the US Food and Drug Administration (FDA) for approval (35). illness. We account for these behavioral changes as described next. Finally, we considered a case where the vaccine is less effective for ages 60+ y. The phase three trials do not fully resolve the effectiveness of the vac- Social Distancing. Expression of symptoms and social distancing policies are cines by age, leading to uncertainty. This scenario represents a worst-case likely to change individuals’ behaviors over time. To model these changes, scenario where the vaccine is much less efficacious for the most sensitive we scaled the contribution of each contact matrix for location x, groups. X rm = αm,x rx . [1] Initial Conditions. Because the expected epidemiological conditions x {Ipre(0), Iasym(0), Isym(0), S(0)}, by the time the initial vaccine doses are ready for deployment, are uncertain, we consider a range of possible values from The weights α depend on disease and symptom status (m) and loca- m,x 1 case per thousand to 20 cases per thousand. These cases were apportioned tion (x) as specified in Table 2. We scaled social contacts for symptomatic between demographic groups to reflect the attack rates of COVID-19 for individuals following changes in behavior observed among symptomatic each group under the given social distancing policy. Alternative levels individuals during the 2009 A/H1N1 pandemic (30). For those without symp- considered for initial conditions are described in SI Appendix, section A and toms (susceptible and asymptomatic), the weights were specified to match appear in Results (Table 1 and Figs. 4 and 5 C). reduced levels of social contacts as the product of social distancing policies. Home contact rates were held constant, and nonhousehold contact rates The planner’s decision problem is to were roughly based on survey data from ref. 15. However, levels of social Vaccine Prioritization Optimization. allocate the daily supply of vaccine ( ( )) across the demographic groups distancing have varied strongly over time and between locations. To account v t according to a given objective. We assume that this group allocation vector, for this variability, we tested a range of alternative levels in addition to the µ, can be chosen on a monthly basis at the beginning of each of the first six base model. The results for these alternative parameter values are discussed decision periods. We also assume that only susceptible individuals are vacci- in SI Appendix, section A. Also, notably, we do not consider the seasonality nated. We numerically solved for vaccine allocation strategies that minimize of contact rates for children in the scenarios where schools are modeled as the total burden associated with three different health metrics: deaths, YLL, closed. This would likely have limited impact on the optimal solutions, but, or symptomatic infections,

  Z T #  X  In the event that vaccination requires two doses over time, we consider an individual deaths: min Isym,i(t)/γsymdt [2] vaccinated upon receipt of the second dose at time t, and we assume that v indicates  0 i∈J  the number of individuals that can be vaccinated with the required number of doses.   Z T kThis vaccine effectiveness is inclusive of any efficiency loss from typical handling in the  X  YLL: min eiδiIsym,i(t)/γsymdt [3] distribution chain.  0 i∈J 

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Wang adviser X. top 23. for Speed Warp allocation Operation vaccine Kelly: Louise Optimal Mary Jr., with Interview Slaoui, Longini M. M. I. 22. Halloran, E. M. Matrajt, L. 21. Zhang J. evaluating 20. when critical is Hogan A. benefits 19. indirect Considering al., et Gallagher E. M. 18. okn ru nCVD1 acns(matMdln subgroup). Modeling Immunization we (Impact on Vaccines comments, SAGE COVID-19 WHO on (MIDAS) helpful the Group Study For Working and Agent seminar 2026797 Disease 130900. modeling Infectious Grants GM COVID-19 of network NSF Models R01 the by Grant in participants supported NIH thank partially Califor- and of is 2034003, University from G.C. the and Award R00RG2419. of Seed Program Grant COVID-19 Research Emergency nia, of Cancer views an Breast the by reflect California supported necessarily the Any this not is do M.R.S. in 1650042. and NSF. authors expressed the Grant the recommendations of under those or are conclusions material Program and Fellowship findings, opinions, Research Graduate NSF ACKNOWLEDGMENTS. (https://github.com/JackBucknerNRM/Vaccine Github in prioritization). deposited been Availability. Data ∗∗ oeue oiiilz n u h oesi vial nGithub, on available is and models data the The run measure. each and for initialize noted to https://github.com/JackBucknerNRM/Vaccine source used the code at available these freely of Access. share relative Data their to to accrue proportion groups in age essen- groups. simply working by to workers differentiated allocated essential (not status)—vaccines groups worker age tial choices and available allocation time. becomes each first over constraining to vaccine constantly allocated the when vaccine applied static once of then The chosen proportion policy. be to the age-only group allowing an age by constructed and found we policy age, was static to policy a addition policies: in constrained status two worker essential by ferentiating iul rmta ru r nta loae oohraegop tart proportional rate a population. at susceptible groups their age of other indi- size to to the allocated allocated to instead been are have group would that that vaccine from then viduals infection), from or vaccine the from falo h ucpilsfo igegopwr xase ete yfl coverage full by (either exhausted were group single a from susceptibles the of all If oass h eet f1 sn yai loainplc n )dif- 2) and policy allocation dynamic a using 1) of benefits the assess To n epne oCVD1 mrec elrtossrnl differ- influenza. strongly of control and transmission declarations for Implications Epidemiol. networks: emergency social on COVID-19 income. to (2020). by responses entiated https://doi.org/10.1101/2020.06.29. future ing 2020. implied its July studying 2020. 1 July and 15 Accessed [Preprint] model 20142851. compartmental medRxiv stochastic trajectories. a 2020). using Press, demic Academies vaccine” National COVID-19 infection, The after of DC: allocation Washington, 2021. equitable March 29 25914, Accessed for vaccine— https://doi.org/10.17226/25914. No. framework COVID-19 preliminary (Report the of of allocation 2020. for December (2021). recommendation States, United interim updated March 29 2021. Accessed Supply uses-of-covid-19-vaccines-in-the-context-of-limited-supply). 2021. Limited of January https://www.who.int/publications/m/item/who-sage-roadmap-for-prioritizing- Context 18 2020). the [Preprint] in medRxiv cines US. the 2021. in February 2 mortality Accessed https://doi.org/10.1101/2021.01.18.21250071. on prioritization risk vaccine. 2020. coronavirus October 3 Accessed a on-the-status-of-a-coronavirus-vaccine. of https://www.npr.org/2020/09/03/909312697/operation-warp-speed-top-adviser- status 2020. the on influenza. pandemic of mitigation (2013). early the China. in outbreak https://doi.org/10.25561/82822). 2020, London, College Imperial 33, https://doi.org/ 2020. August 2020. 11 October 1 [Preprint] Accessed meRxiv 10.1101/2020.08.07.20170456. candidates. vaccine SARS-CoV-2 hne ncnatpten hp h yaiso h COVID-19 the of dynamics the shape patterns contact in Changes al., et tal., et h mat fCVD1 acn iig ubro oe,and doses, of number timing, vaccine COVID-19 of impacts The al., et muooia eoyt ASCV2asse o pt months 8 to up for assessed SARS-CoV-2 to memory Immunological al., et Mdligtealcto n mato OI-9vcie (Rep. vaccine” COVID-19 a of impact and allocation the “Modelling al., et 6516 (2013). 1655–1662 178, oeigteerypaeo h ega OI-9epi- COVID-19 Belgian the of phase early the Modeling al. , et l aaue o nomn h ueia nlssare analysis numerical the informing for used data All h dioyCmiteo muiainPractices’ Immunization on Committee Advisory The al., et Science H AERampfrPirtzn ssO OI-9Vac- COVID-19 Of Uses Prioritizing for Roadmap SAGE WHO oeadprmtrfie o oe nlsshave analysis model for files parameter and Code Science af03(2021). eabf4063 371, hsmtra sbsduo okspotdb the by supported work upon based is material This rc al cd c.U.S.A. Sci. Acad. Natl. Proc. 4118 (2020). 1481–1486 368, ob otl ky Rep. Wkly. Mortal. Morb. ∗∗ https://doi.org/10.1073/pnas.2025786118 Vrin11 ol elhOrganization, Health World 1.1, (Version h g-nyplc ipyinvolves simply policy age-only The September 3 Radio, Public National prioritization. LSCmu.Biol. Comput. PLoS 19658–19660 117, PNAS 1657–1660 69, e1002964 9, | 1o 12 of 11 m J. Am.

ECONOMIC SCIENCES APPLIED BIOLOGICAL SCIENCES 31. A. W. Bartik, Z. B. Cullen, E. L. Glaeser, M. Luca, C. T. Stanton, What jobs are 34. C. Zimmer, Moderna’s Covid vaccine: What you need to know. New York Times, being done at home during the COVID-19 crisis? Evidence from firm-level sur- 30 November 2020. https://www.nytimes.com/live/2020/moderna-covid-19-vaccine. veys. National Bureau of Economic Research Working Paper No. 27422, 2020. Accessed 25 January 2021. https://www.nber.org/papers/w27422. Accessed 15 August 2020. 35. US Food andDrug Administration, “Development and licensure of vac- 32. W. Cook, Many U.S. Workers in Critical Occupations in the Fight Against cines to prevent COVID-19: Guidance for industry” (FDA-020-D-1137, Food COVID-19 (Revised) (Labor and Market Information Institute, 2020). and Drug Administration, 2020, https://www.fda.gov/regulatory-information/ https://www.lmiontheweb.org/more-than-half-of-u-s-workers-in-critical-occupations- search-fda-guidance-documents/development-and-licensure-vaccines-prevent- in-the-fight-against-covid-19/#:∼:text=According%20to%20the%20federal% covid-19). 20standard,Workforce%E2%80%9D%20battling%20COVID%2D19. Accessed 29 36. R. Patel, I. M. Longini, Jr., M. E. Halloran, Finding optimal vaccination strategies March 2021. for pandemic influenza using genetic algorithms. J. Theor. Biol. 234, 201–212 33. M. Herper, D. Garde, Moderna to submit Covid-19 vaccine to FDA as full (2005). results show 94% efficacy. STAT, 30 November 2020. https://www.statnews.com/ 37. J. Bezanson, A. Edelman, S. Karpinski, V. B. Shah, Julia: A fresh approach to numerical 2020/11/30/moderna-covid-19-vaccine-full-results/. Accessed 22 January 2021. computing. SIAM Rev. 59, 65–98 (2017).

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