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PLoS One. 2011; 6(6): e20577.
Published online Jun 8, 2011. doi:  10.1371/journal.pone.0020577
PMCID: PMC3110791
Imperfect Vaccine Aggravates the Long-Standing Dilemma of Voluntary Vaccination
Bin Wu,1* Feng Fu,2* and Long Wang1
1State Key Laboratory for Turbulence and Complex Systems, Center for Systems and Control, College of Engineering, Peking University, Beijing, China
2Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts, United States of America
Matjaz Perc, Editor
University of Maribor, Slovenia
* E-mail: bin.wu/at/evolbio.mpg.de (BW); fengfu/at/fas.harvard.edu (FF)
Conceived and designed the experiments: BW FF LW. Performed the experiments: BW FF LW. Analyzed the data: BW FF LW. Contributed reagents/materials/analysis tools: BW FF LW. Wrote the paper: BW FF LW.
Received April 5, 2011; Accepted May 4, 2011.
Achieving widespread population immunity by voluntary vaccination poses a major challenge for public health administration and practice. The situation is complicated even more by imperfect vaccines. How the vaccine efficacy affects individuals' vaccination behavior has yet to be fully answered. To address this issue, we combine a simple yet effective game theoretic model of vaccination behavior with an epidemiological process. Our analysis shows that, in a population of self-interested individuals, there exists an overshooting of vaccine uptake levels as the effectiveness of vaccination increases. Moreover, when the basic reproductive number, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e001.jpg, exceeds a certain threshold, all individuals opt for vaccination for an intermediate region of vaccine efficacy. We further show that increasing effectiveness of vaccination always increases the number of effectively vaccinated individuals and therefore attenuates the epidemic strain. The results suggest that ‘number is traded for efficiency’: although increases in vaccination effectiveness lead to uptake drops due to free-riding effects, the impact of the epidemic can be better mitigated.
Preemptive vaccination is the principle strategy for the intervention and control of infectious diseases. However, vaccination represents a long-standing social dilemma for public health administration. On the one hand, compulsory vaccination may result in an infringement of civil rights [1]. On the other hand, voluntary vaccination cannot lead to sufficiently high herd immunity for disease eradication. Thus it often fails to protect populations from epidemics [2], [3], [4], [5].
Traditional epidemiological modeling focuses on the pathway of disease transmission, and often does not take into account human strategic behavior in response to the epidemic [6]. However, it is more plausible to integrate human behavior with the epidemiological process. In this sense, voluntary vaccination itself is a social dilemma: vaccinated individuals can escape from the disease with a cost partly incurred by the vaccine side effects; the unvaccinated can also be protected from the epidemics without paying anything provided the population immunity is in effect. In this case, self-interested individuals attempt to shun vaccination while still benefitting from the herd immunity. Such free-riding may lead to a low vaccination level, failing to eradicate the disease, thus a social dilemma [7], [8]. The framework of game theory properly describes how individuals react when facing a dilemma [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. In particular, how the evolutionary outcome of the social dilemma is achieved can be investigated based on the imitation process [20], [21]. Therefore, voluntary vaccination can be studied in this framework and noteworthy there has been an emerging literature of combining epidemiology and game theory [7], [22], [23], [24], [5], [25], [8], [26].
Previous work usually assumes perfect vaccination, i.e., the vaccinated individuals gain perfect immunity against the disease [7], [23], [8]. The effectiveness of vaccination, however, is not An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e002.jpg, such as measles [27], malaria [28] and HIV [29]. Even though the actual vaccination is perfect, the perceived effectiveness can be not. Questionnaire results have shown the perceived effectiveness is often lower than the actual one [24]. This perceived efficacy of vaccination, influenced by psychological effects, plays a determinant role since individuals adjust their strategic behavior based on perceptions of the vaccine efficacy rather than the actual one [5], [24]. Therefore, imperfect vaccination should be taken into account in the game theoretical analysis of the vaccination behavior [30], [31], [32]. Besides, public concern towards the effectiveness of vaccine is so common that it often leads to massive vaccine avoidance. How vaccine effectiveness affects vaccination level and thus the severity of epidemic outbreak has not yet been fully answered. Motivated by these, we study this problem by a minimal model.
For proof of principle, we consider vaccination dynamics in an infinitely large well mixed population. In addition, we assume that individuals have a perfect knowledge on the effectiveness of the vaccination. In this case, there is only one parameter describing both the actual and the perceived effectiveness.
The vaccination game consists of two stages, the yearly vaccination campaign and an epidemic season. During the vaccination campaign, each individual decides whether or not to take vaccination. A vaccinated individual pays a cost An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e003.jpg while an unvaccinated individual pays nothing. This cost An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e004.jpg includes the time spent in taking the vaccination as well as its side effects. During the epidemic season, the population can be divided into two parts: one comprises effectively vaccinated individuals, and the rest is composed of unvaccinated individuals and the vaccinated ones whose vaccinations are not effective. Successfully vaccinated individuals are immune to the seasonal disease, and thus have no risk of getting infected. For the remaining individuals, however, they become infected with a probability An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e005.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e006.jpg is the frequency of effectively vaccinated individuals. In this case the infected bear a cost by An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e007.jpg. This cost An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e008.jpg includes expenses and time for health care as well as mortality. The larger the number of effectively vaccinated individuals is, the less likely an unvaccinated individual gets infected. Thus An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e009.jpg is decreasing with An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e010.jpg.
Let the effectiveness of the vaccination be An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e011.jpg and the vaccine uptake level be An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e012.jpg. The frequency of the effectively vaccinated individuals is An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e013.jpg. The fraction of the vaccinated and healthy individuals is An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e014.jpg, which is composed of two parts: these effectively vaccinated individuals (with frequency An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e015.jpg) and those ineffectively vaccinated individuals (with frequency An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e016.jpg) who are free from the infection (with frequency An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e017.jpg). In this case, each effectively vaccinated individual gets payoff An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e018.jpg. In analogy to this, the frequencies and payoffs for different individuals are given by Table 1.
Table 1
Table 1
The fraction and the payoff for the four types of individuals in the population.
When the epidemic season ends, i.e., the average abundance of infected individuals does not change, individuals adjust their strategies by imitation where successful individual's strategy is more likely to be followed [33], [34]. Here we employ the Fermi update rule to characterize such an imitation process [35], [36], [8], [37], [38]: two individuals An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e027.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e028.jpg are selected randomly; An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e029.jpg learns to behave like An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e030.jpg with probability
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e031.jpg
(1)
where An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e032.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e033.jpg are the perceived payoffs for An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e034.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e035.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e036.jpg is the selection intensity indicating how strongly individuals are responsive to payoff difference.
The dynamics of the vaccination is governed by [20], [39]
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e037.jpg
(2)
It has been suggested that the selection intensity for human imitation is rather weak [21], [34], i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e038.jpg is sufficiently small. We perform the Taylor expansion of the r.h.s of Eq. (2) in the vicinity of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e039.jpg, then after a time rescaling which does not change the dynamics, Eq. (2) can be captured by a much more simple form
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e040.jpg
(3)
In what follows, we investigate how the vaccine uptake evolves by Eq. (3) for general function of infection risk An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e041.jpg. To this end, we focus on how the effectiveness of vaccination has an impact on the the collective outcome of vaccination behavior and the effective vaccination level. Then we incorporate an epidemic dynamics to obtain a specific infection function. Based on this, we provide precise predictions for the two problems. Besides we also study how the effectiveness affects the final epidemic size in this case.
General infection function
For a general function of infection risk An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e042.jpg, when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e043.jpg is valid for all An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e044.jpg lying between zero and one, no one would take vaccination in the long run, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e045.jpg is the unique stable equilibrium for Eq. (3). Since An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e046.jpg is a decreasing function, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e047.jpg is sufficient to ensure An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e048.jpg. In analogy to this, when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e049.jpg is valid, the entire population ends up with full vaccination, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e050.jpg is the unique stable equilibrium. For An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e051.jpg fulfilling An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e052.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e053.jpg, by the monotonicity of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e054.jpg in An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e055.jpg, there is a unique internal equilibrium,
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e056.jpg
(4)
Further, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e057.jpg is decreasing, the derivative at An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e058.jpg, namely An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e059.jpg, is negative. Thus An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e060.jpg is stable, indicating the coexistence of the vaccinated and the unvaccinated. To show how An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e061.jpg is affected by An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e062.jpg requires the exact form of the function of infection risk. We will address it later.
The effective level of vaccination reads
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e063.jpg
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e064.jpg
(5)
By Eq. (5), An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e065.jpg is an increasing function of the effectiveness, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e066.jpg. In other words, the effectively vaccinated level always increases with vaccine efficacy. This result only requires that An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e067.jpg decreases with An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e068.jpg. This is true for most, if not all, known infection functions [22], [23]. Therefore our predictions are robust with respect to variations in specific infection functions.
A specific infection function
In order to give precise predictions, we adopt a simple Susceptible-Infected-Recovered (SIR) model with demographical effects as presented in [7]. In this model, the population is divided into three different compartments: susceptible, who are healthy but can catch the disease if exposed to infected individuals; Infective, who are infected and can pass the disease on to others; Recovered, who are recovered from the infection and gain immunity against the disease. The time evolution of the population states is governed by the following equations
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e069.jpg
(6)
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e070.jpg
(7)
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e071.jpg
(8)
where An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e072.jpg is the birth rate and equal to the mortality rate (for simplicity, we only consider constant population size), An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e073.jpg is the transmission rate, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e074.jpg is the recovery rate, and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e075.jpg is the fraction of effectively vaccinated individuals among newborns.
From Eq. (7), we derive the basic reproduction ratio An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e076.jpg: if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e077.jpg, the time derivative of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e078.jpg is negative, suggesting that the disease cannot persist in the population. The equilibrium state of the population consists of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e079.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e080.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e081.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e082.jpg. By setting An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e083.jpg, we obtain the herd immunity needed to eradicate the disease, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e084.jpg.
Based on this stationary equilibrium, we calculate the probability that an unvaccinated individual gets infected in her life time. The waiting time to acquire infection follows an exponential distributions with rate An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e085.jpg, and so does the waiting time to death but with rate An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e086.jpg. Since infection and death are two independent processes, the probability that infection occurs before death event is the relative ratio of intensities, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e087.jpg. This probability gives the infection risk of an unvaccinated individual, namely, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e088.jpg which is a function of the population level of effective vaccine uptake An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e089.jpg and holds for An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e090.jpg. When An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e091.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e092.jpg, i.e. the disease will be eradicated provided the effective level of vaccination exceeds the critical point An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e093.jpg. Thus we have
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e094.jpg
(9)
Taking this specific infection function Eq. (9) into Eq. (3), we present the full dynamics analysis of the evolution of vaccination behavior in the long run (see Fig. 1). Let the ratio of the vaccination cost versus the infection cost An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e095.jpg be An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e096.jpg. We have (For details, see Text S1)
Figure 1
Figure 1
The vaccination behavior on the basic reproductive ratio An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e097.jpg and the effectiveness An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e098.jpg.
Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e101.jpg: when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e102.jpg, all are unvaccinated for An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e103.jpg.
Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e104.jpg: when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e105.jpg; if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e106.jpg, all are unvaccinated, otherwise there is a unique internal stable equilibrium An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e107.jpg.
Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e108.jpg: when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e109.jpg; if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e110.jpg, all are unvaccinated, if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e111.jpg, there is a unique internal stable equilibrium An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e112.jpg, if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e113.jpg, all are vaccinated, if An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e114.jpg, there is a unique internal stable equilibrium An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e115.jpg.
Where An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e116.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e117.jpg.
Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e118.jpg indicates that for a mild epidemic, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e119.jpg, vaccination behavior is impossible for any vaccination effectiveness. For a more serious epidemic, Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e120.jpg shows, however, there is an overshooting of vaccine uptake: the coexistence of the vaccinated and the unvaccinated emerges as the effectiveness exceeds a threshold. Furthermore, interestingly, the increase in effectiveness does not always promote the vaccination behavior (see the upper panel of Fig. 2). Intuitively, for the vaccinated, increasing the vaccination effectiveness does reduce the infection probability. For the unvaccinated, however, this leads to that they are protected by a even more effective herd immunity. Thus increasing the effectiveness of vaccination is beneficial both to the vaccinated and to the unvaccinated. The two strategies compete with each other and the more beneficial one is more likely to spread through imitation. The result shows, when the effectiveness is below the critical value, the more beneficial one is the vaccinated. When it exceeds the critical value, the more beneficial one is the unvaccinated. Mathematically, the non-monotonicity of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e121.jpg on An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e122.jpg is induced from the non-monotonicity of An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e123.jpg as discussed above. For an even more serious epidemic, Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e124.jpg, the dynamics of the vaccination behavior is qualitatively identical to that of Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e125.jpg. However, in contrast with Case An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e126.jpg, full vaccination can be reached (see the upper panel of Fig. 3).
Figure 2
Figure 2
Fractions of the vaccinated and the effective vaccinated for a disease with a moderate infectiveness.
Figure 3
Figure 3
Fractions of the vaccinated and the effective vaccinated for a serious disease.
Besides the vaccination behavior, by taking Eq. (9) into Eq. (5), the effective vaccination frequency, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e137.jpg is given by
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e138.jpg
(10)
Hence, the effective vaccination frequency increases as the effectiveness increases as predicted (See the lower panels of Figs. 2 and and33).
Further, it is of interest to investigate how the final epidemic size is influenced by the effectiveness of the vaccination. The final epidemic size An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e139.jpg here refers to the average fraction of the infected individuals at the end of the epidemics. For the SIR model with vital dynamics discussed above, when the vaccine uptake reaches a stationary level An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e140.jpg, the final epidemics size of the population is given by
A mathematical equation, expression, or formula.
 Object name is pone.0020577.e141.jpg
(11)
Therefore, An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e142.jpg is a decreasing function with An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e143.jpg. That is to say, the more effective the vaccination is, the smaller proportion is infected eventually. In particular, we find this is true for the flu and the measles (see Fig. 4).
Figure 4
Figure 4
Final epidemic size An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e144.jpg for the flu and the measles.
Voluntary vaccination is the principle strategy to control epidemic outbreaks. Vaccination itself, however, is a social dilemma [8]. Evolutionary game theory, which describes the evolution of strategies in self-interested individuals, is a powerful mathematical framework to study such social dilemmas. Most previous works employing this framework are based on the assumption of perfect vaccination, where epidemics can be eradicated from the vaccinated. The vaccination, however, cannot be so effective [27], [28], [29]. Therefore it is of interest 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. For the spreading of disease, we find that increasing the effectiveness of vaccination always inhibits the prevalence of epidemics. Therefore imperfect vaccine aggravates the long-standing dilemma of voluntary vaccination. Thus to control the epidemics, i.e. to enhance the vaccination effectiveness, there are two ways: one is to improve technology in vaccine: increasing the actual effectiveness of the vaccination. The other is to make use of media: enhancing the perceived effectiveness.
For the vaccination behavior, we find that when the epidemic is sufficiently serious, all the self-interested individuals may take vaccination for an intermediate vaccine efficacy. In other words, increasing effectiveness inhibits the prevalence of the epidemic with a declining vaccination level. For example when An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e152.jpg is larger than An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e153.jpg in Fig. 2 and larger than An external file that holds a picture, illustration, etc.
Object name is pone.0020577.e154.jpg in Fig. 3. This suggests even though the vaccination level decreases with effectiveness sometime, the epidemic is still better controlled than before, thus it is not necessary to be panic. Besides, all the above results are robust to general imitation processes [34].
Here we study the simplest possible case, i.e., well-mixed populations, for proof of principle. A natural extension of the present analysis is to take population structure into account. For instance, we can consider spatial structure, which restricts the neighborhood of individuals whom one can infect or imitate. In doing so, however, the evolutionary dynamics of vaccination behavior become more complex and require separate, in-depth studies. In essence, the vaccination game is similar to the well-studied snowdrift game [40]. Therefore, spatial structure acts as a “double-edged sword” [8]. In particular, spatial structure promotes vaccination behavior for small vaccination costs, and thus we expect that the critical efficacy of vaccination above which vaccination behavior persists should be smaller compared to the well-mixed case. These extensions are promising areas for future research.
Supporting Information
Text S1
Dynamics analysis.
(PDF)
Acknowledgments
We thank D. Zhou and J. Wang for comments on an early version of this manuscript.
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: The authors acknowledge support by the National Natural Science Foundation of China Grants No. 10972002 and No. 60736022. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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