Imperfect Vaccine Aggravates the Long-Standing Dilemma of Voluntary Vaccination

Transcription

1 Impefect Vaccine Aggavates the Long-Standing Dilemma of Voluntay Vaccination Bin Wu 1 *, Feng Fu 2 *, Long Wang 1 1 State Key Laboatoy fo Tubulence and Complex Systems, Cente fo Systems and Contol, College of Engineeing, Peking Univesity, Beijing, China, 2 Pogam fo Evolutionay Dynamics, Havad Univesity, Cambidge, Massachusetts, United States of Ameica Abstact Achieving widespead population immunity by voluntay vaccination poses a majo challenge fo public health administation and pactice. The situation is complicated even moe by impefect vaccines. How the vaccine efficacy affects individuals vaccination behavio has yet to be fully answeed. To addess this issue, we combine a simple yet effective game theoetic model of vaccination behavio with an epidemiological pocess. Ou analysis shows that, in a population of selfinteested individuals, thee exists an oveshooting of vaccine uptake levels as the effectiveness of vaccination inceases. Moeove, when the basic epoductive numbe,, exceeds a cetain theshold, all individuals opt fo vaccination fo an intemediate egion of vaccine efficacy. We futhe show that inceasing effectiveness of vaccination always inceases the numbe of effectively vaccinated individuals and theefoe attenuates the epidemic stain. The esults suggest that numbe is taded fo efficiency : although inceases in vaccination effectiveness lead to uptake dops due to fee-iding effects, the impact of the epidemic can be bette mitigated. Citation: Wu B, Fu F, Wang L (2011) Impefect Vaccine Aggavates the Long-Standing Dilemma of Voluntay Vaccination. PLoS ONE 6(6): e doi: / jounal.pone Edito: Matjaz Pec, Univesity of Maibo, Slovenia Received Apil 5, 2011; Accepted May 4, 2011; Published June 8, 2011 Copyight: ß 2011 Wu et al. This is an open-access aticle distibuted unde the tems of the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal autho and souce ae cedited. Funding: The authos acknowledge suppot by the National Natual Science Foundation of China Gants No and No The fundes had no ole in study design, data collection and analysis, decision to publish, o pepaation of the manuscipt. Competing Inteests: The authos have declaed that no competing inteests exist. * bin.wu@evolbio.mpg.de (BW); fengfu@fas.havad.edu (FF) Intoduction Peemptive vaccination is the pinciple stategy fo the intevention and contol of infectious diseases. Howeve, vaccination epesents a long-standing social dilemma fo public health administation. On the one hand, compulsoy vaccination may esult in an infingement of civil ights [1]. On the othe hand, voluntay vaccination cannot lead to sufficiently high hed immunity fo disease eadication. Thus it often fails to potect populations fom epidemics [2,3,4,5]. Taditional epidemiological modeling focuses on the pathway of disease tansmission, and often does not take into account human stategic behavio in esponse to the epidemic [6]. Howeve, it is moe plausible to integate human behavio with the epidemiological pocess. In this sense, voluntay vaccination itself is a social dilemma: vaccinated individuals can escape fom the disease with a cost patly incued by the vaccine side effects; the unvaccinated can also be potected fom the epidemics without paying anything povided the population immunity is in effect. In this case, selfinteested individuals attempt to shun vaccination while still benefitting fom the hed immunity. Such fee-iding may lead to a low vaccination level, failing to eadicate the disease, thus a social dilemma [7,8]. The famewok of game theoy popely descibes how individuals eact when facing a dilemma [9,10,11,12,13,14,15,16,17,18,19]. In paticula, how the evolutionay outcome of the social dilemma is achieved can be investigated based on the imitation pocess [20,21]. Theefoe, voluntay vaccination can be studied in this famewok and notewothy thee has been an emeging liteatue of combining epidemiology and game theoy [7,22,23,24,5,25,8,26]. Pevious wok usually assumes pefect vaccination, i.e., the vaccinated individuals gain pefect immunity against the disease [7,23,8]. The effectiveness of vaccination, howeve, is not 100%, such as measles [27], malaia [28] and HIV [29]. Even though the actual vaccination is pefect, the peceived effectiveness can be not. Questionnaie esults have shown the peceived effectiveness is often lowe than the actual one [24]. This peceived efficacy of vaccination, influenced by psychological effects, plays a deteminant ole since individuals adjust thei stategic behavio based on peceptions of the vaccine efficacy athe than the actual one [5,24]. Theefoe, impefect vaccination should be taken into account in the game theoetical analysis of the vaccination behavio [30,31,32]. Besides, public concen towads the effectiveness of vaccine is so common that it often leads to massive vaccine avoidance. How vaccine effectiveness affects vaccination level and thus the seveity of epidemic outbeak has not yet been fully answeed. Motivated by these, we study this poblem by a minimal model. Analysis Fo poof of pinciple, we conside vaccination dynamics in an infinitely lage well mixed population. In addition, we assume that individuals have a pefect knowledge on the effectiveness of the vaccination. In this case, thee is only one paamete descibing both the actual and the peceived effectiveness. The vaccination game consists of two stages, the yealy vaccination campaign and an epidemic season. Duing the vaccination campaign, each individual decides whethe o not to PLoS ONE 1 June 2011 Volume 6 Issue 6 e20577

2 Impefect Voluntay Vaccination take vaccination. A vaccinated individual pays a cost Vw0 while an unvaccinated individual pays nothing. This cost V includes the time spent in taking the vaccination as well as its side effects. Duing the epidemic season, the population can be divided into two pats: one compises effectively vaccinated individuals, and the est is composed of unvaccinated individuals and the vaccinated ones whose vaccinations ae not effective. Successfully vaccinated individuals ae immune to the seasonal disease, and thus have no isk of getting infected. Fo the emaining individuals, howeve, they become infected with a pobability f (y), whee y is the fequency of effectively vaccinated individuals. In this case the infected bea a cost by Cw0. This cost C includes expenses and time fo health cae as well as motality. The lage the numbe of effectively vaccinated individuals is, the less likely an unvaccinated individual gets infected. Thus f (y) is deceasing with y. Let the effectiveness of the vaccination be e and the vaccine uptake level be x. The fequency of the effectively vaccinated individuals is y~ex. The faction of the vaccinated and healthy individuals is x½ez(1{e)(1{f (ex)) Š, which is composed of two pats: these effectively vaccinated individuals (with fequency xe) and those ineffectively vaccinated individuals (with fequency x(1{e)) who ae fee fom the infection (with fequency 1{f (ex)). In this case, each effectively vaccinated individual gets payoff {V. In analogy to this, the fequencies and payoffs fo diffeent individuals ae given by Table 1. When the epidemic season ends, i.e., the aveage abundance of infected individuals does not change, individuals adjust thei stategies by imitation whee successful individual s stategy is moe likely to be followed [33,34]. Hee we employ the Femi update ule to chaacteize such an imitation pocess [35,36,8,37,38]: two individuals a and b ae selected andomly; a leans to behave like b with pobability 1 1z exp ½{k(f b {f a ) Š whee f a and f b ae the peceived payoffs fo a and b, and k is the selection intensity indicating how stongly individuals ae esponsive to payoff diffeence. The dynamics of the vaccination is govened by [20,39] _x~x(1{x)½ðez(1{e) ð1{f (ex) ÞÞ(1{f (ex)) tanh ( k 2 ({V)) zðez(1{e)(1{f (ex)) Þf (ex) tanh ( k 2 ({VzC)) z(1{e)f (ex)(1{f (ex)) tanh ( k 2 ({V{C)) z(1{e)f 2 (ex) tanh ( k 2 ({V))Š It has been suggested that the selection intensity fo human imitation is athe weak [21,34], i.e. k is sufficiently small. We pefom the Taylo expansion of the.h.s of Eq. (2) in the vicinity of k~0, then afte a time escaling which does not change the dynamics, Eq. (2) can be captued by a much moe simple fom _x~x(1{x)(ef (ex)c{v) In what follows, we investigate how the vaccine uptake evolves by Eq. (3) fo geneal function of infection isk f. To this end, we focus on how the effectiveness of vaccination has an impact on the ð1þ ð2þ ð3þ Table 1. The faction and the payoff fo the fou types of individuals in the population. Vaccinated & Healthy Vaccinated & Infected Unvaccinated & Healthy Unvaccinated & Infected Faction x½ez(1{e)(1{f (ex)) Š x(1{e)f (ex) (1{x)(1{f (ex)) (1{x)f (ex) Payoff {V {V{C 0 {C They ae the vaccinated and healthy, the unvaccinated and healthy, the vaccinated and infected and the unvaccinated and infected. doi: /jounal.pone t001 PLoS ONE 2 June 2011 Volume 6 Issue 6 e20577

3 Impefect Voluntay Vaccination the collective outcome of vaccination behavio and the effective vaccination level. Then we incopoate an epidemic dynamics to obtain a specific infection function. Based on this, we povide pecise pedictions fo the two poblems. Besides we also study how the effectiveness affects the final epidemic size in this case. Geneal infection function Fo a geneal function of infection isk f (y), when ef (ex)c{ Vv0 is valid fo all x lying between zeo and one, no one would take vaccination in the long un, i.e. x ~0 is the unique stable equilibium fo Eq. (3). Since f (y) is a deceasing function, ef (0)C{Vv0 is sufficient to ensue ef (ex)c{vv0. In analogy to this, when ef (e)c{vw0 is valid, the entie population ends up with full vaccination, i.e. x ~1 is the unique stable equilibium. Fo e fulfilling ef (0)C{Vw0 and ef (e)c{vv0, by the monotonicity of ef (y)c{v in y, thee is a unique intenal equilibium, f {1 ( V x ~ ec ) : ð4þ e Futhe, ef (y){(v=c) is deceasing, the deivative at x, namely ef (x ), is negative. Thus x is stable, indicating the coexistence of the vaccinated and the unvaccinated. To show how x is affected by e equies the exact fom of the function of infection isk. We will addess it late. The effective level of vaccination eads y ~ex dr dt ~myzci{mr, whee m is the bith ate and equal to the motality ate (fo simplicity, we only conside constant population size), b is the tansmission ate, c is the ecovey ate, and y is the faction of effectively vaccinated individuals among newbons. Fom Eq. (7), we deive the basic epoduction atio :if ~b=(czm)ƒ1, the time deivative of I is negative, suggesting that the disease cannot pesist in the population. The equilibium state of the population consists of (S,I,R ), with S ~1=, I ~m½ (1{y){1Š=b and R ~1{S {I. By setting I ~0, we obtain the hed immunity needed to eadicate the disease, y h ~1{1=. Based on this stationay equilibium, we calculate the pobability that an unvaccinated individual gets infected in he life time. The waiting time to acquie infection follows an exponential distibutions with ate bi, and so does the waiting time to death but with ate m. Since infection and death ae two independent pocesses, the pobability that infection occus befoe death event is the elative atio of intensities, bi =(bi zm). This pobability gives the infection isk of an unvaccinated individual, namely, f (y)~1{1= ½ (1{y) Š which is a function of the population level of effective vaccine uptake y and holds fo 0ƒyvy h. When y h ƒyv1, f (y)~0, i.e. the disease will be eadicated povided the effective level of vaccination exceeds the citical point y h. Thus we have ð8þ 0 1{ 1 if 0ƒyv1 { 1 R f (y)~ B 0 0 y 1 { 1 : ð9þ ~f {1 ( V ): ð5þ ec By Eq. (5), y is an inceasing function of the effectiveness, e. In othe wods, the effectively vaccinated level always inceases with vaccine efficacy. This esult only equies that f (y) deceases with y. This is tue fo most, if not all, known infection functions [22,23]. Theefoe ou pedictions ae obust with espect to vaiations in specific infection functions. A specific infection function In ode to give pecise pedictions, we adopt a simple Susceptible-Infected-Recoveed (SIR) model with demogaphical effects as pesented in [7]. In this model, the population is divided into thee diffeent compatments: susceptible, who ae healthy but can catch the disease if exposed to infected individuals; Infective, who ae infected and can pass the disease on to othes; Recoveed, who ae ecoveed fom the infection and gain immunity against the disease. The time evolution of the population states is govened by the following equations ds dt ~m(1{y){bsi{ms, di dt ~bsi{ci{mi, ð6þ ð7þ Taking this specific infection function Eq. (9) into Eq. (3), we pesent the full dynamics analysis of the evolution of vaccination behavio in the long un (see Fig. 1). Let the atio of the vaccination cost vesus the infection cost V=C be v1. We have (Fo details, see Text S1) Case 1: when ƒ 1, all ae unvaccinated fo e [ (0,1). 1{ 1 Case 2: when 1{ vr 1 0ƒ( p 1{ ffiffi ) 2 ; if eƒ 1{ 1, all ae unvaccinated, othewise thee is a unique intenal stable equilibium x. 1 Case 3: when w( p 1{ ffiffi ) 2 ;ifeƒ 1{ 1, all ae unvaccinated, if 1{ 1 veƒe 1, thee is a unique intenal stable equilibium x,ife 1 veƒe 2, all ae vaccinated, if e 2ve, thee is a unique intenal stable equilibium x. 1{ 1 Whee x ~ (1{ e ) e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 0 (1{)2 {2 (1z)z1. 2, e 1,2 ~ 1z 2 { PLoS ONE 3 June 2011 Volume 6 Issue 6 e20577

4 Impefect Voluntay Vaccination Figue 1. The vaccination behavio on the basic epoductive atio and the effectiveness e. Hee ~0:1, whee ~V=C. See main text fo details. doi: /jounal.pone g001 Case 1 indicates that fo a mild epidemic, 1v ƒ 1 1{, vaccination behavio is impossible fo any vaccination effectiveness. Fo a moe seious epidemic, Case 2 shows, howeve, thee is an oveshooting of vaccine uptake: the coexistence of the vaccinated and the unvaccinated emeges as the effectiveness exceeds a theshold. Futhemoe, inteestingly, the incease in effectiveness does not always pomote the vaccination behavio (see the uppe panel of Fig. 2). Intuitively, fo the vaccinated, inceasing the vaccination effectiveness does educe the infection pobability. Fo the unvaccinated, howeve, this leads to that they ae potected by a even moe effective hed immunity. Thus inceasing the effectiveness of vaccination is beneficial both to the vaccinated and to the unvaccinated. The two stategies compete with each othe and the moe beneficial one is moe likely to spead though imitation. The esult shows, when the effectiveness is below the citical value, the moe beneficial one is the vaccinated. When it exceeds the citical value, the moe beneficial one is the unvaccinated. Mathematically, the nonmonotonicity of x on e is induced fom the non-monotonicity of f {1 V =e as discussed above. Fo an even moe seious ec epidemic, Case 3, the dynamics of the vaccination behavio is qualitatively identical to that of Case 2. Howeve, in contast with Case 2, full vaccination can be eached (see the uppe panel of Fig. 3). Besides the vaccination behavio, by taking Eq. (9) into Eq. (5), the effective vaccination fequency, y ~ex is given by 0 1{ 1 y (1{ if 0ƒ (e)~ e ) e v1{ 1 B : 0 e 1{ 1 Hence, the effective vaccination fequency inceases as the effectiveness inceases as pedicted (See the lowe panels of Figs. 2 and 3). Futhe, it is of inteest to investigate how the final epidemic size is influenced by the effectiveness of the vaccination. The final epidemic size I hee efes to the aveage faction of the infected individuals at the end of the epidemics. Fo the SIR model with vital dynamics discussed above, when the vaccine uptake eaches a stationay level x, the final epidemics size of the population is given by 0 m if 0ƒ I b(e{) e v1{ 1 (e)~ m( {1) b e 1{ 1 : ð11þ Theefoe, I is a deceasing function with e. That is to say, the moe effective the vaccination is, the smalle popotion is infected PLoS ONE 4 June 2011 Volume 6 Issue 6 e20577

5 Impefect Voluntay Vaccination Figue 2. Factions of the vaccinated and the effective vaccinated fo a disease with a modeate infectiveness. The uppe panel shows the stationay fequency of the vaccinated with espective to the effectiveness. No one takes vaccination until it is sufficiently efficient, ew0:45. Then the vaccine uptake level inceases with the effectiveness. When the effectiveness exceeds a theshold, e~0:65, howeve, the vaccination level deceases with the effectiveness. The lowe panel shows the stationay abundance of the effectively vaccinated individuals with espect to the effectiveness. It is shown the efficient vaccinated individual inceases with the effectiveness all the time. Thus the behavio of vaccination and the impact of the vaccination against epidemic ae not in ageement: fo high effectiveness, even though vaccination ate is deceasing, the numbe of effectively vaccinated individual inceases as the effectiveness e inceases. Hee ~3, ~0:3 satisfying 1 1{ vr 1 0ƒ( p 1{ ffiffi ) 2. doi: /jounal.pone g002 Figue 3. Factions of the vaccinated and the effective vaccinated fo a seious disease. The uppe panel shows the stationay fequency of the vaccinated with espective to the effectiveness. Compaed to Fig. (2), the whole population could take vaccination povided the effectiveness is modeate, 0:5vev0:6. The lowe panel indicates the stationay abundance of the effectively vaccinated individuals with espect to the effectiveness. Compaed with Fig. (2), the fequency of the effective vaccinated individual also inceases with the effectiveness, but it is highe than 1 that of Fig. (2). Hee ~5, ~0:3 satisfying w( p 1{ ffiffi ) 2. doi: /jounal.pone g003 PLoS ONE 5 June 2011 Volume 6 Issue 6 e20577

6 Impefect Voluntay Vaccination Figue 4. Final epidemic size I fo the flu and the measles. The final epidemic size hee is the aveage abundance of the infected individual in the long un. Fo both the flu and the measles, the final epidemic size deceases with the effectiveness. Hee m~1 is the bith ate of the population. Fo the measles, ~0:1, R~16 and b~3: ; Fo the flu ~0:001, R~2:5 and b~2: [6,7]. doi: /jounal.pone g004 eventually. In paticula, we find this is tue fo the flu and the measles (see Fig. 4). Discussion Voluntay vaccination is the pinciple stategy to contol epidemic outbeaks. Vaccination itself, howeve, is a social dilemma [8]. Evolutionay game theoy, which descibes the evolution of stategies in self-inteested individuals, is a poweful mathematical famewok to study such social dilemmas. Most pevious woks employing this famewok ae based on the assumption of pefect vaccination, whee epidemics can be eadicated fom the vaccinated. The vaccination, howeve, cannot be so effective [27,28,29]. Theefoe it is of inteest to ask how the effectiveness of the vaccination has an impact on the vaccination. To this end, we combine the SIR model with the imitation dynamics. Fo the speading of disease, we find that inceasing the effectiveness of vaccination always inhibits the pevalence of epidemics. Theefoe impefect vaccine aggavates the longstanding dilemma of voluntay vaccination. Thus to contol the Refeences 1. Colgave J (2006) State of immunity: the politics of vaccination in twentiethcentuy Ameica. Bekeley, CA: Univesity of Califonia Pess. 2. Fine P, Clakson J (1986) Individual vesus public pioities in the detemination of optimal vaccination polices. Am J Epidemiol 124: Bauch CT, Galvani AP, Ean DJD (2003) Goup inteest vesus self inteest in smallpox vaccination policy. Poc Natl Acad Sci USA 100: Jansen VA, Stollenwek N, Jensen HJ, Ramsay ME, Edmunds W, et al. 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Fo the vaccination behavio, we find that when the epidemic is sufficiently seious, all the self-inteested individuals may take vaccination fo an intemediate vaccine efficacy. In othe wods, inceasing effectiveness inhibits the pevalence of the epidemic with a declining vaccination level. Fo example when e is lage than 0:7 in Fig. 2 and lage than 0:6 in Fig. 3. This suggests even though the vaccination level deceases with effectiveness sometime, the epidemic is still bette contolled than befoe, thus it is not necessay to be panic. Besides, all the above esults ae obust to geneal imitation pocesses [34]. Hee we study the simplest possible case, i.e., well-mixed populations, fo poof of pinciple. A natual extension of the pesent analysis is to take population stuctue into account. Fo instance, we can conside spatial stuctue, which esticts the neighbohood of individuals whom one can infect o imitate. In doing so, howeve, the evolutionay dynamics of vaccination behavio become moe complex and equie sepaate, in-depth studies. In essence, the vaccination game is simila to the wellstudied snowdift game [40]. Theefoe, spatial stuctue acts as a double-edged swod [8]. In paticula, spatial stuctue pomotes vaccination behavio fo small vaccination costs, and thus we expect that the citical efficacy of vaccination above which vaccination behavio pesists should be smalle compaed to the well-mixed case. These extensions ae pomising aeas fo futue eseach. Suppoting Infomation Text S1 (PDF) Dynamics analysis. Acknowledgments We thank D. Zhou and J. Wang fo comments on an ealy vesion of this manuscipt. Autho Contibutions Conceived and designed the expeiments: BW FF LW. Pefomed the expeiments: BW FF LW. Analyzed the data: BW FF LW. Contibuted eagents/mateials/analysis tools: BW FF LW. Wote the pape: BW FF LW. 10. Taulsen A, Nowak MA (2007) Chomodynamics of coopeation in finite populations. 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8 Repoduced with pemission of the copyight owne. Futhe epoduction pohibited without pemission.