2010_Optimal Vaccine Stockpile Design for an Eradicated Disease

8/12/2019 2010_Optimal Vaccine Stockpile Design for an Eradicated Disease

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Vaccine 28 (2010) 43124327

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Vaccine

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / v a c c i n e

Optimal vaccine stockpile design for an eradicated disease: Application to polio

Radboud J. Duintjer Tebbens a,b,, Mark A. Pallansch c, James P. Alexander c, Kimberly M. Thompson b

a Delft University of Technology, Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD, Delft, Netherlandsb Kid Risk, Inc., P.O. Box 590129, Newton, MA 02459, USAc Centers for Disease Control and Prevention, National Center for Immunization and Respiratory Diseases, Division of Viral Diseases, Atlanta, GA 30333, USA

a r t i c l e i n f o

Article history:

Received 3 September 2009

Received in revised form 31 March 2010Accepted 3 April 2010Available online 27 April 2010

Keywords:

Vaccine stockpileOptimizationDynamicsPolio eradication

a b s t r a c t

Eradication of a disease promises significant health and financial benefits. Preserving those benefits,hopefully in perpetuity, requires preparing for the possibility that the causal agent could re-emerge(unintentionally or intentionally). In the caseof a vaccine-preventable disease, creation and planning fortheuse ofa vaccine stockpilebecomesa primaryconcern. Doingso requires consideration ofthe dynamicsat different levels, including the stockpile supply chain and transmission of the causal agent. This paperdevelops a mathematical framework for determining the optimal management of a vaccine stockpileover time. We apply theframeworkto thepoliovaccine stockpilefor thepost-eradication eraand presentexamples of solutions to one possible framing of the optimization problem. We use the framework todiscuss issues relevant to the development and use of the polio vaccine stockpile, including capacityconstraints, production and filling delays, risks associated with the stockpile, dynamics and uncertaintyof vaccine needs, issues of funding, location, and serotype dependent behavior, and the implications oflikelychanges over timethat might occur. This framework serves as a helpful context fordiscussions andanalyses related to the process of designing and maintaining a stockpile for an eradicated disease.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Global eradication of infectious diseases offers the promise ofsubstantial health and financial benefits due to the preventionof cases of disease and the reduction of costs for disease controland treatment aftereradication. Recognizingthe humanitarian andeconomical benefits of eradication, the international public healthcommunity successfully eradicated smallpox, and is currentlyattempting to eradicate the transmission of wild polioviruses, dra-cunculiasis(guineaworm),andrinderpest(ananimaldisease)[1,2].Eradication of otherdiseases,includingmeasles and malaria,repre-sents a continuing topic of discussion [35]. Preserving the benefitsachieved through eradication requires preparing for the possi-bility that the causal agent could re-emerge (unintentionally orintentionally). In the case of a vaccine-preventable disease, cre-ation and planning for the use of a vaccine stockpile becomes aprimary concern, particularly when achieving the goal of erad-ication leads to expectations and demands to stop vaccination.For example, following smallpox eradication, national and inter-national public health agencies prepared to respond to potentialbut unlikely reintroductions of smallpox after the cessation ofvaccination for smallpox by creating vaccine stockpiles. Milstien

Corresponding author at: Kid Risk, Inc., 2 Seaport Lane, 11th Floor, Boston, MA02210. Tel.: +1 857 383 4235; fax: +1 617 385 3295.

E-mail address:[email protected](R.J. Duintjer Tebbens).

conducted an extensive review of the smallpox vaccine stockpileand other stockpiles for non-eradicated diseases (i.e., meningitis,yellow fever, influenza, and anthrax), and offered lessons learnedrelated to the establishment, maintenance, governance, financing,regulation, implementation, and use of a post-eradication stockpile(Table 1)[6].Emergency antigen and vaccine banks also exist forfoot and mouth disease and other animal diseases[7].

Stockpiles for eradicated diseases, for which routine vaccina-tion with a vaccine used to achieve eradication will cease, differfrom those for non-eradicated diseases in two important ways.First, while the perceived risk of reintroduction of the causal agentremains small for an eradicated disease, population immunity willlikely decrease substantially after reduced agent vaccination, lead-ing to a situation in which a reintroduction can potentially spreadvery rapidly. This means that the speed of deployment of the vac-cine becomes a key requirement of the stockpile and that thepossible consequences of insufficient quantities of vaccine in thestockpile become very important. Second, the stockpile may con-tain a no longer routinely used vaccine, making a rotating stockprocurement strategy impossible, with implications for vaccinelicensing, testing, expiry, and storage. Thus,managing the stockpilerequires consideration of the dynamic interactions between thevaccine supplychainand disease transmissionin a highlyuncertainenvironment. The complexity of the issues suggests the need forquantitative analysis to inform the process of developingand main-taining a post-eradication stockpile for a vaccine that will ceaseto be used for routine vaccination following eradication. While

0264-410X/$ see front matter 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.vaccine.2010.04.001
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Table 1

Lessons learned from prior stockpiles (adapted from Milstien[6]).

Aspect of the stockpile Lessons learned

Establishment Need for more than one manufacturerNeed to keep stocks in more than one format

Maintenance Need to make provisions for potential advances inthe fieldNeed for strategies to handle intellectual propertyrights issues

Governance Need of an independent oversight committeeNeed for permanent institutional memoryprovisionsManufacturing facilities may provide the beststorage possibilities in terms of oversightImportance of well-designed and implementedquality assurance and quality control

Financing Need for advance funding for procurementFunding must cover storage, transport, qualityassurance and quality control, and other costs

Regulation Possible need for innovative regulatory approachesWHO prequalification is essentialImportance of national acceptance of the stockpile

Implementation Need for liability indemnificationNeed for criteria for trigger events and

distribution, including epidemiological andlaboratory confirmation

several mathematical models of stockpiles of antiviral drugs forinfluenza contributed to the evolving discussions of preparednessfor an influenza pandemic[810],to our knowledge no quantita-tive model exists for analyzing stockpiles for eradicated diseases.This paper formulates a mathematical framework for optimizingthe management of a stockpile for an eradicated disease. We focuson the current development of the polio vaccine stockpile for thepost-eradication era[11],and anticipate that this framework willprovide helpful context for discussions and analyses related to theprocess of its design, use, and maintenance. Building on our prior

work exploring responses to outbreaks[12,13],we use the frame-work to discuss numerous issues, including capacity constraints,production and filling delays and the expiry process, risks associ-atedwith the stockpile,dynamicsand uncertainty of vaccine needs,issues of funding, location, and serotype dependent behavior, andthe implications of likely changes over time that might occur.

The following section presents a general framework forstockpile optimization, with two distinct formulations of the opti-mization problem. The next section provides background relatedto the polio vaccine stockpile and then presents the supply chainmodel for this study. We then present a hypothetical example toillustrate some of the dynamics and present variations of the opti-mization model that highlight different issues with respect to apolio vaccine stockpile to demonstrate the value of the framework.

Finally, we discuss insights we derived from developing the frame-work and analyzing the example, and we suggest the need forcontinued work on this topic.

2. General optimization framework

While optimizing the use of scarce health resources typicallyrelies on relatively static tools such as cost-effectiveness analyses,mathematical optimization offers opportunities to further improveresourceallocationdecisions[1417]. Inthecaseofavaccinestock-pile,optimization requires an understanding of the dynamics of thesupply chain and the nature of the demand[1820].In this sec-tion, we formulate optimization problems for a simplified, abstract

stockpile to illustrate the general framework.

We consider the simplified stock-and-flow diagram for thestockpile depicted inFig. 1.A stock-and-flow diagram consists ofstocks(shownasboxes)whoselevelscanchangeovertime,inflowsand outflows (shown as arrows with valves) that control the levelsin the stocks, and intermediate variables (shown as text connectedwith arrows) that may contain constants or calculations and influ-ence the flows[21].The arrows show direct influences betweenstocks, flows, and intermediate variables. In text discussing thestock-and-flow figures, we refer to stocks using bold text andflowsor other variables using italics. The boxstockpileinFig. 1repre-sents thequantityof vaccine readily availablefor use. Thestockpilesize gets drawn down as a result ofexpiry and lossdue to a finiteshelf-lifeorwastageinstorage,or deploymentinresponsetovaccinedemand. Vaccine demand depends on how quickly we can vacci-nate people (captured in thedistribution constraints). The amountof vaccine in thestockpileand the time needed to deploy vaccinefromthe stockpile,capturedinthedeployment constraints,togetherdeterminethemaximum deploymentrate (e.g.,themaximumdeploy-ment ratebecomes 0 if the amount of vaccine from thestockpileis 0). Thestockpilesize may increase as a result of a nonnegativeorder rate, but due to the production process, newly ordered vac-cine first accumulates in the stockvaccine in productionthroughtheproduction startsflow before arriving in thestockpilethrough

the production flow. The stock vaccine in production in realityincludes many intermediate stocks representing different stages inthe production pipeline, including vaccine orders waiting to enterthe production pipeline, production of bulk, storage of bulk, vac-cine being filled, and vaccine waiting to be tested. For simplicity,we represent them here all in one stock,vaccine in production,and show the box in bold face to indicate that this stock in real-ity consists of multiple stages. We capture the delays and capacityconstraint of the production process inproduction constraints. Theproduction constraints andamount ofvaccine in production deter-mine themaximum production rate,and loss mayoccurasaresultofwastage during the production process. Managers place orders at acertain order rate following theirordering strategy, although a delaymay exist between setting the strategy and executing it, as shown

by the delay mark (double line) in the arrow fromordering strat-egyto order rate. Theordering strategyat any given timetmay besubject tofinancial constraints anduseinformationaboutthe stock-pile size at time t(both shown by the dotted arrows) and otherinformation about the current state of the stockpile (arrows notshown).Productionleads tovaccine costsandunmet vaccine needslead to sub-optimal outbreak response and thus to public healthcosts. Wheneverdeploymentcannot meet thevaccine demanddueto insufficient vaccine available from thestockpile, this leads tounmet vaccine needs.

Optimizing the vaccine stockpile involves balancing the trade-off between vaccine costs andpublic health costs. We formulate twodifferent framings of the optimization problem.

Framing 1: Minimize thepresent value of total costs over allfeasible

ordering strategies,assuming no financial constraints:

Minimize C=

t0

(cp(t)+ cv(t))ertdt

s.t. o(t) 0

whereCis the net present value of the total costs,cpandcvare theorder ratedependent annualpublic health costsand annualvaccinecosts, respectively,ris the discount rate, ando is theorder rate. Itfollows from the constraint that anyordering strategythat leads tononnegative order rate is feasible. If theorder rate exceeds the max-imum production rate, excess orders will simply remain in the stockvaccine in productionas backlog and theproduction constraintswill govern the rate at which these orders ultimately flow to the

stockpile.


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Fig. 1. Stylized general stockpile model.

Framing 2:Minimize the present value of public health costs overall feasible ordering strategies subject to the financial constraints:

Minimize Cp =

t0

cp(t)ertdt

s.t. o(t) 0

0

t0

cv(t)ert dt F

where Cp is the net present value of the public health costs, Frepresents thefinancial constraintsas a total amount of availablefunds (alternatively, the financial constraint could consist of vari-able maximum funds available over different time periods), andall other symbols are as in framing 1. Under framing 2 the avail-able funds determine the lowest possible health costs, while underframing 1 the potential public health costs determine the amountof funds available for the stockpile.

3. Vaccine stockpile model: disease-specific context for

polio

Before formulating the polio vaccine stockpile model and pre-

senting a hypothetical example, the following subsection providesbackground on polio eradication and stockpiles.

3.1. Background

3.1.1. Global polio eradication

The Global Polio Eradication Initiative (GPEI) anticipates suc-cessful interruption of wild poliovirus transmission in the nextfew years, given sufficient financial resources, at which time itwill begin implementation of post-eradication risk managementstrategies[11].The benefits of the successful polio immunizationprogram in the United States include preventing large numbersof cases of paralytic polio and overwhelming economic savingsdue to avoided treatment costs [22]. Similarly, finishing polio

eradication globally promises large long-term net benefits [23]as long as the GPEI can successfully contain the post-eradicationoutbreaks that likely will occur due to the possibility of circu-lating vaccine-derived polioviruses (cVDPVs)[24].Other potentialoutbreak risksinclude,unintentional or intentional poliovirusrein-troductions, VDPVs from immunodeficient long-term excretors,and any polioviruses circulating after apparent interruption oftransmission or remaining in the environment[24]. The WorldHealth Organization (WHO) currently plans for the cessation ofroutine use of the live, oral poliovirus vaccine (OPV) several yearsafter wild poliovirus eradication to minimize the risk of generatingnew VDPVs[25].Following global eradication, national and globalpolicy makers will face numerous choices[26,27],including cre-ation and maintenance of one or more vaccine stockpiles. All of

these choices, including whether to use IPV for routine immuniza-

tion, and the surveillance and response policies, influence the risksand epidemiology of outbreaks and hence the size of the optimalstockpile.

3.1.2. US polio vaccine stockpile

The current vaccine tools to respond to polio outbreaks includeOPV, with trivalent (tOPV), bivalent (bOPV, types 1 and 3), and

monovalent (mOPV) OPV formulations currently available, inac-tivated poliovirus vaccine (IPV), a combination of these, or doingnothing. Jenkins and Modlin performed a decision analysis to eval-uate these options in the context of a potential future outbreakin the US and they identified mOPV as the vaccine of choice ifa supply exists, although they recognized that IPV is currentlythe only licensed polio vaccine in the US and carries no risk ofcausing vaccine-associated paralytic polio (VAPP) or cVDPVs[28].In February 2004, a joint National Vaccine Advisory Committeeand Advisory Committee on Immunization Practices work groupconcluded that an 8 million dose stock of IPV represents a nec-essary component of a US stockpile for polio, and that this sizewould be adequate in the event of a short-term disruption of theroutine IPV supply or to control an outbreak given current and

anticipated continuedhigh population immunity[29,30]. However,questions remain about the efficacy of IPV in outbreak response,and as a result the committee also stated that prudent prepared-ness requires access to 8 million doses of tOPV or 8 million dosesof mOPV of each serotype. It recommended that the US work withthe WHO and other international partners to help finance, create,and maintain a global poliovirusvaccine stockpilethatprovidestheUS with immediateand guaranteed access. [29, p. 1110]Given thedesire to contain or avoid the spread of viruses derived from OPVused in outbreak response, a report from the international Com-mittee on Development of a Polio Antiviral and Its Potential Rolein Global Poliomyelitis Eradication recommended rapid develop-ment of at least one and preferably two polio antiviral compoundsas an additional tool for outbreak response[31].The US now haspartiallyfilledthe recommended stockpileof 8 million doses of IPV,andthe current plan foran international stockpileincludesensureduniversal access to mOPV[32].The size and composition of the USstockpile, however, might need to change over time, as the risks ofpolio outbreaks and the availability of IPV and OPV change.

3.1.3. Global polio vaccine stockpile

WHO plans to establish a global polio vaccine stockpile as aprerequisite for OPV cessation [33,34]. Currently, developmentof the global polio vaccine stockpile centers on the requirementthat the stockpile contain sufficient doses either to re-interruptpoliovirus transmission following any post-eradication outbreaks,or if unableto stop transmission, to vaccinate until OPV productionrestarts assuming insufficient IPV production capacity for univer-

sal use[32].Based on analysis and a review of prior stockpiles,


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Fig. 2. High-levelstock-and-flow representationof the poliovaccine and antiviral stockpile.Acronyms: AV1, AV2: two different(trivalent) antiviral products; IPV: (trivalent)

inactivated poliovirus vaccine; mOPV1, mOPV2, mOPV3: monovalent oral poliovirus vaccine types 1, 2, and 3, respectively; tOPV: trivalent oral poliovirus vaccine.

recommendations in 2004 for a polio vaccine stockpile includedestablishment of an independent oversight commission, interna-tional financing, periodic assessment of costs, ensured access forall countries without regulatory hurdles or import/export barriers,andliabilityprotection[6]. CurrentWHOstandardoperatingproce-dures (SOPs) call for a stockpile of 750 million doses of each mOPVserotype, including 250 million doses of each serotype stored asfinished product at the time of OPV cessation and a minimum bal-ance of 100 million at any time, with the remaining mOPV storedas bulk [32]. The SOPs further state that UNICEF would main-tain ownership of the international stockpile to ensure universalaccess and coherent use for response to an outbreak. The SOPs

also established release criteria involving timely assessment of thetrigger event by the independent oversight commission andsubse-quent recommendations regarding release of vaccine by the WHODirector-General.

The process of filling OPV vaccine involves handling livepolioviruses. In an increasingly susceptible world after OPV ces-sation, reintroduction of any live poliovirus presents an importantrisk to maintaining a polio-free world[35].To minimize this risk,strict biocontainment requirements and the requirement of veryhigh population immunity in the area surrounding the productionfacility couldsubstantiallyincrease the productionand filling costs.In this context, the SOPs recommend physical storage and mainte-nance of the stockpile at two or more distinct vaccine productionfacilities that comply with these requirements to reduce the risk of

an inadvertent poliovirus reintroduction[36].Consequently, somecurrent conceptions of the stockpile envision all procurement tooccurpriortoOPVcessationwhilecoveringtheanticipatedneedstorespond to likely cVDPV outbreaks in the time period immediatelyfollowing OPV cessation.

While current efforts represent significant progress, manyissuesrelatedto this stockpileremain open, andoptimization of thestockpile design offers the potential to reduce costs while ensur-ing preparedness. Developed countries continue their efforts toevaluate the adequacy and accessibility of the global stockpile fordomestic needs. Presumably, WHO might also consider the exis-tence of national vaccine stockpiles in evaluating development ofthe global vaccine stockpile. We previously concluded that inter-national cooperation represents a key requirement to optimize theglobal use of resources for polio vaccine stockpiles[37].

3.2. Polio vaccine stockpile supply chain

Fig. 2visualizes the different stocks and flows in a polio vac-cine stockpile at the highest level. The stocks of mOPV bulk (anytype) and IPV bulk increase through orders resulting in productionand decrease either through filling or loss (wastage). We assumethat the shelf-life of the vaccine bulk product is infinite for all prac-tical purposes[32].Filling is the only flow into mOPV final (anytype) or IPV final, while the outflows from final vaccine includedeployment, loss, and expiry. In a post-OPV cessation context, OPVdeployment would only resultin theevent of outbreaks, butfor IPVsome deployment may also result from routine usage in the event

of routine vaccine shortage. As shown, tOPV does not exist in bulkform, but gets filled from the three mOPV bulk types. In addition,Fig. 2shows two stocks of antiviral drugs, which may become apossibility in the future[31].

Fig. 3shows the stock-and-flow diagram for optimizing fram-ing 1, focusing on a single mOPV type. We focus on framing 1 forthe example below and refer to Appendix A.3 for a similar diagramreflecting framing 2, and expand on the general model shown inFig.1 to explicitlydistinguish bulkand finalvaccine components. InFig.3, boththevaccine in bulk production pipeline andvaccine infilling pipeline stocks use bold faced boxedges to indicate the pos-sible subdivision into multiple stocks reflecting different stages inthe production process. The number of subdivided stages dependson the actual properties of the bulk production and filling pro-

cesses, including the number and location of production and fillinglines and procedures for virus growth and testing. These proper-ties determine the appropriate way to model the delay, rangingfrom a simple first-order delay (reflecting no intermediate stagesand multiple parallel filling and production lines) and yielding thelargest variance around the mean of the eventual outflow from theprocess, to a fixed (pipeline) delay yielding no variance around themean[21]. In addition, capacity constraints in any of the stages inthe pipeline can increase the actual duration of the process.Fig. 3also explicitly shows the accumulation of vaccine costs from bulkproduction, filling, and maintenance of the stockpile, the accumu-lation ofpublic health costsdue to excess cases and political costsassociated with unmet vaccine needs,andalltheinputsdeterminingthe flows (e.g., the different wastage rates that determine loss out-flows). Theordering strategyhere determines bothbulk production


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Fig. 3. Stock-and-flow diagram for the minimization of the total costs with no financial constraints (framing 1) for a single serotype stockpile.

starts andfilling starts andmay useinformationabout current levelsofvaccine bulkandvaccine finalas well as the predictableexpiryofvaccine final.Finally, Fig.3explicitlyshowstheinteractionofthestockpilesupplychainwithanoutbreaksub-modelthatdeterminesthe vaccine needs and deploymentresulting from virus introductionsas well as the excess paralytic casesin the event ofunmet vaccineneeds. Consistent with the general framings, optimization fram-ing 1 for this model entails minimizing the total costs (i.e., sumofcumulative vaccine costsandcumulative public health costs)over all feasibleordering strategies.

Trade-offs inthe model centeraround thecosts of producingandfilling bulk vaccine doses vs. the costs associated with excess par-alytic polio cases that might occur in the event of an outbreak due

to a delayed or reduced outbreak response[12].This includes boththe medical andsocietalcosts of excess cases andthe politicalcostsof stockpile failure (i.e., of facing unmet vaccine needs during anoutbreak). The number of excess paralytic cases must also includecases of VAPP resulting from OPV use, which may increase as thefraction of susceptibles in the population increases and protectiondue to maternal antibodies declines[24].

4. Hypothetical example for polio

This section presents an example of solutions for framing 1based on hypothetical model inputs to demonstrate the utility ofthe framework. The example focuses on the primary objective of

the global polio vaccine stockpile to respond to outbreaks duringthe time immediately after OPV cessation.Appendix Aincludesexamples for framing 2 and some variations on the assumptions.The example illustrates the key dynamics and raises a number ofimportantconsiderationsin thedesign of an optimal vaccine stock-pile for polio, although future analyses should address the specificchoices made as the process for developing the poliovirus vaccinestockpile evolves. We use an outbreak sub-model that generatesvaccineneedsandexcessparalyticcasesassociatedwithanyunmetvaccine needs implemented using MathematicaTM (seeAppendixA.13for details of the outbreak sub-model and derivations of thesolutions to the different framings of the optimization problemusing linear programming methods). For simplicity, the exampleassumes a deterministic stream of vaccine needs that result from a

deterministic number of virus introductions over time. For ease of

presentation, we further assume no wastage or maintenance costs,because these only scale the total costs but have essentially noimpact on the optimization algorithm.Table 2provides the hypo-thetical values of constants used in the example for the stockpilesupply chain model. While we assume first-order delay processesin the outbreak sub-model, we assume fixed delaysin the stockpilesupply chain. We further consider first-order delays inAppendixA.2 to explore the possible spectrum of delay types in the stockpile.

4.1. Illustration of outbreak sub-model dynamics

The outbreak sub-model simulates the diffusion of poliovirus

following an initial introduction through a finite number of popu-lation blocks. Once infected, population blocks eventually controlthe outbreak by increasing the level of population immunity. Thisoccurs due to either natural burnout, which leads to many para-lytic polio cases, or due to outbreak response immunization, whichcontrols the outbreak more rapidly and requires vaccine from thestockpile. Given that population blocks interact dynamically, theamount of vaccine available from the stockpile itself determinesthe future vaccine needs. For example, with insufficient vaccinefrom the stockpile, an outbreak may spread to new populationblocks and consequently generate additional vaccine needs. Thus,the outbreak sub-model requires input from the stockpile supplychain model (i.e., vaccine final) and at the same time generatesoutput that feeds into the stockpile supply chain model (i.e., distri-

bution as a result of vaccine demand, vaccine needs, and excessparalytic cases), as shown inFig. 3. Virus introductions in thissimple model consist of exogenously generated increments in thenumbers of infected populations. To illustrate the dynamics in thesub-model,Fig. 4shows the vaccine needs arising from an expo-nentially decreasing number of virus introductions starting in year3 with a half-life of 2 years and totaling 10 introductions as a sim-plified scenario roughly consistent with the expected decrease ininitiating cVDPV events[24,27,38].Year 3 represents the point intime when outbreak response activities would begin using vaccinefrom the stockpile (i.e., OPV cessation), while bulk production andfilling may already occur before year 3. Fig. 4shows the vaccinedemand over time, either with no stockpile or response, or withsufficient vaccine in the stockpile to cover all vaccine needs. With

sufficient vaccine, the response leads to rapid control of all the out-


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Table 2

Stockpile supply chain model constants used for the demonstrative example (including initial values for vaccine stocks) a.

Symbol in equations Name inFig. 3 Value

Vpb

Vaccine in bulk p roducti on pipeline 0 doses (initial va lue)Vb Vaccine bulk 0 doses (initial value)Vp

f Vaccine in filling pipeline 0 doses (initial value)

Vf Vaccine final 0 doses (initial value)tb Minimum bulk production time 18 monthsbc Bulk production capacity 100 million doses/month

tf Minimum filling time 3 monthsfc Filling capacity 10 million doses/monthts Shelf-life 60 monthsw1,w2,w3,w4 Wastage rates 0 1/monthr Discount rate 0.0025 1/month (=3% per year)cb Bulk production costs per dose 0.2$/dosecf Filling cost per dose 0.1$/doseccase Cost associated with excess cases 1000$/casecpol Political costs of failure 100 million $/month with unmet vaccine needscm Maintenance cost rate 0$/month

a While we emphasize the hypothetical nature of the examples and choices of input values, most values represent informed choices based on prior work[27].

breaks. With no stockpile, virus exportations to other populationsoccur over time and outbreaks continue to increase until no moreat-risk populations remain (i.e., populations with sufficient sus-ceptibles to sustain outbreaks); correspondingly, vaccine demandrises until it eventually plateaus. Thus, if we stop routine vaccina-tion and do nothing to respond to outbreaks, then any small risk ofoutbreaks results in an eventual return to uncontrolled polioviruscirculation.

4.2. Illustration of optimal solution for framing 1

Appendix A.2describes how we obtained an optimal solutionfor framing 1 using standard linear programming methods. Thesolution relies on the plausible assumption that the penalty (i.e.,costs) associated with each dose of unmet vaccine needs exceedsthecostsofstockpilingadose.Ifthisassumptiondoesnothold,thenno economic justification exists for the stockpile, because the costsof the consequences do not exceed the stockpile costs. In addition,

we ignored expiry by assuming that we use vaccines for outbreakresponse soon after they enter the stock of final vaccine, which ispossible if vaccine demand is deterministic and filling capacity issufficiently high.

Fig. 5shows the optimal solution assuming fixed delays for thebulk production, filling, and expiry processes. Due to discountingin the objective function, the optimal solution delivers vaccine aslate as possible to incur costs as late as possible. This means thatit does not start accumulating vaccine final until year 2, althoughit must start filling before the actual vaccine needs arise in year 3

Fig. 4. Forecast of the vaccine demand assuming an exponential decay in virus

introductions, starting in year 3.

because of the filling capacity constraint and delay. Similarly, theoptimal solution requires that we build the stock of vaccine bulkas late as we can and only up to the level required to be able tofill at maximum capacity when needed. Once we start bringingthe outbreak under control, the vaccine needs decrease (Fig. 4),deployment of vaccine final diminishes, and as a result filling candecrease andboth vaccine stocks deplete to (almost)0. The amountof vaccine bulk needed depends on the time period over which weplan orders (i.e., the planning period). In the example, the planningperiod equals half a month, so the stock of vaccine bulk at any timemust hold sufficient vaccine to cover the total filling needed for thenext half month (e.g., 0.5 months10 million doses per month= 5million doses when filling occurs at maximum capacity in year 3(Table 2)). Thus, the choice of planning periodimpacts the requiredlevel of vaccine bulk, which in the case of deterministic demandwith fixed delays becomes zero as the planning period approaches0. The (discounted) cumulative costs reflect the same two wavesof production and filling activity and in this example amount to

approximately $55million over 23 years based on the hypotheticalmodel inputs fromTable 2.

5. Variations of the optimization models

The example in the prior section represents only one of manypossible variations of the optimization models. These variationsmay reflect different practical realities and we discuss these

Fig. 5. Vaccine stocksand cumulative costs (discounted)for theoptimal solutionofframing 1 assuming fixed delays in the stockpile supply chain and an exponential

decay in virus introductions, starting in year 3.


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Fig. 6. Optimal solution assuming a sudden and permanent doubling of the bulk production and/or filling costs in year 3: (a) only bulk production cost doubled; (b) bothbulk production and filling costs doubled; (c) both bulk production and filling c osts doubled and filling capacity constraint relaxed.

qualitatively here to expose importantissues thatneed explicit dis-cussion in the process of designing thepoliovirus vaccine stockpile.

5.1. Risks and costs of vaccine production after OPV cessation

As mentioned, the risks associated with OPV production afterOPV cessation might effectively imply an increase in vaccineproduction costs due to constraints on the physical location of pro-duction facilities as well as containment requirements. To explorethis possibility,Fig. 6shows the effect of a sudden and permanentdoubling of the production costs per dose in year 3. InFig. 6a, weonly doubled the bulk production costs, presumably because bulkproduction requires the most stringent and expensive biosafetyprecautions. As a result of the cost increase, in the optimal solutionall bulk accumulation occurs before year 3, while the vaccine finallevelsremain unchanged compared toFig. 5. In Fig. 6b, we exploredthe effect of simultaneously doubling the filling costs from year 3forward.This results in an optimal solution that acquires allneededvaccine final before year 3, which due to the capacity constraint

involves earlier accumulation of vaccine final than inFig. 6a.Thisin turn requires production of most vaccine bulk early on, with onelast order of vaccine bulk occurring just before year 3. If we relaxthe filling constraint, then the optimal solution orders as much aspossible of both bulk and final vaccine just before year 3 (Fig. 6c).

While the practical effect of the risk associated with producingpoliovirus after OPV cessation is a sudden increase in productioncosts at OPV cessation, in reality there exists a trade-off betweenthe costs of containment measures and the risk associated withproduction after OPV cessation. An extension of the model mightconsider whether the expected excess cases resulting from therisk of releases from vaccine production facilities justify the costsof containment. Such an extension would explicitly consider geo-graphical locations of the stockpile to link virus introductions to

production and would need to consider immunity level in the pop-

ulation surrounding the production facility as it affects the risk ofnew virus introductions and excess cases.

5.2. Expiry of filled vaccine

The assumption of a fixed delay process for expiry, as in theexample, is appropriate if disposal of expired vaccine lots occurson the basis of expiry dates set at the time of filling, which iscurrent practice. Alternatively, in the future randomtestsmay pos-sibly determine the disposal of lots or batches. In that case, a moreappropriate model would disaggregate the stock by age, with thehazard of expiry increasing with age. In the example, we neglectedexpiry by assuming that we could receive newly filled vaccine onlya short time before its required use, but this assumption does nothold if filling capacity is low relative to vaccine demand, filling andproduction costs increase over time, or vaccine demand is highlystochastic. As an extreme departureaway from the assumption of afixed delay in expiry,Appendix A.2.1considers a first-order expiryprocess and shows that this drives up the costs, because a fixed

fraction of filled vaccine expires before deployment. The expiryof vaccine occurs primarily due to the loss of efficacy resultingfrom a decrease in vaccine virus titers over time, and consequentlyone approach to increasing the vaccine shelf-life might includebeginning with higher initial titers. Modeling this option wouldnecessitate including a decision about the desired titer for fillingin the diagram along with the associated costs, and possibly someinfluence of the titers on the risks associated with the vaccine.

5.3. Nature of production and filling delays

Like the nature of the expiry process, the technical character-istics of the bulk production and filling processes also determinethe appropriate model for the filling and bulk production delays

(see Appendix A.4. From theperspective of thestockpile owner, the


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production delay might look like a fixed delay, with no distributionaround the time that doses arrive due to a single time of deliv-ery. From the manufacturers perspective, however, a given orderof bulk vaccine might potentially get produced over time, withproduction of some doses completing before others, which wouldsuggest a first- or low-orderdelay. Filling involves a relatively fixeddelay associated with testing batches of final product, but if manyfilling lines exist, each line might become available at a differentpoint in time such that the delay could look more like a low-orderdelay. Given that the exact time that vaccine becomes availableplays an important role in the event of potential shortage of vac-cine, thechoiceof thedelayin themodelshould reflectthe physicalreality of the process and the relevant perspective of the analy-sis (i.e., stockpile user vs. manufacturer). We mention it explicitlyhere since different stakeholders involved in discussions about thestockpile design might implicitly assume different types of delays,and we anticipate that explicit discussions might prove helpful.

5.4. Production and filling capacity over time

Depending on the implementation of OPV cessation, a warmbase of OPV production may or may not continue to exist (e.g.,continuedroutine use of IPVbasedon Sabin OPVseed strains might

offer a warm base)[39].Our model did not explicitly include thecost of maintaining or increasing this capacity, but at the very leastmaintaining capacity to produce and fill a vaccine no longer rou-tinely used presents economic challenges from the manufacturersperspective. To model the manufacturers perspective, the costsof maintaining capacity would be linked to the expected demand.Fromthe stockpileowners perspective, things couldlook different.While all bulk production may occur before OPV cessation (givenpractical unlimited shelf-life of the bulk product), the only way tomaintain a long-term OPV stockpile, if so desired, involves fillingat multiple points after OPV cessation, unless the shelf-life of filledvaccine increases dramatically. Given the substantial societal costof excess cases, the stockpile owners incentives to maintain somefilling capacity remains much stronger than those of the manu-

facturer. The difference in incentives calls for continuing dialoguebetween owners and manufacturers and contractual mechanismsto better align the incentives.

5.5. Forecasting vaccine demand

The forecast of vaccine demand represents the key driver ofdecisions regarding the stockpile. Prior work suggests that thelikelihood of outbreaks remains greatest immediately after OPVcessation [24], but small risks exist of potentially much largeroutbreaks in the long-term [12,27]. Combined with changes incosts and risks of filling and production, changing vaccine demandimplies that the optimal allocation policy will also change overtime (seeAppendix A.5.Thus, the optimization problem becomes

stochasticand dynamic.Moreover, the vaccine demand depends onboth the stochastic risks and the stock of vaccine final, and in twodifferent ways. First, if the vaccine demand exceeds the maximumoutputof filled vaccine, then this will likelycreate newdemand dueto the natural expansion of the ongoing outbreak. This leads to apositive (reinforcing) feedback loop around vaccine demand, sincethe likelihood of unmet vaccine needs lead to more demand whichleads to greater likelihood of unmet vaccine needs. Second, the useof OPV to respond to outbreaks carries a risk of generating newcVDPV outbreaks, leading to another positive feedback loopinvolv-ing vaccine demand, since more vaccine demand leads to moredistribution, which leads to a higher probability of virus introduc-tions, creating new vaccine demand. The strength of this feedbackincreases over time as global population immunity decreases and

depends on the specifics of the outbreak response. This remains a

topic of further research, including more mathematical modelingof poliovirus outbreak spread.

5.6. Excess paralytic cases

Realistically estimating the excess paralytic cases associatedwith unmet vaccine needs requires a more detailed dynamic out-break and response model[12,13].Such a model requires specificassumptions about the response strategy, including the time fromdetection until the first mass immunization response, target pop-ulations and coverage of the immunization rounds, and numberof immunization rounds. The model must include this informationboth for the response strategy when sufficient vaccine is availableand for the response strategy in the event of unmet vaccine needs.Thus, discussions of the different response strategies in differentoutbreak scenarios remain an important factor that will determinethe ultimate quality of the stockpile optimization model.

5.7. Funding stream

Framing 2 minimizes the cumulative public health costs overpossible streams of the use of funds over time (see example inAppendix A.3).Due to the risks mentioned above associated with

production and filling after OPV cessation, the costs of filling andprocurement will likely increase over time. Furthermore, a possi-ble decreasing number of competitors in the market might alsoreduce supply and increase prices. While this makes upfront fund-ing appear attractive, the finite shelf-life of finished vaccine meansthat filling must occur to facilitate access to vaccine as needed overtime, which necessitates some funds to replenish the stock of finalproduct at a later stage depending on the long-term demand fore-cast. Under framing 2, the decision maker can flexibly decide whento spend funds over time, but the funding stream may not in factcome with such flexibility. For example, the situation may arise inwhich the donors of stockpile funds do not provide 100% fundingupfront but distribute funds over time. At the other extreme, set-ting aside funds from donors for an extended period of time may

not represent a feasible option. This situation would require use ofa minimum useof funds variableinthemodelthatfurtherconstrainstheutilizationof funds over time. Moreover,optimizing theutiliza-tion of some initial funds in practice involves working with a finitetime horizon, which means that thestockpile only covers this finitetime period. In reality, provisions are needed beyond any practicaltime horizon for as long as a risk of virus introductions exists.

Various mechanisms in the model might influence the avail-ability of new funds and would involve new feedback loops in themodel.Forexample,unusuallyhighvaccineneedsduetotheoccur-rence of outbreaks might deplete the stocks of vaccine to such lowlevels that replenishment of the stockpile becomes desirable. Therelationship could be a threshold relationship (i.e., if bulk vaccinedecreases below a certain level we can raise funds for new vac-

cine) or more continuous (i.e., the lower the stock of bulk the morepressure on donors to supply new funds). Alternatively, new fundsmight not actually become available until excess paralytic casesactually occur.A reluctance to use OPVin thelong term might favortheuseofIPVforoutbreakresponse.Thiswouldimpactthedemandfor IPV vs. OPV over time and the availability of new funds.

5.8. Country-dependence

The formulation of the optimization model does not explic-itly include the physical location of the stockpile as an additionaldimension of the problem. However, the trade-off between costsandrisksof thestockpile maydepend on thelocation(s)of thefacil-ities. For example, manufacturing facilities in countries of higher

income will most likely offer better secondary safeguards, such as


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high IPV coverage[36],but typically produce vaccine at a highercost than lower income countries[40].The current SOPs place agreater weight on safety than on the possibility of less expensivevaccine by requiring vaccine storage at manufacturing facilities inlocations that comply with the secondary safeguards[32].Inputsthat depend on the location of the outbreak include the forecastof vaccine needs (since risks vary by country [24]), the excesscases associated with unmet vaccine needs (since outbreaks indifferent countries may vary widely in magnitude[13]),and thevaluation of future outcomes[27].Given the shared global objec-tive of the stockpile[37],one approach might first disaggregatethe models by income level to estimate outcomes for each incomegroup and then sum over the income groups to get the totals[13,24,27].

5.9. Serotype dependence

Different poliovirusserotypes varywithrespect to vaccine sero-conversion, transmission characteristics, and rate of paralysis perinfection[41,42].This means that the vaccine needs and excessparalytic cases depend on serotype, and thus that the optimal con-tent of mOPV stocks may differ by serotype. Under framing 1, the

lack of a stockpile cost constraint means that the optimizationproblems for each serotype remain essentially independent. Theoptimal quantity of mOPV over time may vary by serotype, but theamount of vaccine of one serotype does not directly impact theresources available for the other serotypes. Under framing 2, theexplicit stockpile cost constraintimplies that the problems becomeinterdependentandoptimalratiosoftypes1,2,and3willexistovertime.

IPV protects against all 3 serotypes, butwith different effective-ness[42].One approach to optimize an IPV stockpile would basetheforecastoftheneedsandexcessparalyticcasesontypicalaver-age serotype behavior [13], or alternatively index by serotype andaggregatethe total publichealth andstockpile cost of each serotypeas thebasisfor theobjective functions. Although tOPV also protectsagainst all 3 serotypes, it appears an unlikely candidate for a globalpoliovaccine stockpilebecause mOPVyields higherseroconversionthan tOPV[13]and using tOPV to respond against an outbreak ofone serotype would unnecessarily reintroduce two otherserotypesof live poliovirus. Thus, the full optimization problem most likelyinvolves a combined stockpile of mOPV (all three types) and IPVvaccine. For example, with mOPV as the main tool for rapid out-break response, especially immediately after OPV cessation, IPVmight play a role in responding at the edges of the mOPV targetpopulation to stop transmission of OPV-derived viruses, perhapsusing a form of ring vaccination[43].Projected future routine useofIPVwillalsoaffectoutbreakresponsedemandsandoptimization.In this case, what might seem like a relatively expensive use of IPVcould potentially serve to mitigate possible future increases in theneeds for more mOPV and IPV, such that the optimization prob-lem will need to determine the optimal balance not only of eachserotype of mOPV, but also of mOPV and IPV. Similarly, if antiviralsbecome available, the models must consider their potential rolesas well.

6. Discussion

We present a framework for optimizing the supply chain of apolio vaccine stockpile aimed at facilitating the response to poliooutbreaks after polio eradication and OPV cessation. This frame-work helps create a common platform for discussions among thevarious stakeholders and decision makers who must ultimatelydesign and implement the stockpile. The risks associated with vac-

cine production after OPV cessation and delays in the production

and filling processes necessitate creation and management of thestockpile in advance of OPVcessation. We demonstrate with a sim-ple example how optimization may lead to useful results in termsof the ordering strategy that minimizes the present value of publichealth and vaccine costs, although we emphasize that these hypo-thetical results depend on simplifying assumptions in the stockpileand outbreak model.

We emphasize the need to address various issues in order tofully optimize the stockpile in the context of all its complexities.First, the technical details of the stockpile, such as capacity con-straints and delays in the production and expiry processes, impactthedynamics within thesupply chain andrequire careful consider-ation. Second, the relationships between vaccine production risksandvaccine demandas well as between vaccine demandand finan-cial constraints lead to additional feedback loops that merit furtherexploration. Third, the perspective (e.g., stockpile owner vs. man-ufacturers, short term vs. long term) impacts the objectives andtherefore the optimal policy for a stockpile. Explicit understand-ing of the perspectives will help discussions. Fourth, the vaccinedemand is inherently stochastic, which implies some probabilityof unmet vaccine needs even for a very large stockpile. The extentto which the stockpile must cover all possible scenarios dependson the true costs of excess cases and the political costs of failure.

Fifth, the serotype dependence and availability of multiple prod-ucts (mOPVs, IPV, antivirals) add another layer of complexity, asdoes the geographical dimension. Finally, the use of OPV after OPVcessationcarries itsown risk, leading to another feedback loop backto the demand. The specifics of the outbreak response remain criti-cal to the actual use and benefits of the stockpile. Further modelingin this area provides an opportunity to better anticipate this riskand determine policies that maximize the probability of successfulcontrol and minimize the probability of generating new outbreakswith the response vaccine. We propose a flexible framework thatcan incorporate all of these complexities, although optimization ofthe full stochastic and dynamic problem will most likely requireheuristics based on simulation rather than relatively straightfor-ward linear programming (see Appendix A.6). Careful examination

and discussion of the assumptions and their implications mustoccur before actual optimization of the full stockpile problem.

We expect that political factors might also play a role in deci-sions about financial investment in a global stockpile. For example,although we discussed how the location of the stockpile(s) andserotype-variability present additional levels of complexity, we didnot explore the political issues that arise in the context of nationalorotherpreferencesforvaccinesuppliers.Logisticsrelatedtodistri-bution of the stockpile resources and campaign operational issuesalso remain an important issue.We implicitlyassumed operationalreadiness and global access to the stockpile, but such access mayrequire negotiation, which we emphasize, must occur early in theprocess of creating and designing the stockpile. In this regard, wenote that the current plans for an international stockpile strive to

ensure that vaccine can be available, if necessary, to any countryin the world within 48 hours[32, p. 2].While this paper focuses on a polio vaccine stockpile, this

approach might prove useful for the design of future stockpilesfor eradicated diseases and potentially emerging pathogens. Forexample, in the case of pandemic flu, the stockpile might containantivirals and a prototype pandemic flu vaccine to provide first-response interventions during the time needed to develop a moreeffective vaccine targeting the newly emerged pandemic strain.Explicit consideration of thesupply chainand transmission dynam-ics might also provide helpful input to stockpile decisions in thiscontext. We anticipate that providing a coherent frameworkrepre-sentsa significantcontribution thatwill facilitate discussionsaboutmany of the assumptions and numerical estimates for key inputs

required to perform optimization.


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Fig. A1. Stock-and-flow diagram of the simplified outbreak sub-model.

Acknowledgments

Drs. Duintjer Tebbens and Thompson acknowledge support fortheir work from CDC Grant U01 IP000029 and under NVPO N37(FY2005). The findings and conclusions in this article are those ofthe authors and do not necessarily represent the views of the Cen-tersforDiseaseControlandPrevention.TheauthorsthankMargaretWatkins,SteveWassilak,andSteveCochiforhelpfulcomments.Wealso thank two anonymous reviewers for excellent comments thatsignificantly improved the manuscript.

Appendix A.

This appendix presents technical details about the outbreaksub-model and stockpile supply chain model, derives linear pro-gramming formulations of the optimization problems wherepossible, presents framing 2 of the optimization problem, providessome additional results, and discusses solution strategies for sit-uations in which a standard linear programming approach is notpossible.

A.1. The outbreak sub-model

Fig. A1shows the structure of the outbreak sub-model, whichcaptures the key dynamics at a highly aggregate level but doesnot account for many of the more detailed level dynamics thatmust be considered for a more realistic and data-based analysis[13].Table A1provides the symbols and hypothetical values of theconstants we used for the demonstrative examples.

The input to the sub-model is a given number of virus introduc-tions over time and the main output is the incidence of paralyticpolio cases. The sub-model depends on the availability of vac-cine final in the stockpile supply chain model, but also determinesthe deployment of vaccine final in response to vaccine demand,which in turn depends on the diffusion of outbreaks. Unlike a tra-ditional SIR model, to capture the dynamics of virus transmissionbetween populations and vaccine needs at the global level the

stocks here represent numbers of population blocks rather than

numbers of individuals. Populations at risk are those in whichthe proportion of susceptibles is high enough to allow significantpoliovirus transmission. Infected populationsare those with anongoing outbreak.Populations not at risk are populations withsufficient herdimmunityto prevent outbreaks.New births increasethenumberofpopulations at risk,and populations not at riskcanagainbecomesusceptiblethroughwaningofimmunity.Weassumedeaths reduce the number of population blocks in each stock at asimilar rate.

The rate at whichpopulations at riskbecomeinfected popu-lationsdepends on theforce of importation, which is proportional

Table A1

Sub-model constants (including initial values at the top of the table) used for thedemonstrative examplesa.

Shortsymbol

Name inFig. A1 Value (hypothetical)

P Population blocks (initial value) 600b (initial value)PAR Populations at risk (initial value) 0 (initial value)IP Infected populations (initial value) 0 (initial value)PNAR Populations not at risk (initial value) 60 (initial value)DV Deployed vaccine (initial value) 0 (initial value)b brate 0.025/12 per month drate 0.01/12 per month Importation rate 0.64 per month Needs per block 10 million doses per blocktdepl Vaccine deployment time 1 monthtdist Vaccine distribution time 1 monthtr Response effect time 4 monthstb Burnout time 16 monthsw Waning time 1 million monthsc

fsb Fully susceptibles per block 5 million peoplecfs Cases per fully susceptible exposed 1/200 cases/people

fse Fraction of fully susceptibles exposed 0.5 Relative paralysis risk OPV vs. WPV 200/750,000d

a While we emphasize the hypothetical nature of the examples and choices ofinput values, most values represent informed choices based on prior work[27].

b This value implies that the sub-model characterizes the world as 600 identicalpopulation blocks with approximately 10 million people.

c This value essentially ignores waning of immunity in the model.d Ratio of typical rate of paralysis per WPV infection to typical rate of VAPP per

first OPV dose[42].


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toinfected populations, withimportation rateas the proportion-ality constant. In addition, virus introductions move populationsat risk to the stock of infected populations, and the sum ofvirusintroductions andforce of importation determines the flow infection.When infected, populations become more immune and eventuallybecomepopulations not at riskas a result ofnatural burnoutandoutbreak response. Natural burnoutdepends on the average burnouttime(i.e., the average time a population would remain infected inthe absence of a response before natural infections reduce the pro-portion susceptible to a sufficiently low level to stop the outbreakand become a population not at risk). Similarly, outbreak responsedepends on the average response effect time, which we assume isshorter than the naturalburnout time. Everyinfected populationgives rise to vaccine needsaccording to an assumed constant (i.e.,needs per block).

Ideally, an increase in vaccine needs immediately leads tovaccination of the newly infected population blocks. In reality,however, detection of the outbreak and subsequent deploy-ment of vaccine takes time, and to represent this delay inthe model vaccine resides in a stock deployed vaccine beforebeing used for outbreak response. The available amount ofvac-cine final limits the distribution (d) of vaccine, which equalsd =Min(dmax,d*)=Min(Vf/tdepl,Vn/tdist), where dmaxis the maximumdeployment rate, d* the vaccine demand (or desired deploymentrate), Vfthe amount of vaccine final, Vn the vaccine needs, tdeplthevaccine deployment time, andtdistthevaccine distribution time.This formulation is consistent with a first-order delay plus a capac-ity constraint. While the vaccine deployment time represents theminimum time to deploy vaccine from the stockpile, the vaccinedistribution time represents the minimum time to distribute thevaccine in the field. If Vn/tdist > Vf/tdepl, then this leads tounmetvaccine needs, which trigger political costs of failure in the stock-pile supply chain model. Forsimplicity,we assumed that tdist = tdepl(Table A1)so unmet vaccine needseffectively arise whenevervac-cine needsexceedvaccine final.New paralytic casesoccur both dueto natural burnout(wildcases) anddue to OPVinfections associatedwithoutbreak response(VAPP cases). Even with sufficientvaccinefinal, wild polio cases occur due to the delay in the response andVAPP cases due to the vaccine used. A model identical to that inFig. A1, except that d =Min(,d*) = Vn/tdist, determines the inci-dence of paralytic cases given sufficient vaccine (i.e.,new paralyticcases 0). The incidence ofexcess paralytic cases is the differencebetweennew paralytic casesandnew paralytic cases 0.

The stock-and-flow diagram inFig. A1corresponds to the fol-lowing set of differential equations:

d PAR(t)dt

= b P(t)+PNAR(t)

w PAR(t) (t)PAR(t)

d IP(t)dt

= (t)PAR(t) IP(t)IP(t)

tb

DV(t)tr

d PNAR(t)

dt =

IP(t)

tb +

DV(t)

tr PNAR(t)PNAR(t)

w

d DV(t)dt

= Min

Vftdepl

, IP(t)

tdist

DV(t)tdist

(t) = IP(t)

P(t) +(t)

Inc(t)=

IP(t)tb

fsbcfsfse+

DV(t)tr

fsb cfsfse

(A1)

Here, we use abbreviations for constants and stocks fromTable A1and Incdenotes the incidence of paralytic cases (new paralytic casesinFig. A1), denotes the force of infection (the flow infectioninFig. A1), and represents an exogenous function that describes therate of virus introductions per population at risk over time (virus

introductionsinFig. A1),appropriately scaled relative to the time

step of theintegration scheme(we useEulerintegration with a timestep of= 0.5 months for all analyses).

A.2. Solving framing 1 (no financial constraint)

We can solve the optimization problem for framing 1 by mak-ing the realistic and mathematically weak assumption that thepublic health costs associated with each dose of unmet vaccineneedsexceed the costs of purchasing the vaccine. In other words,we assume that the opportunity costs of not having invested ina needed dose of vaccine for the stockpile outweigh the costs ofstockpiling this vaccine dose. If this condition does not hold, then

justification for the stockpile becomes questionable in the firstplace.Thus,we canreformulatethe optimizationproblemas that offinding feasiblebulk productionandfillingflows that minimize thecumulative vaccine costswhile avoidingunmet vaccine needs. Fora given time stepand using Euler integration, thevaccine needsvniat step i followby solving Eqs. (A1) iteratively,assuming infinitevaccine final (note that the solution does not depend onvaccinefinal):

DVi = DVi1 +

IPi1tdist

DVi1

tdist

PARi=PARi1+

b Pi1+PNARi1

w PARi1 i1PARi1

IPi = IPi1 +

i1 PARi1 IPi1 IPi1

tb

DVi1tr

PNARi=PNARi1+

IPi1tb

+DVi1tr

PNARi1 PNARi1

w

vni = IPi

(A2)

As we will see below, the requirement that unmet vaccine needsbe avoided imposes linear constraints onbulk productionand fill-ing, allowing us to formulate linear programming (LP) problems,for which well-known iterative solution algorithms exist[44].Thelinear programming problem depends on the formulation of thedelaysin thestockpile supplychainmodel(Fig.3). Wefirstdescribe

the case of first-order delays given that it allows the most robustformulation (i.e., requiring no further assumptions), although thecase of fixed-order delays probably most closely matches realityand thus appears in the main paper. To simplify equations, wedrop terms involving wastage, consistent with the assumption thatw1 = w2 = w3 = w4 = 0 inTable 2.

A.2.1. Optimal solution for framing 1 assuming first-order delays

With first-order delays, vaccine in filling pipeline in Fig. 3vanishes and the pipelinevaccine in bulk production pipelineconsists of one stock. The equations of the stockpile supply chainmodel for optimization problem 1 become:

dVf(t)

dt

=f(t) d(t)Vf(t)

tsdVb(t)dt

= b(t)f(t)

dVpb (t)

dt = bs(t) b(t)

totalcosts (t) = (b(t) cb +f(t) cf+ cpol 1uvn(t) ccase epc(t))ert

(A3)

Here, we use abbreviations for constants and stocks fromTable 1andfdenotesfilling, b denotes bulk production,ps denotes bulk pro-duction starts,epcdenotesexcess paralytic cases, and 1uvn(t)equals1 ifunmet vaccine needs > 0 and 0 otherwise. Also,d(t) is distribu-tionwhich equalsdeployment. The constants in Eqs.(A3)may alsochangeover time without loss of generality (e.g.,cband cf).Weseek

to find filling flows (f(t)) and production flow (b(t)),t= 0,, 2,. . .,


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wheredenotes the time period over which we plan orders (i.e.,the planning period). Given first-order delays, any desired fillingrate (f*) is constrained by the maximum filling rate (fmax), whichdepends on the filling capacity (fc), the minimum filling time(tf),and the level ofvaccine bulk(Vb):

f(t) = Min(fmax(t), f(t)) = Min

Min

fc,Vb(t)

tf

, f(t)

(A4)

The formulation off* determines the decision rule for filling, whichmight occur as a result of gaps between a desired level ofvaccinefinal and the actual level ofvaccine final, as well as the anticipateddrawdown ofvaccine finaldue to routine use and expiry (in thecase of rotating stocks):

f = Max

0, ur + e+

Vf Vf

tvf

whereurdenotes routine usage,e denotes expiry,Vfdenotes thedesired level of vaccine final, and tvfdenotes the desired stockadjustment time forvaccine final, which determines how quickly

we respond to gaps between desired and actualvaccine final. Sim-ilarly, we could formulate the desired bulk production rate (b*) asfollows:

b = Max

0, ur + e+

Vb Vb

tvb

where Vbdenotes thedesired level ofvaccine bulk,and tvbdenotesthe desired stock adjustment time forvaccine bulk. These decisionrules reflect a stockpile management strategy aimed at avoidingsteady state error that might arise as a result of predictable out-flows from the stockpile supply chain. However, since we optimizedirectly to solve for thefillingandbulk productionflows, the actualformulation of desired filling and desired bulk production rates do

not influence the solution. Instead, once we obtain a solution forfillingand bulk production, we could backtrack to determine therequired minimum desired stock levelsVf andV

bthat would leadto the same optimal solution, contingent on the formulation off*

andb*.To simplify notation, we choose= , although we emphasize

that our results do not depend on the choice of the numerical inte-gration step size as long as is sufficiently small. Using Eulerintegration and the requirement that for the optimal policy weavoidunmet vaccine needs, we can rewrite Eqs.(A3)as:

Vf(t) =

1

ts

nVf(0)+

t

j=1

1

ts

tj(f(j 1) d(j 1))

Vb(t) = Vb(0)+

tj=1

(b(j 1)f(j 1))

totalcosts(t) = (b(t) cb +f(t) cf)ert

(A5)

The requirement that we avoidunmet vaccine needstranslates intothe requirement thatdistribution (=Min(Vf/tdepl,Vn/tdist))isnotcon-strained by the stockpile, or Vf(t)Vn(t) tdepl/tdist at all times.Notethatifthisconditionholds,thend(t) = Vn(t)/tdist.Moreover,perEq. (A4)fillingcannot exceedfilling capacity (fc)atanytime.Inaddi-tion, it follows from Eq.(A4)that given a filling ratef(t), we musthavethat Vb(t)f(t) tf.Finally, bulk productioncannot exceedbulkproduction capacity(bc) at any time. Combining these conditions

with the Eqs.(A5),we obtain the following linear programming

problem:

Minimize

endt=1

(b(t 1) cb +f(t 1) cf)ert subjectto(for t= 1, . . . , end) :

tj=1

1

ts

tjf(j 1)

Vn(t)tdepltdist

(1 (/ts))

tVf(0)

+

tj=1

1

ts

tj Vn(j 1)

tdist

f(t 1) fct

j=1

b(j 1) f(t)tf

Vb(0) +

tj=1

f(j 1)

b(t 1) bc

We solve the LP problem in MathematicaTM [44] usingVf(0)= Vb(0)=0, = = 0.5 months, and an analytical time horizonof 276 months (23 years), yielding solutionsf(t) andb(t), fort= 0,. . .,tend1, withtend = 552. To obtain the actualordering strategyfromFig. 3,we must also determine the stock ofvaccine in bulk

production and the flow bulk production starts using the constraintthat Vpb (t) = b(t) tb,where tbis the minimum bulk productiontime,with an objective function minimizing the total bulk production(although technically, this need not be minimized for given that inthe model inFig. 3,we only associate costs with bulk productionand not withbulk production starts). However, given our focus onpresenting the levels ofvaccine bulk andvaccine final as the mainoutcome of the optimization, we did not compute the actualbulkproduction startsflow.

Fig. A2shows the optimal solution assuming first-order delaysin the stockpile supply chain. Comparing this result toFig. 4, whichassumes fixed delays but is otherwise equal, we find that the first-order delayassumptionentails theneedto buildmorevaccine bulkstock than with fixed delays to meet the filling requirements and

becausethestockofvaccine final continuously losesvaccine duetoexpiry. This leads to somewhat higher cumulative costs, as shownin relation to thebulk production andfillingflows andin direct com-parison with the model with fixed delays inFig. A3. Thus, althoughthe time of orders and the stock levels differ to some extent, theoptimal solution varies only minimally in terms of costs betweenthe extremes of the range of possible delay types.

Fig.A2. Theoptimalsolutionforframing1 assuming first-orderdelays inthe stock-pile supply chain and an exponential decay in virus introductions, starting in year

3.


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Fig.A3. Comparisonof thecumulativecostsand bulkproduction andfilling strategyfortheoptimalsolutionofframing1withfixeddelays(solutionfrom Fig.3) andfirst-orderdelays (solution fromFig. A2) duringthe first 5 years (nofurtherfilling orbulkproduction occurs beyond that time).

A.2.2. Optimal solution for framing 1 assuming fixed delays

In the case of fixed delays, we represent the stockpile model asa set of difference equations rather than differential equations. Thefollowing equations describe the stockpile model for optimizationproblem 1 (with= to simplify notation):

Vf(t) = Vf(t 1)+ (f(t 1) d(t 1)) = Vf(0)+

tj=1

(f(j 1) d(j 1))

Vb(t) = Vb(t 1)+ (b(t 1)fs(t 1)) = Vb(0)+

tj=1

(b(j 1)fs(j 1))

(A6)

Here, in addition to the abbreviations used above, fs(t) =f(t+ f)denotes filling starts and f= tf/ (rounded to nearest integer)denotes the number of time steps in theminimum filling time. Tosimplify calculations, we omitexpiryfrom Eqs.(A6).This assump-tion remains valid if we can deploy vaccine before it expires,corresponding to a situation in which we fill vaccine a short timebefore it is needed for deployment. This in turn is possible only iffilling capacity is sufficiently large (otherwise we would have tostock up vaccine well ahead of its use) and future demand knownwith sufficient certainty. These conditions hold for the hypothet-ical example in the main paper. If they do not hold, we can nolonger write the problem as an LP problem and we must model

the process for expiry in more detail, as discussed in the mainpaper.Thetotal costsremain as in Eqs.(A5)and the condition forvac-

cine finalalso remains Vf(t)Vn(t) tdepl/tdist, as with first-orderdelays, because we did not alter the delay type for the deploymentand distribution processes. However, the condition for vaccinebulkthat it contains sufficient stock to satisfy the outflow trans-lates intoVb(t) fs(t) for any t. As with first-order delays, wewill optimize over f and b (i.e., not fs and bs), allowing us tokeep the same capacity conditions and prompting us to furtherimpose thatb(t) = bs(t b)=0 for t= 0,. . .,b1, whereb = tb/(rounded to nearest integer) denotes the number of time steps inthe minimum bulk production time, and that f(t) =fs(tf)= 0 fort= 0, . . ., f1. Using all conditions, we obtain the following LP

problem:

Minimize

endt=1

(b(t 1) cb +f(t 1) cf)ert subjectto(for t= 1, . . . , end) :

tj=1

f(j 1) Vn(t)tdepl

tdist

Vf(0) +

tj=1

Vn(j 1)tdist

f(t 1) fct

j=1

b(j 1) Vb(0)

+

t+1

j=1

f(j + f 1)

b(t 1) bcand :f(t) = 0 for t= 0, . . . , f 1b(t) = 0 for t= 0, . . . , b 1

In solving the problem, we shall further effectively assume thatf(t)=0 for t endby omitting terms offbeyond the analytical timehorizon as they arise in the third condition above.

A.3. Formulation of framing 2 (with financial constraint)

Fig. A4 shows a possible model diagram corresponding to fram-ing 2 of the optimization problem, which seeks to find the feasibleordering strategy that minimizes the total public health costs. Fea-

sibility of the ordering strategy here means that orders satisfy theproduction and financial constraints (Fig.1). Given that theproduc-tionconstraints arealready embeddedin the stockpilesupply chainmodel, the problem essentiallyconsists of determining the optimalfunds allocation strategy given a certain amount oftotal initial stock-pile funds, the initial value ofthe stock ofstockpile funds.Thefundsallocation strategy determinesboth theuseof funds (i.e., the fractionofremainingstockpilefunds usedforpurchaseofbulkoffilledvac-cine),and thefractionof funds forbulkproduction vs.filling.Stock-pile fundsmay increase as a result of interest on the funds as wellas the addition ofnew funds. In this formulation, the latter remainsan exogenous variable, although future models might examine thedependence ofnew fundson the levels in the stockpile and pub-lic health outcomes. The allocation of funds for bulk production

triggers the start of production, but the funds accumulate first inthe stock offunds for bulk production, before actualspending onbulk production as bulk production completeswith a delay.A similarprocess occurs for filling, although importantly the flowfilling startsis constrained byvaccine bulk, and thusnew funds for fillingandthe funds allocation strategy must depend on the level ofvaccinebulk. The remaining structure remains similar to that for framing1(Fig. 3),although the cumulative vaccine costs become obsoleteas we minimize only over thecumulative public health costs.

Thestrategyforsolvingframing2dependsonwhethersufficientfunds exist to avoid unmet vaccine needs.Ifthisisthecase,thenoneor more solutions exist that minimize the public health costs to 0,and we can directly determine an optimal solution by computingthefunds allocation strategy from the optimal solution of framing 1.

Usingtheoptimal bs(t)andfs(t)fromframing1,weworkbackwardsto get:

Newfunds forbulk production : nfb(t) = bs(t) cbNewfundsforfilling : nff(t) =fs(t) cfUseof funds : uf(t) = nfb(t)+ nff(t)Fundsfor bulkproduction : Fb(t) = Fb(t 1)+ (nfb(t 1) b(t 1) cb)Fundsforfilling : Ff(t) = Ff(t 1)+ (nff(t 1)f(t 1) cf)Stockpilefunds : F(t) = F(t 1)+ (nf(t 1) uf(t 1))

where nfdenotes theflow appreciation of funds.Thefunds allocationstrategymight consist of a certain utilization rate of the remainingstockpile fundsand a certain fraction of theuse of fundsthat goesto filling vs. bulk production. Thus, with sufficient funds to avoidunmet vaccineneeds, this solution simply tells managers how tospend the funds to accomplish the optimal strategy determined in

framing 1, as shown inFig. A5.


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Fig. A4. Stock-and-flow diagram for the minimization of the total costs with no financial constraints (framing 1) for a single serotype stockpile.

Any optimal solution when the stockpile funds remain insuf-ficient to accommodate the vaccine needs invokes the feedbackbetween vaccine final andvaccine needs.As a result, in this case we

can no longer transform the optimization problem to a linear pro-gramming problem. Thus, determining the optimal solution in thiscase requires non-linear programming methods or some genericglobal optimization methods (e.g., evolutionary algorithms, simu-lated annealing, or other sampling-based optimization methods).Finding or approaching the optimal solution for this case remainsbeyond the scope of this paper, and we note that the complexityof these problems necessitates development of methods to ensurethat the algorithm finds the true global optimum instead of one of

Fig.A5. Solutionof framing 2 in theformof required funds,assuming sufficient ini-tial stockpilefunds ($56 million)to avoidunmetvaccineneeds, andno appreciation

of funds. The solution directly follows from the optimal solution of framing 1.

many local optima. In exploratory analyses that we performed, wefound that even with initial stockpile funds well above the thresh-oldfor maintaining zeropublic health costs, the conventional global

optimization methods used did not find the global optimum, mostlikely dueto thechallenges that arise in searching a combination oftwo noisy dynamic functions within a very large space of possiblefunctions.

If the stockpile funds remain only slightly below the threshold,intuitionsuggeststhattheoptimalsolutionmightinvolvefollowingtheoptimalfundutilizationpathfromtheprevioussectionuntilthestockpile funds are exhausted. This would ensure thatunmet vac-cine needsoccuronlyattheendofthetimehorizon,andtheeffectofexportations would not kick in within the time horizon. However,in reality thestockpile must cover more than a limited time horizon(especially if virus introductions may continue to occur). More-over, given decreasing global population immunity and potentiallyincreasing procurement or filling costs per dose over time, lower

cumulative public health costs may occur by tolerating unmet vac-cine needs early on and building up a stockpile for later use whenpopulation immunity becomes lower and the ability to respondmight prevent more excess paralytic cases. Such a policy obviouslyimplies enormous potential political ramifications, with the risk ofescalating the situation and permanent re-establishment of virustransmission. As thelatter depends on thedynamicsof virus expor-tations, the dynamics of the sub-model will ultimately determinethe optimal spending path in a situation of insufficient stockpilefunds.

A.4. Possible types of delays in the stockpile supply chain model

Differenttypes ofdelays exist indynamicmodels,and thenature

of the delay often significantly impacts results[21].A fixed delay


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Fig. A6. Kinetics of vaccine bulk and vaccine final in response to an instantaneous infusion of 100 million doses of vaccine bulk at time 2 months, assuming: (a) a first-orderfilling delay with time constant 3 months and (b) a fixed filling delay with time constant 3 months.

implies that all materials ordered at a given point in time arriveat exactly a fixed amount of time later. In contrast, a first-orderdelay spreads out the arrival of the materials exponentially, withthe mean arrival time equal to the specified delay time. Thus, thematerialsmix in theprocess such that some arrive before andsome

after the average delay time according to an exponential distribu-tion. Higher-order delays may also occur, which will narrow theshape of the distribution of the arrival of materials to concentratethis increasingly around the average delay time, until the distribu-tion approaches a fixed delay. Given the importance of delays inthe supply chain of the stockpile and their impact on the supplychain dynamics, we illustrate the difference between first-orderand fixed (or pipeline) delays. We consider the effect of an instan-taneous infusion of 100 million doses into thestockofvaccine bulkin month 2. We assume that at any given time we order as muchbulk as available forfilling, but thatfillingoccurs with a delay withtime constant equal to tf= 3 months.Fig. A6ashows that with afirst-order delay, vaccine final immediately starts to increase assoon as bulk becomes available, while the amount of vaccine bulkexponentially decreases to zero. In contrast,Fig. A6bshows thatwith the fixed delay, the 100 million doses simply do not get filleduntil 3 months after they enter the stock ofvaccine bulk. For bothdelays, the average time at which vaccine gets filled equals month5 (i.e., 3 months after the time of the infusion).

A.5. Alternative virus introduction scenarios

For illustrational purposes, Fig. A7 shows the optimal solu-tion for framing 1, but with a virus introduction scenario that

Fig.A7. Optimal solutionfor framing 1 assuming fixeddelaysin thestockpilesupply

chain, with an additional single virus introduction assumed in year 15.

includes one additional introduction in year 15 compared to thebase case (Fig. 5).However, we note that this solution depends onthe assumption that we can produce and fill vaccine at any time atthe same cost. If this is not the case, then we must stock up vaccineearlier, and this raises the possibility of expiry before deployment

of vaccine final. Thus, solving forthis scenario under realisticcondi-tions requires an extension on the model used for the hypotheticalexamples (see below).

A.6. Strategy for full optimization

As discussed in the main paper, many potential variations andextensions exist on the model for the hypothetical examples,including additional feedbacks, stochastic variables, and more pre-cise modeling of the expiry process and delays in the stockpile. Inaddition,the assumption thatany optimal policy mustavoid unmetvaccine needs implies a loss of generality of the model, for examplein the case of stochastic demand or insufficient upfront funding. Inlightof the limitations on existing global optimization methods dis-

cussed above, we suggest the use of safety stocks and heuristics tooptimize more general models. A possible heuristic could include:

(1) Find the optimal solution using the deterministic model forframing 1 using linear programming.

(2) Choose a safety stock levelS, either as constant or fraction ofoptimal stockpile level over time.

(3) Compute the total costs for 1 stochastic iteration of the full,stochastic model.

(4) Repeat step (3) Ntimes to determine the expected total costsfor given safety stock level S.

(5) Repeatsteps (2)(4)to findtheexpectedtotalcostsas a functionofS.

(6) Find the value ofSthat minimizes the total expected costs.

The heuristically-approximated optimal solution then consists ofthe optimal stockpile levels from the deterministic model plus theoptimal safety stock. While this represents a very simple heuristic,it might offer more robust solutions and be amenable to logicalextensions, such as distinct safety stocks for bulk and final, policiesusing adaptive safety stocks, etc.

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