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http://mdm.sagepub.com/content/early/2013/05/21/0272989X13487946The online version of this article can be found at:

 DOI: 10.1177/0272989X13487946

published online 21 May 2013Med Decis MakingMatan J. Cohen, Mayer Brezis, Colin Block, Adele Diederich and David Chinitz

Vaccination, Herd Behavior, and Herd Immunity  

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Vaccination, Herd Behavior, andHerd Immunity

Matan J. Cohen, Mayer Brezis, Colin Block, Adele Diederich, David Chinitz

Background: During the 2009 outbreak of novel influenzaAH1N1, insufficient data were available to adequatelyinform decision makers about benefits and risks of vaccina-tion and disease. We hypothesized that individuals wouldopt to mimic their peers, having no better decision anchor.We used Game Theory, decision analysis, and transmissionmodels to simulate the impact of subjective risks and pref-erence estimates on vaccination behavior. Methods: Weasked 95 students to provide estimates of risk and healthstate valuations with regard to AH1N1 infection, complica-tions, and expectations of vaccine benefits and risks. Theseestimates were included in a sequential chain of models:a dynamic epidemic model, a decision tree, and apopulation-level model. Additionally, participants’ inten-tions to vaccinate or not at varying vaccination rates weredocumented. Results: The model showed that at low vacci-nation rates, vaccination dominated. When vaccinationrates increased above 78%, nonvaccination was the domi-nant strategy. We found that vaccination intentions did

not correspond to the shift in strategy dominance and seg-regated to 3 types intentions: regardless of what others do29/95 (31%) intended to vaccinate while 27/95 (28%) didnot among 39 of 95 (41%) intention was positively associ-ated with putative vaccination rates. Conclusions: Somepeople conform to the majority’s choice, either shifting epi-demic dynamics toward herd immunity or, conversely, lim-iting societal goals. Policy leaders should use modelscarefully, noting their limitations and theoretical assump-tions. Behavior drivers were not explicitly explored in thisstudy, and the discrepant results beg further investigation.Models including real subjective perceptions with empiricor subjective probabilities can provide insight into devia-tions from expected rational behavior and suggest interven-tions in order to provide better population outcomes. Keywords: vaccination; infectious disease; state public healthinitiatives; decision analysis; simulation methods; MonteCarlo methods; expected utility theory. (Med Decis MakingXXXX;XX:XXX–XXX)

Vaccinations against infectious diseases areamong the most successful interventions mod-

ern medicine has to offer, improving the welfare of

humankind, increasing life expectancy, and decreas-ing morbidity. For most vaccines available, there areabundant valid and reliable data regarding the prev-alence, incidence, natural history, prognosis, andattack rate of their respective diseases and causativeagents. There is also sound evidence concerningvaccine benefits, effectiveness, and risks.1 Indivi-duals and health care providers have concrete foun-dations of knowledge from which shared decisionscan be made.

There is a rich body of research on collectivebehavior dynamics in response to social influence2

and threshold analyses models that investigatebehavior change across populations.3 The applica-tion of group behavior models to vaccination scenar-ios has become of interest in recent years and more soas the World Health Organization recommends thatglobal policy should encourage the use of models toprovide instruments for decision making andimplementation.4,5

Traditionally, communicable disease transmis-sion models of outbreak and vaccination simulate

Received 24 January 2012 from Center for Clinical Quality and Safety,Hadassah-Hebrew University Medical Center, Jerusalem, Israel (MJC,MB); Division of Internal Medicine Ein Kerem campus, Hadassah-Hebrew University Medical Center, Jerusalem, Israel (MJC);Department of Clinical Microbiology and Infectious Diseases, Hadas-sah-Hebrew University Medical Center, Jerusalem, Israel (MJC, CB);Braun School of Public Health and Community Medicine, Hadassah-Hebrew University Medical Center, Jerusalem, Israel (MJC, DC); andJacobs University, Bremen, Germany (AD). MJC is partly supportedby the Danny Faulkner trust. Presented at the 2010 biennial Europeanmeeting of the Society for Medical Decision Making, Hall, Austria.Recipient of Lee Lusted Student Prize–1st prize for the oral presentationof a paper. Revision accepted for publication 30 March 2013.

Address correspondence to Matan J. Cohen, Center for Clinical Qualityand Safety, Hadassah-Hebrew University Medical Center, Ein KeremCampus, POB 12000, Jerusalem, 91120, Israel; e-mail: matanc123@gmail.com.

DOI: 10.1177/0272989X13487946

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disease dynamics and incidence. In recent years,individual behavior has been introduced into suchmodels’ equations, varying with disease risk and inte-grating imitation strategies.6–12 Fu and others10 gen-erated an agent-based model that indicates thatimitation-based vaccination behavior reaches subop-timal solutions, even when individuals imitate ratio-nal and successful strategies. People can use andimitate preventive measures other than vaccination,such as limiting contacts or improving hygiene hab-its. This behavior has been modeled and shown tohave a potential role in outbreak evolution.13 Thereare even models that demonstrate spread of behaviorpatterns alongside the spread of disease.14 The afore-mentioned models all simulate behavior and dynam-ics in hypothetical populations/cohorts.

Vaccination games are mathematical simulationsin which competing strategies are evaluated.15 Thesegames have evolved into complex models of transmis-sion among humans, taking account of other vectorssuch as animals and both inter- and intrapopulationdynamics.16–18 Access to and availability of vaccines,both within countries and in the international arena,have tremendous implications, and simulations dem-onstrate the self-serving gains that countries mightobtain by aiding others.19 To some extent, all of theaforementioned models and their results are incom-prehensible to most nonspecialists, departing fromthe original function that models are generated to per-form—to depict reality in a coherent and tangible waythat can aid decision making.

The global outbreak of novel influenza A H1N1 2009and the opportunity to provide vaccination wereunusual in that there was a dearth of data to informdecision making and formulate recommendations.This presented an opportunity to explore a decision-making process in which individuals held varyingbeliefs regarding disease risks (of and from infection)and vaccine effectiveness and risks. Lacking hard dataas an anchor, we hypothesized that individual behaviormight have been influenced by mimicry, seeking secu-rity in assimilation and herd behavior. Our interest wasin both group and individual behaviors. Therefore, wedecided to use a Game Theory platform for combiningformal decision analysis (individual-level models)with transmission models (group-level models).

METHODS

Survey Participants and Data Collection

We ran a simulation with a class of 95 undergrad-uate medical students during November and

December 2009. These students participated in a com-pulsory course in evidence-based medicine taught bythe authors (M.J.C., M.B.). In the course we used aninteractive teaching system in which each studenthad an identified unique radio-frequency responsepad (Classroom Performance System, eInstructionCorporation, Denton, Texas). These response padswere numbered and randomly handed out to the stu-dents. Consequently, each student had a unique ran-dom number between 1 and 100. The students wereasked to provide intelligent ‘‘guesstimates’’ of influ-enza A H1N1 2009, regarding basic reproductiverate, risks of infection consequences, vaccine effec-tiveness, and potential risks of mild and severeadverse reactions. There was no opportunity to pre-pare for these questions. Participants were encour-aged to provide realistic guesses by awarding thosewhose estimates were within ranges published inthe literature with extra course credit points. Utilitiesof health states were generated with health-state val-uation scores for the health consequences of mild orsevere occurrence of a clinical influenza syndrome,hospitalization due to contracting A H1N1, and vac-cine complications.20 The quantitative answers givenby the participants were used to generate a model ofparticipants’ payoffs from either vaccination or non-vaccination at various population vaccination rates.

The model is based on rational principles anddetermines the expected payoff for vaccination.Assuming that decision makers are rational, strivingto maximize benefit, the model utilities are inter-preted as motivators/drivers of expected vaccinationbehavior.

Additionally, for each student, we constructeda set of 4 unique putative classmate immunizationrates, 1 in each percentage quartile between zeroand 100%. For each immunization rate, we askedthe student’s intention to vaccinate. For example,the student with response pad 23 was asked, in sepa-rate questions (presented in 4 sequential weeks),whether she would vaccinate if 23%, 48%, 73%, or98% of her peers were vaccinated. The studentswere instructed to consider and appreciate that theywere a group who spend many hours together inclosed spaces on a daily basis. The responses, thatis, their intended behavior, were compared andthen combined with the model’s predictions onexpected behavior. (Note that throughout this paperwe label behaviors as expected and intended.)

Preference valuation method and reasoning.Respondents were asked to value health statesbetween 0 (death) and 100 (perfect health). These

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estimates were used in the analysis with no timedimension. Originally, we had hoped to estimatethe utility generated by the act of vaccination. How-ever, we found this methodologically challenging,and after deliberation we chose to valuate healthstates rather than actions. Generating utility ofactions cannot be done with time-tradeoff methodsor with the standard gamble procedure, becauseactions are not exchangeable with health states.This led to another difficulty; some of the healthstates were expected to be temporary and to lasta negligible time compared with long-lasting eventsin the scenario we presented (a short spell of influ-enza v. perpetual neurological damage, unlikely asit may be). These differences are conceptually hardto grasp and, using traditional measures of utility,which rely on the exchangeability of health states,seemed not to serve the purpose of our study. There-fore, we do not report true expected utilities butrather present payoffs and valuations as utilityproxies.

Analysis

Outline. Respondents’ unique estimates of riskwere generated from individual estimation of thebasic reproductive rate with dynamic epidemicmodels. Their risk estimates were used in decisiontrees to calculate expected valuation of vaccinationand nonvaccination. Mean and weighted populationpayoffs provided the basis for a population utilitymodel (Figure 1). As mentioned above, this modelwould serve to depict expected behavior, assumingeach participant wished to maximize his or her pay-off/utility. This process was then repeated with var-iation in model inputs.

Dynamic epidemic model. The risk of contractinginfluenza A H1N1 2009 was derived from a Reed-Frost dynamic epidemic model. The probability ofa susceptible individual contracting the disease dur-ing time period t was determined by 1) the probabil-ity that any two individuals selected at randomwould come into effective contact and 2) the numberof infected cases at time t.21 The most basic model isthe SIR compartmental model (Susceptible,Infected, Resistant compartments), in whichindividuals are either susceptible or immune to aninfectious agent. To move from the susceptible com-partment to the immune compartment, individualshave to be infected or immunized. Infected individ-uals, depending on the nature of the disease, can bevectors of transmission. In the model we used, there

95 respondents, each providing personal and unique risk and preference estimates.

0 immunized 0.9 0.1 0.85 0.78

15 immunized 0.87 0.14 0.8 0.8

30 immunized 0.8 0.3 0.75 0.84

45 immunized 0.78 0.45 0.65 0.87

Each respondent’s generated risk of infection was included in a decision model and the expected utilities of vaccination and non vaccination at different vaccination rates.

Uvacc Unon-vacc Uvacc Unon-vacc

0 immunized 0.872 0.77 0.77

15 immunized 0.874 0.785 0.80

30 immunized 0.878 0.80 0.82

45 immunized 0.88 0.82 0.85

For each vaccination rate, mean utilities for vaccination and non vaccination and weighted utility are calculated together for the population model (�igure 3).

Mean Uvacc Mean Unon-vacc Weighted U

0 immunized r = 0.95 r=0.47

15 immunized r = 0.91 r=0.27

30 immunized r = 0.84 r=0.1

45 immunized r = 0.68 r=0.06

Each respondent’s basic reproductive rate was used to estimate the risk of infection given 96 vaccination rates, from zero to 95.

R0=3 R0=1.5

Figure 1 Analysis flow diagram. The analysis process is pre-

sented, beginning with use of basic reproductive rates and putative

vaccination rates to generate infection risk estimates for each

respondent. These were included in the decision tree in order tocalculate respondent-unique expectation of utility of vaccination

and nonvaccination, in each potential vaccination rate. Mean

and weighted utilities were then calculated and included in the

Schelling model (see Figure 3).

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is a preset proportion of immunized people in thebeginning of each run; we assumed random mixingand infectiveness lasting 1 day. Dynamics of the dis-ease over time resulted from disease and populationcharacteristics: transmissibility (R0, the basic repro-ductive rate), duration of infectiousness, and theproportion of susceptible individuals. Respondents’subjective estimates of these estimates were used formodeling the expected risk of infection (see Appen-dix A).22–24 We modeled 96 risk estimates of infec-tion for each respondent: from 0% vaccination,where none was vaccinated, to 100%, where all 95people were vaccinated (0/95, 1/95, 2/95 . . . 94/95, 95/95).

Individual-level decision analysis. We analyzedeach individual’s payoffs with a decision tree (Fig-ure 2). The tree begins with a decision node, offeringthe choice of either taking the vaccine or not. Takingthe vaccine might be followed by an adverse event ornot. The adverse event might be mild or severe. Theclinical pathways of either decision (vaccinating ornot) are then similar: either contracting influenzaA H1N1 2009 or not; if infected, one might experi-ence mild, moderate, or severe symptoms, includinghospitalization or death. Expected preference valua-tions of vaccination and nonvaccination were gener-ated from individualized decision trees. We ran thedecision model 96 times for each proportion of vac-cination. In each run, the risk of infection was gener-ated from the Reed-Frost model.

Schelling model. The data derived from the deci-sion tree model were then entered into a model, orig-inally proposed by Thomas Schelling and describedas a uniform multiperson prisoner’s dilemma—uniform in so far that only the number of individuals(not their identity) matters.25 This model depictspersonal utilities/payoffs that depend on the choiceof action of the individual and on the proportion ofother individuals who behave similarly (see exam-ple in Appendix B). The weighted utility reflects‘‘society’s average utility,’’ since it is weightedaccording to the proportion of individuals whoperform the act and the proportion of those whodo not. When the population becomes homoge-neous, the weighted utility of the dominant groupbecomes society’s utility. For every proportion ofvaccinees, we calculated the average utility of eitherstrategy (vaccinating or not). The overall weightedexpected utility (representing society’s utility) wasweighted according to the proportion of vaccinees(Appendix C).

In epidemic models with random mixing, thegreater the proportion of immune individuals (eitherafter infection or after vaccination), the lesser of thechance of being infected. The extent of that changein infection risk depends on the insulting agents’reproductive rates. Vaccination payoffs are derived,among other things, from the risk of infection andtherefore are associated with the proportion ofimmunized people. Conversely, vaccine risks arenot dependent on the risk of infection. Therefore,nonvaccinators’ expected utility is dependent onthe risk of infection, whereas that of vaccinatorsresults from the interplay of vaccine benefits andcomplications and the predisposing risk of infec-tion. A goal of our study is to model and investigatethe dynamics of the expected utilities of vaccinatorsand nonvaccinators.

Additional analyses. During the analysis of theresults, we found that participants were segregatedinto 3 distinct groups according to their intentionto vaccinate: strict vaccinees, who stated that theywould vaccinate in all 4 putative vaccination rates;strict nonvaccinees, who stated that they wouldnot vaccinate in all 4 putative vaccination rates;and flexible vaccinees, who were not persistent intheir stated vaccination/nonvaccination intentionat different vaccination rates (see Results section).We then proceeded to combine the results of vacci-nation intent with the results of the model depictingexpected behavior (Appendix D describes thedetailed calculations). Finally, we repeated the anal-ysis with inclusion of disease and vaccine risk esti-mates, abstracted from professional literature longafter the pandemic abated and when publicationsstarted to appear. The purpose of this analysis wasto compare the outputs derived from models with‘‘real’’ data inputs with those of the models thatused perceived estimates.

RESULTS

A summary of the respondents’ subjective esti-mates is presented in Table 1. Study participants esti-mated a high mean basic reproductive rate of 5.24with an even larger standard deviation. In compari-son, estimates for highly transmissible disease,including SARS, are most often no higher than 4.26

Vaccination was estimated to decrease risk of infec-tion by half, and expected mild and severe adversevaccine-associated event rates were 38% and 4%.Infection after vaccination was associated with anincreased probability of death, a decreased

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probability of severe disease not requiring hospitali-zation and an unchanged probability of hospitaliza-tion. With regard to disease severity and vaccineadverse events, worse clinical scenarios were ratedwith greater variability.

Among all individuals, the expected payoff ofeither choice increased in positive association withthe proportion of vaccinated people (Figure 3). Theexpected payoff from vaccination showed a mildincline, the expected payoff from not taking the vac-cine had a steeper incline, and the weighted expectedpayoff consistently increased. At low vaccinationrates, the vaccination strategy dominated and pro-vided higher payoffs. The increase for not vaccinatingwas such that the difference in payoffs between thestrategies decreased as vaccination rates increased.In the model, when the vaccination rates reached78%, the strategies equalized. When vaccinationrates increased above 78%, nonvaccination domi-nated; this strategy provided a higher expected payoff

compared with people who vaccinated. These modelpredictions were generated from the subjective esti-mates of risks, benefits, and preferences of the partic-ipants and did not include any real establishedestimates of disease and vaccine risks and benefits.

As previously indicated, stated vaccination inten-tions under varying putative rates of vaccinationwere segregated into 3 groups: 27 of 95 (28%)reported that they had not intended to vaccinate atall (strict nonvaccinees), and 29 of 95 (31%) reportedthat they intended to take the vaccine no matter whatothers did (strict vaccinees). The remaining respond-ents, 39 of 95 (41%), were labeled as flexible vaccin-ees. In this latter group, uptake of vaccine waspositively associated with putative peer vaccinationrates: At putative vaccination rates of 0% to 20%,a quarter answered that they would vaccinate; atputative rates of 21% to 40%, 37% answered thatthey would vaccinate; between 41% and 60% puta-tive rates, 51% answered that they would vaccinate;

Figure 2 Decision tree. The decision tree depicts either taking the H1N1 influenza virus vaccine or not and the subsequent consequences.

SE = side effects.

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and at higher rates, 57% answered that they wouldvaccinate.

The results suggest that no less than 32% and nomore than 72% of the participants intended to vacci-nate. We repeated the analysis in this range. In thisway we combined the findings of the model depictingexpected behavior and the questions about intendedbehavior, in order to zero in on the group of individ-uals for whom the model might be more relevant andto demonstrate the impact on the entire population.When we included all participants in the model,including the strict vaccinees and nonvaccinees (Fig-ure 4a), vaccination dominated across the wholerange of vaccination rates. Modeling the flexiblegroup alone (Figure 4b), we found that nonvaccina-tion dominated from 60% vaccination rates onward.Similar to the crude results depicted in Figure 3, pay-off from vaccination increased among both vaccineesand nonvaccinees, and the payoff incline amongthose not taking the vaccine was steeper and eventu-ally dominated.

Finally, we repeated the simulations and gener-ated a Schelling diagram with estimates similar tothose in the medical literature about vaccine risks ofadverse events (40% mild and 0.001% severe),27,28

effectiveness (90%),29 and the novel H1N1 influenzabasic reproductive rate (R0 = 1.31),16,30 keeping theindividual subjective valuation estimates (Figure 5).The crude payoff model, presented in Figure 5a wascalculated as it was in the original model (Figure 3),with replacement of the real risk estimates. It demon-strated that nonvaccination was dominant from a 5%

Table 1 Descriptive Data of Students’ Responses (N = 95)

Mean Standard Deviation

Reproductive rate 5.28 6.24Relative risk of H1N1 infection with vaccine compared with no vaccine .50 .815Probabilities

Probability of mild side effects of vaccine .38 .29Probability of severe side effects of vaccine .04 .081

Given H1N1 infectionProbability of severe disease, no hospitalization .21 .14Probability of hospitalization .06 .06Probability of death .01 .02

Given H1N1 infection after vaccinationProbability of severe disease, no hospitalization .09 .15Probability of hospitalization .05 .13Probability of death .03 .13

UtilitiesUtility of easy flu 87.35 8.81Utility of difficult flu 69.86 16.68Utility of flu hospitalization 49.09 20.14Utility of mild vaccine side effects 91.76 7.03Utility of difficult vaccine side effects 56.5 22.02

Figure 3 Expected utility for each strategy and weighted utility at

varying rates of vaccination in the population. This diagram

depicts the mean expected utility of vaccinees [EU(vaccine)], themean expected utility of nonvaccinees [EU(no vaccine)], and the

weighted expected utility [EU(weighted)]. The x-axis represents

the proportion of the population who are vaccinated. The y-axis

represented depicts the expected utility. For explanation of calcu-lations, refer to the text and Appendix A.

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vaccination rate threshold. The model presented inFigure 5b was analogous to the model generatedfrom flexible participants (Figure 4b) with the sameinput data replacements. It showed that nonvaccina-tion was dominant through the whole range of vacci-nation proportions, despite their declared intentionto vaccinate even at high rates of population coverage.

DISCUSSION

Explaining Vaccination Behavior

A good vaccine should benefit both the individualand the surrounding community. The vaccine shoulddecrease an individual’s chance of contracting thedisease (preferably with negligible risks of adverseevents) and provide the community with anothermember whose immune status decreases his or herpropensity to be a vector for transmission. Ideally,when enough individuals are vaccinated, herdimmunity emerges; the proportion of nonsusceptibleindividuals increases to an extent that the infectiousagent is significantly limited in its ability to spreadamong the remaining susceptible individuals, out-breaks are aborted, and pandemics abate. Individualscan take advantage of herd immunity and not vacci-nate themselves or their relatives. If too many indi-viduals seek this course of action, the collaborativeeffort to curtail outbreaks, pandemics, and epidemicsfails. This is a source of tension between individuals’preferences and society’s well-being and prosperity.

We report a study in which participants providedsubjective estimates of risk and benefit from diseaseand vaccination that were used to generate a popula-tion utility model. Initially, this model identifieda population vaccination rate threshold from whichnonvaccination dominated (Figure 3). The revealedchange in strategy dominance, from vaccination tononvaccination, prompted us to consider various fac-tors that might affect behavior. This might haveresulted from an anticipation by the participants,conscious or subconscious, that herd immunitywould emerge. Following the answers regarding vac-cination intent, to further explore this line of think-ing, we limited the analysis to flexible respondents(Figure 4b), that is, those who were neither strictlyvaccinees nor strictly nonvaccinees. Although thethreshold for nonvaccination dominance decreased,the dynamics were the similar to the initial model(shown in Figure 3). We were therefore left with sig-nificant discrepancies between expected behaviorand intended behavior that beg explanation.

(a)

(b)

Figure 4 Expected utility for each strategy and weighted utility atvarying rates of vaccination in the population, based on partici-

pant segregation. (a) This diagram shows the Schelling models

for varying proportions (between 32% and 72%) of vaccinees inthe population and depicts the combined utilities for all vaccinees

(strict and flexible, dashed line), all nonvaccinees (strict and flex-

ible, dotted line), and the overall weighted utilities (solid line). (b)

This diagram provides the utilities of the members of the ‘‘flexible’’group for those vaccinated (dashed line) and those not vaccinated

(dotted line).

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The results of our inquiry regarding vaccinationintent were not reflected in the initial model. Theexpected behavior model included no estimates or

valuations that explicitly reflect intent, fear, or otherbehavior motivators. This could explain the discrep-ancies. Keeping that in mind, we direct attention toseveral findings that might represent psychologicalundercurrents. We realized that as vaccination ratesincreased, so did the payoff from vaccination. Thiswas true even when vaccination rates were higherthan the threshold from which nonvaccination dom-inated. Additionally, the decrement in payoffs amongnonvaccinators lessened as the proportion of vac-cines decreased. These phenomena did not mirroreach other and could indicate that within the subjec-tive estimates of risks, health state payoffs, and theirratios, there were at work factors that increase thewillingness to assimilate into and follow the group.Keeping with this theme, we also propose that theresulting dominance of vaccination up to a thresholdand the following shift in dominance to nonvaccina-tion could reflect the nature of a relationship that fearand overt free-riding have with the decision process.

Another factor that suggested itself was undue fearof the pandemic. This was unveiled when we per-formed a repeated analysis including published riskestimates of disease communicability and complica-tions (Figure 5). Having brought together subjectivevaluations of health states with true estimates ofrisk, we revealed that nonvaccination would havedominated at very low vaccination rates, far lowerthan those required to curtail the AH1N1 pandemic.We believe that even today, if people were asked toestimate the risks associated with AH1N1 virus,they would overestimate the risks of severe diseaseand mortality.

As mentioned above, the uniqueness of the AH1N12009 outbreak and vaccination opportunity was thelack of concrete evidence upon which decisionscould be based. Thus, waiting to see what happenedbefore deciding to vaccinate could just as well havebeen a strategy for dealing with the new outbreak.31

We did not allow for differential vaccination timesin our model.

These possible interpretations of individual-levelincentives, misconceptions, and dynamics arealigned with studies that investigated the motivationunderlying voluntary vaccination. For instance, cog-nitive biases and flawed risk perception might leadpeople who vaccinated, and eventually did not con-tract influenza, to be less inclined to vaccinate thenext year.32,33 In fact, Perisic and Bauch34 haveshown that the ability to eradicate disease under vol-untary vaccination policy requires unique circum-stances, most notably limited transmissibility ofinfection and disease. There is previous evidence

(a)

(b)

Figure 5 Expected utility for each strategy and weighted utility at

varying rates of vaccination in the population, based on partici-

pant segregation and evidence-based estimates. (a) This diagram

shows the model for the entire range of population vaccinationrates, whereas (b) is limited to the range of varying vaccination

rates, between 32% and 72%, and to those members of the flexible

group. In both (a) and (b), vaccines are depicted in the dashed line

and nonvaccinees in the dotted line. The solid line in (a) depictsthe weighted utility.

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that discrepancies between optimal public healthprograms/recommendations and personally per-ceived gains could lead to extremely suboptimalequilibria, and free riding has been shown to main-tain these equilibria.13,35–37

People not taking the vaccine can explain theirbehavior through various avenues of rationales andfeelings, as would be expected of modern-dayhumanistic citizens.38 These are, perhaps, the pos-tures of those who do not sense a great threat to them-selves and their loved ones. In Israel, during 2010,there were low compliance rates with national rec-ommendations to vaccinate against AH1N1. Thiswas the year following the original AH1N1 2009 out-break. Yet, a single case of mortality of an otherwisecompletely healthy 15-year-old boy, attributed toAH1N1, was the trigger for increased fears and anxi-ety, resulting in complete depletion of the vaccinereserves within a few weeks. To stress this point,we doubt whether any person bitten by a known rabidmammal, except for those practicing strict andextreme ideologies, would contemplate refusal ofrabies vaccination.

Policy Implications

In many walks of life, there are potential tensionsbetween individual preferences and the preferencesof society, the aggregate of all individuals. This age-old tension generated instruments such as rules andinstitutions, allowing societies to realize mutualgoals. The mass of people who do vaccinate are a com-mon pool resource, although this perception is lesstangible, and self-governance, which has been pro-posed to be a naturally occurring phenomenon,39 isunlikely. Therefore, governing bodies need to inter-vene to protect the population they represent andserve.

Financial incentives can shape policy and influ-ence vaccination rates and outbreak progression.40

However, there are inherent disincentives to eradi-cate diseases, requiring concentrated efforts andcoordination of key players in order to achieve com-plete vaccine coverage.41 Free-riding can occurbetween countries,42 and competing societies mightcurb the potential of each to optimize its own well-being. Universal vaccination coverage probablydoes not maximize social utility, and balancingcost-effectiveness and herd immunity might proveto be too great a challenge. In contrast, utilitarian pol-icies might not be adequate when deliberating theallocation of public resources and social justice.Politically, selective immunization of certain groups

might not be appealing and could be difficult toimplement. Furthermore, given transparencies inpublic debates and policy making, some people willnecessarily know that they are being asked to carrythe burden for others and will oppose this. Deonto-logical alternatives are appealing, designating vacci-nation as the ‘‘right thing to do,’’ making provisionsso that all people vaccinate, and justifying the finan-cial costs.

The issues of selective immunization direct atten-tion to 2 groups: those at greatest risk from disease(the elderly and immune suppressed) and thosewho are the prominent vectors of viral transmission(the young).43 Shim and others44 developed a modelthat presumed selfish behavior and showed that dif-ferent age groups would be best self-served by vacci-nating at different stages of the AH1N1 2009pandemic. Oliver45 has shown that selective incen-tives might be counterproductive and might notalways deliver the desired effects to society. How-ever, Galvani and others46 have shown that utilitarianoptimum and Nash equilibrium can be achieved withpandemic influenza as opposed to epidemic influ-enza. The incentives in pandemic influenza aresuch that self-interest groups, which would other-wise not be eager to vaccinate, receive sufficientreward and concomitantly provide a protective bar-rier for viral transmission.46 We offer anotherapproach to selective vaccination policies, namelydirecting interventions that increase vaccinationuptake among people who are flexible in their inten-tions to vaccinate. The findings of this paper suggestthat providing dynamic knowledge to people aboutpopulation vaccination rates could ‘‘nudge’’ themtoward vaccinating. Whether efforts should be madeto identify flexible individuals and to incentivizethem to vaccinate is beyond the scope of this paper.

Lessons for Modeling

In this report, we present and propose a platformfor modeling society’s overall utility (expected pay-offs), given varying vaccination rates. The model pro-vides grounds for strategic planning of vaccinationpurchasing. It is novel, incorporating both decisiontrees and dynamic epidemic models into the GameTheory framework. These tools allow simultaneousappreciation of how micro-motives and macro-behavior interact. The overall structure of the modelis relatively simple and intuitive, and the results arereadily interpreted. This framework could prove tobe valuable for decision makers aiming to identifywhere society’s utility can be maximized. There are

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few publications that attempted to apply ThomasSchelling’s threshold theory to infectious diseaseand vaccination issues,47,48 and there is scant litera-ture about studies that used the Schelling modelused in this report.49 The Schelling model has notbeen previously used for analysis of vaccinationgames, although the significance of binary behavioralpatterns and their bearing on outbreak dynamics havebeen acknowledged.50 We present detailed modelformulas, enabling others to repeat and improve ourmethods (see the appendices).

Avenues for better modeling of vaccination gamesshould focus attention on placing the scenarios in theappropriate mathematical frameworks of Game The-ory lore. It should be considered whether the model-ing should be of cooperative v. noncooperativegames. Critical factors include the availability ofcomplete, partial, or no information about one’sown and others’ payoffs; whether participants accrueknowledge and experience (repeated games, finite orinfinite); and participants’ ability to communicate.Such games could also model public policy interven-tions that redefine the game environment by provid-ing data about public vaccination rates and byproviding health or financial incentives. Anotherframework that could prove interesting is whethervaccination games can be simulated using common-pool resource models and experiments, vaccineesbeing both providers and appropriators to and of theresource.

In our initial hypothesis, we expected to find thatthe self-perceived utility from vaccination was posi-tively associated with the putative proportion ofvaccinees in the population. We found this to betrue among a third of the participants. Two-thirdsstated that their choice was insensitive to vaccinationrates. The quantitative model, generated initiallyfrom risk estimates and preferences (Figure 3), gaveno clue to this phenomenon. Asking participantsdirectly whether they would or would not vaccinate,under putative vaccination rates of the population,revealed this finding and demonstrated the fragilevalidity of modeling methods. We acknowledge thereviewers of this manuscript who recommendedthat we highlight these discrepancies throughoutthe paper by designating results that influenceexpected and intended behavior. Incorporating frag-ile data into models, themselves an intentional sim-plification of reality, contributes to the potentialdiscrepancies between reality and model estimates.Measuring utilities or valuing payoffs of mimicryand of intentional or unintentional free-ridings andadding these to the model might have suggested

that this segregation of vaccination intent exists. Fur-thermore, incorporating aspects of fear of disease/vaccination and their associated complications eitheras part of their valuations or as a different, comple-mentary dimension of these states could haveenriched the model. To correctly define and uniquelyabstract such data from individuals and providevalid, reliable, and coherent estimates are thought-provoking issues. However, abandoning data collec-tion to guide decision making is unacceptable. Pro-moting and advancing policy with no preliminarydata collection will not be an accepted standard forpolicy making. Scientists in the field of decisionmaking, in particular medical decision making,should make use of the best scientific evidenceavailable, always keeping in mind that ‘‘researchlimitations’’ and ‘‘model simplifications’’ are notsimply academic vernacular, and should practicevigilance when proposing recommendations.

Limitations

The findings in this study are to be appreciated incontext. We used simple assumptions and parameter-ization for generation of utilities/payoffs and risks.Furthermore, the most basic dynamic epidemicmodel and decision tree were used in this study.More advanced stochastic and dynamic models existthat can be used to the same end, thereby better cap-turing the complexity of reality. The sample of partic-ipants (undergraduate medical students) was a selectgroup who might not be representative of other popu-lations, particularly with respect to their attitudetoward the value of medical evidence. However, theadvantage of this sample was their shared everydaylife, simulating a population in which individualscan appreciate repercussions the group might experi-ence following individual actions. During the peerreview process, it was remarked that the respondents’estimates of disease and vaccine characteristics werevery far from the best available evidence. Addition-ally, infection among the vaccinated was associated,on average, with increased probability of death butdecreased probability of severe disease andunchanged probability of hospitalization. Post hocrationalization can be postulated for these results,although they might just as well reflect human irratio-nality and inconsistency in subjective numerical esti-mation of complementing risks. One can suppose thatif a vaccinated person has clinical infection, hewould more likely succumb, rather than just be ill,because of his failure to develop adequate protection.Ibuka and others51 have shown that risk perceptions

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had their own dynamics during the AH1N1 pan-demic. More research should be directed towardunderstanding the causes for discrepancies andinconsistencies of these quantitative appreciations.Another insightful commentator noted that in thisstudy we missed the opportunity to dissect the deci-sion process drivers. Our data collection could notprovide insight as to the extent to which decisionsare based on rational thought and maximizing utilityv. behavioral mimicry, intentional free-riding, andother motivators. Hershey and others52 found thatframing by surveyors influenced selfish answersand not altruistic answers, demonstrating the powerof data collection method. This course of researchdeserves further study and novel research designs,with emphasis on the neutrality and impartiality ofthe scientists, who most often are vaccination propo-nents. Others have demonstrated that trust in theinformation provider and altruism are also part ofthe thought process of vaccination decisions.33,53–55

CONCLUSION

Our results suggest that at least some portion of thepopulation is influenced by the actions of others,although many persons’ intentions are not. Had wenot asked specific questions about intentions andonly collected risk estimates and utilities, we wouldhave erroneously generalized the results of ourmodel. Although our methods did not focus onunearthing and differentiating between decisiondrivers, we feel that our results suggest that mimicrydoes play a role in driving vaccination decisions.Understanding the dynamics of individual decisionmaking and its association with the decisions ofothers can enrich the array of policy options underconsideration. We believe that modeling simulationsthat analyze the behavior of hypothetical cohortshave become redundant and that the time is ripe forcombining real-life perception and estimations ofprobability and value.

APPENDIX A

Estimation Infection Risk from the Reed-FrostModel

n = population size.x = number of individuals immunized in the

population.t = days since outbreak began.It = number of expected infected on day t.

R0 = basic reproductive rate.At t = 0: It = 1.When t . 0: It = [n – x – S(I0 1 . . . 1 It–1)] * [1 – (1 – R0/

n)It–1].Risk of infection is S(I0 1 . . . 1 It)/n.

APPENDIX B

Conceptual Schelling Diagram

The x-axis of the figure represents the proportionof individuals who perform a specific strategy (p),in this hypothetical example, the proportion of ice-hockey players who wear a helmet. The y-axis isa measure of expected utility (EU). The EU (helmet)line (dotted line) represents the expected utility fora player who wears a helmet at a given proportionof players who wear helmets. The EU (no helmet)line (dashed line) represents the expected utility ofa player who does not wear a helmet. The weightedEU line (solid line) represents the average utility ofa player in this population: that is, the weightedsum of the two expected utilities, calculated asweighted EU = EU(helmet) * p 1 EU(no helmet) *(1 – p). This diagram is a qualitative graphic presenta-tion of the dilemma provided in the opening ofThomas Schelling’s25 paper.

Propor�on of players who wear a helmet

Expe

cted

u�l

ity

10 0.5

APPENDIX C

Utility Calculations for the Schelling Diagram

EUip(v) = vaccination expected utility of person i at the

given proportion of vaccines – p.SEUi

p(v)/n = mean expected utility of a vaccine.EUi

p(nv) = nonvaccination expected utility of person i atthe given proportion of vaccines – p.

SEUip(nv)/n = mean expected utility of a nonvaccinee.

EUp(W) = weighted expected utility at a given propor-tion of vaccinees – p.

p * SEUip(v)/n 1 (1 – p) * SEUi

p(nv)/n.

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APPENDIX D

Utility Calculation, Accounting for RespondentSegregation

� EUsvp (v) = expected utility of the strict vaccinee partic-

ipant given proportion of vaccinees – p.� a = SEUsv

p (v)/29 = mean expected utility, strict vacci-nee group (the group comprised of 29 respondents)given proportion of vaccinees – p.

� EUsnvp(nv) = expected utility of the strict nonvaccinee

participant given proportion of vaccinees – p.� b = SEUsnv

p(nv)/27 = mean expected utility, strictnonvaccinee group (the group comprised of 27respondents) given proportion of vaccinees – p.

� EUfp(v) = expected utility of vaccination in the flexi-

ble participant given proportion of vaccinees – p.� g = SEUff

p(v)/39 = mean expected utility, vaccinatedflexible participants given proportion of vaccinees – p.

� EUfp(nv) – expected utility of nonvaccination in the

flexible participant given proportion of vaccinees – p.� d = SEUf

p(nv)/39 = mean expected utility, nonvaccinatedflexible participants given proportion of vaccinees – p.

� Nfp(v) = p * 95 – 29; number of flexible vaccinees at

given proportion of vaccinees – p.� Nf

p(nv) = 39 – Nfp(v); number of flexible nonvaccinees

at given proportion of vaccinees – p.� EUp(v) = [a* 29 1 g * Nf

p(v)]/[p * 95]; the expected util-ity among vaccinees given proportion of vaccinees – p.

� EUp(nv) = [b * 27 1 d * Nfp(nv)]/[(1 – p) * 95]; the

expected utility among nonvaccinees given propor-tion of vaccinees – p.

EUp Wð Þ5 a�291g�Npf vð Þ1b�271d�Np

f nvð Þh i

=95

5a�0:311b�0:281g� p� 0:31ð Þ1d� 0:72� pð Þ5 theweightedexpectedutilityatagivenproportion

of vaccinees � p:

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