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Conceived and designed the experiments: CTB. Performed the experiments: CTB SB. Analyzed the data: CTB SB. Contributed reagents/materials/analysis tools: CTB SB. Wrote the paper: CTB SB.

The authors have declared that no competing interests exist.

Immunization programs have often been impeded by vaccine scares, as evidenced by the measles-mumps-rubella (MMR) autism vaccine scare in Britain. A “free rider” effect may be partly responsible: vaccine-generated herd immunity can reduce disease incidence to such low levels that real or imagined vaccine risks appear large in comparison, causing individuals to cease vaccinating. This implies a feedback loop between disease prevalence and strategic individual vaccinating behavior. Here, we analyze a model based on evolutionary game theory that captures this feedback in the context of vaccine scares, and that also includes social learning. Vaccine risk perception evolves over time according to an exogenously imposed curve. We test the model against vaccine coverage data and disease incidence data from two vaccine scares in England & Wales: the whole cell pertussis vaccine scare and the MMR vaccine scare. The model fits vaccine coverage data from both vaccine scares relatively well. Moreover, the model can explain the vaccine coverage data more parsimoniously than most competing models without social learning and/or feedback (hence, adding social learning and feedback to a vaccine scare model improves model fit with little or no parsimony penalty). Under some circumstances, the model can predict future vaccine coverage and disease incidence—up to 10 years in advance in the case of pertussis—including specific qualitative features of the dynamics, such as future incidence peaks and undulations in vaccine coverage due to the population's response to changing disease incidence. Vaccine scares could become more common as eradication goals are approached for more vaccine-preventable diseases. Such models could help us predict how vaccine scares might unfold and assist mitigation efforts.

“Herd immunity” is a phenomenon whereby an entire population—including unvaccinated individuals—can be protected from infection by vaccinating only a certain percentage of the population. This suggests that immunization programs can be victims of their own success: past vaccinations can drive disease incidence to such low levels that as-yet unvaccinated individuals feel no incentive to get vaccinated, which creates conditions for future outbreaks. “Behavior-incidence” models capture this interplay between disease dynamics and vaccinating behavior. However, the predictive and explanatory value of these models is rarely tested against empirical data, and it is not clear whether the implied strategic interaction between individuals drives vaccinating behavior in real populations. Here we develop a behavior-incidence model based on evolutionary game theory and social learning. We show it often explains vaccine coverage data during a vaccine scare better than most competing models without strategic interactions and/or social learning. It can also predict future vaccine coverage and disease incidence peaks to a significant extent. Thus, strategic interactions between individuals via herd immunity appear to be a significant driver of behavior during a vaccine scare. It may be possible to harness behavior-incidence models to predict how future vaccine scares might unfold and possibly also to mitigate them.

Vaccine coverage in England & Wales during the whole cell pertussis vaccine scare in the 1970s and the measles-mumps-rubella (MMR) vaccine scare in the 1990s share a common pattern of decline and recovery over many years (

Media reports of alleged vaccine risks began in 1974 for pertussis and 1998 for MMR

Theory suggests that vaccine scares exemplify a “free-rider problem”: vaccine-generated herd immunity can reduce disease incidence to such low levels that vaccine risks appear large in comparison, causing some individuals to cease vaccinating. Hence, these non-vaccinators effectively “free ride” on the herd immunity generated by vaccinators. Game theory analyzes situations where the outcome of an individual's choice depends on the choices made by other individuals. Thus, game theory can be used to analyze free-rider problems such as vaccine scares. A growing literature combines mathematical models of disease transmission with game theory or other behavioral models to explore the feedback loop that connects disease incidence and vaccinating behavior among individuals: disease incidence influences vaccinating behavior through individuals wanting to avoid health risks, and vaccinating behavior in turn influences disease incidence through herd immunity generated by vaccination

A crucial assumption of these “behavior-incidence” models is that disease incidence feeds back on vaccinating behavior: a surge in disease incidence can convince individuals to start being vaccinated again. However, it is not immediately clear whether feedback is necessary to explain the time series of vaccine coverage in

In both vaccine scares, the publication of alleged vaccine risks was followed by a media firestorm in national newspapers, television, and radio

For significant parts of many historical vaccine coverage time series, vaccine coverage is roughly constant if a vaccine scare is not occurring. It is relatively easy to make behavior-incidence models reproduce constant vaccine coverage because there are sufficient degrees of freedom in parameter space

We tested the behavior-incidence model in two stages. In stage one, we tested just the explanatory power of the behavioral component of the model on its own: we formulated a behavioral model based on a social learning process where vaccinating behavior depends on the disease incidence, and where disease incidence comes from the empirical data rather than being generated by a model. In stage two, we tested both the explanatory and predictive power of the full behavior-incidence model: we formulated a mathematical model of disease transmission and connected it to the behavioral model by making vaccinating behavior depend on disease incidence generated by the transmission model.

In stage one, we formulated a social learning process based on the imitation dynamic of evolutionary game theory _{v} is the penalty to vaccinate, _{i} is the penalty for becoming infected, _{v}+_{i}_{i}m_{v}/_{i}_{v} which, unlike other parameters, evolves over time as the perceived vaccination penalty changes during the vaccine scare.

We wanted to determine whether adding social learning and feedback in this way to some underlying description of how the perceived vaccination penalty evolves over time can better explain _{pre} until the vaccine scare, then climbs linearly for D_{increase} years to reach a maximum of σω_{pre} (where σ>1) and remains there for D_{max} years before declining linearly back to ω_{pre} over a period of D_{decrease} years. We explored five possible shapes for ω(t):

Curve #1: instantaneous increase in perceived vaccine risk followed by linear decline: set D_{increase} = D_{max} = 0 and fit ω_{pre}, σ, D_{decrease};

Curve #2: instantaneous increase followed by plateau followed by instantaneous decline: set D_{increase} = D_{decrease} = 0 and fit ω_{pre}, σ, D_{max};

Curve #3: instantaneous increase followed by plateau followed by linear decline: set D_{increase} = 0 and fit ω_{pre}, σ, D_{decrease}, D_{max};

Curve #4: linear increase followed by plateau followed by instantaneous decline: set D_{decrease} = 0 and fit ω_{pre}, σ, D_{increase}, D_{max};

Curve #5: linear increase followed by plateau followed by linear decline: fit ω_{pre}, σ, D_{decrease}, D_{increase} , D_{max}.

A diagram of ω(t) appears in Supporting Information (

These curves were not motivated by a specific mechanistic model of risk perception, but rather were intended to describe a wide range of possible functional forms requiring differing numbers of parameters, thus enabling the explanatory power of the behavioral model to be tested against a broad range of potential competing candidates, as opposed to a single candidate. Public health efforts to restore faith in a safe and efficacious vaccine are represented as the eventual decline in perceived vaccine risk in these risk evolution curves.

For each curve, we compared the parsimony (explanatory power) of the behavioral model with both social learning and feedback—Equation (2)—to three reduced behavioral models with: (a) social learning but no feedback:

We used the AICc—a modified Akaike Information Criterion

In stage 2, we evaluated the parsimony of the full behavior-incidence model. We augmented our behavioral model with a Susceptible-Infectious-Recovered (SIR) compartmental model that captures disease transmission processes. Despite their simplicity, similar models have been shown to capture pertussis and measles dynamics relatively well

The design of the parsimony analysis for the behavior-incidence model was similar to that of the behavioral model (see Supporting Information,

In stage 2 we also tested the predictive power of the behavior-incidence model, under risk evolution curve #1. The slope of curve #1 is fixed at the start of the scare and does not change thereafter. This allowed us to fit the behavior-prevalence model under curve #1 to the early data points on both vaccine coverage and disease incidence in _{fit}), to see whether it can predict later data on vaccine coverage and disease incidence (_{fit}>

We analyzed both the whole cell pertussis vaccine scare and the MMR vaccine scare. For pertussis, the behavioral model with social learning and feedback fit the vaccine coverage data quite well under all risk evolution curves (

Red lines are 50 bootstrapped samples. Numerical values in subpanels are AICc scores: lower values indicate greater parsimony.

We repeated the parsimony analysis using the full behavior-incidence model, finding some further improvement in fit and parsimony relative to the three reduced behavioral models. For pertussis, the behavior-incidence model again achieves a better AICc score in all cases except for the model with neither social learning nor feedback under curve #5 (Supporting Information

The numerical value in the figure inset is the AICc score.

By comparing the fit of the behavior-incidence model to the fit of the reduced model with neither social learning nor feedback, under curve #1 for MMR (

The results for the reduced model with feedback but no social learning are also telling (Supporting Information

In principle, a good fit could occur because the model is underdetermined: there are too many parameters for the amount of available data and thus the model is able to fit any arbitrary pattern by adjusting the parameter values appropriately. To rule out this possibility, we also fitted the model to randomly generated time series (correlated white noise) for the case of MMR. If the model were underdetermined, then the model should also be able to fit these arbitrary time series. In Supporting Information

In stage 2 we also evaluated the predictive power of the behavior-incidence model by fitting the model to the first part of vaccine coverage and disease incidence time series (_{fit}) to see how well it predicts the second part (_{fit}). For pertussis, the model has little predictive power in the first few years of the scare: the best-fitting solution fails to capture the long-term dynamics of either vaccine coverage or disease dynamics, and the sampled realizations of the PSA are likewise inaccurate and widely scattered (_{fit} = 1973; _{fit} = 1977, _{fit} = 1978;

Best fitting model (blue dots), 50 realizations from PSA (red lines), vaccine coverage and disease incidence data used to fit model (_{fit}; thick black lines), and data used to evaluate model predictions (_{fit}; dashed black lines) are shown.

The model also qualitatively captures the subtle undulations in vaccine coverage between 1982 and 1987 that are superimposed on the longer-term trend (_{fit} values). However, even when _{fit} = 1988 and the whole time series is used to fit the model, it continues to over-predict the magnitude of the first incidence peak; this may be partly explained by under-reporting of pertussis incidence in the early years of the vaccine scare when misdiagnosis would have been more likely. The model also places the first incidence peak in 1975, instead of 1974 when it actually occurred.

In the years preceding the time window shown in

The results are qualitatively similar under the bootstrapping analysis: the bootstrapped predictions change abruptly in 1978, generating coherent and accurate predictions through 1988 (Supporting Information _{fit} = 1988), from the bootstrapping analysis we estimate that σ = 27 (95% CI: 19, 35), corresponding to a 27-fold increase in the perceived vaccine risk at the start of the vaccine scare. Other confidence intervals and best-fitting parameter values appear in Supporting Information

Predicting behavior-incidence dynamics during the MMR vaccine scare is more challenging. Vaccine coverage declined less. Measles did not become endemic until 2008 _{fit} values). This appears to be stimulated by an unmistakable rebound of vaccine coverage, rather than by incidence peaks. Despite this limitation, by 2005, the model predicts vaccine coverage in 2009 relatively well. It also captures qualitatively the subtle undulations caused by feedback—the sudden deceleration of coverage in 2006–2007 and the subsequent acceleration in 2008–2009. Bootstrapping again yields similar results to PSA (Supporting Information

Best fitting model (blue dots), 50 realizations from PSA (red lines), vaccine coverage and disease incidence data used to fit model (_{fit}; thick black lines), and data used to evaluate model predictions (_{fit}; dashed black lines) are shown.

Using the whole time series to fit the model (_{fit} = 2009), from the bootstrapping analysis we estimate that σ = 3.9 (95% CI: 3.1, 4.6), corresponding to a 4-fold increase in the perceived vaccine risk at the start of the vaccine scare. This value is much less than the 27-fold increase estimated for pertussis. For the delay δ, we estimate a biologically plausible value of 1.2 years (95% CI: 0.6, 1.8). The main effect of δ is to improve model fit by allowing peaks in the incidence data to stimulate correctly timed surges in the vaccine coverage data. If the delay is fixed at δ = 0, the alignment becomes worse. Other confidence intervals and best-fitting parameter values appear in Supporting Information

In both vaccine scares, the fit to vaccine coverage is better than the fit to disease incidence data. This occurs because individuals weigh both infection risks and vaccine risks in their vaccinating decisions (Equation (6)), therefore vaccine coverage is determined both by disease incidence feedback and by the risk evolution curve. As a result, if the transmission model over-predicts incidence in some part of the time series, vaccine coverage can still be made to fit well by increasing the perception of vaccine risk during the same time period, such that an increase in the prevalence of infection is balanced by an increase in the perception of vaccine risk. For instance, in the first six years of the pertussis scare (where the model over-predicts the size of the incidence peak relative to subsequent incidence peaks), this can be accomplished by increasing the value of σ such that perceived risk jumps more significantly at the start of the vaccine scare. For risk evolution curve #1, this also elevates perceived vaccine risk later on in the time series, but not as much since vaccine risk tends to return to baseline over time and therefore the resulting incremental change in vaccine risk is smaller during the later years of the vaccine scare. Something similar can be said of MMR, which is why the timing of the incidence peaks appears to be more important for model fit than the relative size of incidence peaks.

Here we analyzed a relatively simple mathematical model of behavior-incidence dynamics. The model was based on evolutionary game theory, included both social learning and feedback of disease incidence on vaccinating behavior, and also included an exogenous description of how perceived vaccine risk evolves during a vaccine scare. We showed that the behavior-incidence model explains vaccine coverage data more parsimoniously than most reduced models with the same risk evolution curve but without social learning and/or feedback. More interestingly, in some circumstances, the behavior-incidence model can predict future vaccination coverage and disease incidence in a population where a vaccine scare has taken hold. These results suggest that strategic (game theoretical) interactions between individuals and social learning may be crucial governing mechanisms of the population response to a vaccine scare, in addition to changes in subjective vaccine risk perception.

The models with both social learning and feedback (both the behavioral model and the behavior-incidence model) were significantly more parsimonious than most other candidates. The exception was the reduced model with neither social learning nor feedback under curve #5, which did better in 3 of the 4 comparisons. In some sense, our experimental design “stacks the cards” against the behavior-incidence model: by adding a sufficient number of free parameters to the risk evolution curve it will always be possible to achieve an arbitrarily good AICc score without adding social learning or feedback (see Supporting Information,

Considering these issues, it may not be appropriate to interpret our results in terms of a classical model selection exercise (where the model with the best AICc score is adopted). Additionally, we have little idea of how perceived vaccine risk actually evolved during these vaccine scares and hence it is difficult to construct a mechanistic risk evolution model in the first place, which makes a true model comparison elusive. Because of the apparent difficulties in teasing out the effects of the inherent dynamics of a vaccine scare from those of social learning and feedback, we refrain from interpreting our results as a classical model selection exercise. Rather, we choose to emphasize that a theoretically motivated approach consistent with human behavior improves model fit with little or no parsimony penalty, even when the underlying risk evolution curve is very crude (such as curves #1–#4).

Adding layers of sophistication to the model by including serious outcomes, combination versus single vaccines, age structure, spatial structure, or stochasticity may further improve the model's predictive power. These aspects represent opportunities for future work. Likewise, introducing a mechanistic model of how risk perception evolves instead of imposing risk evolution curves is worth pursuing, particularly in light of the interpretation caveats described in the previous paragraph. For example, this could take the form of a more mechanistic description of the impact of public health efforts such as information campaigns. However, the parsimony and predictive power of the model even without these extensions is considerable, and may be attributable to tight coupling between vaccinating behavior and disease incidence.

This research illustrates the importance of choosing the right transmission model when constructing a behavior-incidence model. Whooping cough incidence during the whole cell pertussis vaccine scare entered the regime of deterministic dynamics (widespread and unbroken chains of transmission), meaning that a simple, deterministic SIR model could capture the incidence peaks relatively well. However, measles incidence during the MMR scare was in a highly stochastic regime for most of the vaccine scare, which may explain the worse fit of the deterministic SIR model in that case.

A significant model limitation is the necessity to choose a weight governing how much the overall goodness of fit is determined by model fit to vaccine coverage versus the model fit to disease incidence. In the case of MMR, the fit to disease incidence was not weighted very strongly, on account of the poor ability of the deterministic model to fit stochastic disease dynamics. When model fit to both incidence and vaccine coverage is good, then the choice of _{fit}, which means the predictive capacity of the model is less.

Another model limitation is that, in the predictive analysis, the behavioral model is ‘trained’ on modeled incidence for _{fit}, rather than on actual incidence. This amounts to assuming that individuals were making vaccinating decisions based on modeled incidence, rather than on the incidence dynamics that the population actually experienced. One way to avoid this would be to fit the behavioral model to historical incidence data (_{fit}) and then rely on modeled incidence data for projections into the future (_{fit}). However, there are technical difficulties arising from the switch at _{fit} that would make this approach problematic. In particular, because of under-reporting in the empirical data, it would be easy to ‘confuse’ the behavioral model by switching its dependence from empirical incidence data to modeled incidence data at _{fit}. Moreover this model limitation is not a problem if agreement between modeled and empirical incidence is sufficiently close. Hence, ideally, it is better to train the behavioral model on modeled incidence for _{fit}. In any case, the issue of how to design good tests of the predictive ability of behavior-incidence models requires more thought.

The model cannot predict when a vaccine scare will occur since this presumably depends on singular historical events, such as publication of a study linking a vaccine to health risks. The model also requires data from the first years of a vaccine scare to predict subsequent years. In our analysis, we fitted the parameter σ that determines how much the vaccine penalty jumps when the scare starts. The predictive power of the model could increase if σ were known from the start. This is possible in principle, since it could be estimated from population surveys after a vaccine scare begins. This also represents opportunity for future work.

In 2003, polio was on the verge of global eradication when a vaccine scare in northern Nigeria caused an international resurgence of the disease

Schematic diagram of risk evolution curves.

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Parsimony analysis of the four behavioral models (horizontal dimension) under five evolution curves (vertical dimension) for pertussis vaccine scare. Solid black line is whole cell pertussis vaccine coverage. Dashed blue line is best fit of model to data. Red lines are bootstrapped fits. Numerical values in inset are AICc values of the best-fitting model.

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Parsimony analysis of the four behavioral models (horizontal dimension) under five evolution curves (vertical dimension) for MMR vaccine scare. Solid black line is MMR vaccine coverage. Dashed blue line is best fit of model to data. Red lines are bootstrapped fits. Numerical values in inset are AICc values of the best-fitting model.

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Parsimony analysis of behavior-incidence model, pertussis vaccine scare. Best fitting model (red) versus data (black) on whole cell pertussis vaccine uptake, for 5 risk evolution curves and 4 cases, using the behavior-incidence model. The numerical value in the inset of each subpanel is the corresponding AICc value for the fit. See page 2 for definition of risk evolution curves.

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Parsimony analysis of behavior-incidence model, MMR vaccine scare. Best fitting model (red) versus data (black) on MMR vaccine uptake, for 5 risk evolution curves and 4 cases, using the behavior-incidence model. The numerical value in the inset of each subpanel is the corresponding AICc value for the fit. See page 2 for definition of risk evolution curves.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #1. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #2. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #3. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #4. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #5. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #6. Also shown are goodness-of-fit and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009. Model was not fitted to vaccine coverage data using this risk evolution curve since a constant perceived vaccine penalty (curve #6) would correspond to no vaccine scare having occurred.

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Best fit of behaviour- incidence model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #1. Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour- incidence model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #2. Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour- incidence model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #3. Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour- incidence model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #4. Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour- incidence model (red) to MMR vaccine coverage data and 10 sets of correlated white noise data and (black), for risk evolution curve #5. Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal from 1995 to 2009.

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Best fit of behaviour-incidence model (red) to 10 sets of correlated white noise data and (black), for risk evolution curve #6 Also shown are log of maximum likelihood function and AICc of best fit (figure inset). Vertical scales range from 0.7 to 1.0; horizontal 1995 to 2009. Model was not fitted to vaccine coverage data using this risk evolution curve since a constant perceived vaccine penalty (curve #6) would correspond to no vaccine scare having occurred.

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PSA Results, Pertussis, _{fit} from 1975 to 1988. Solid black line represents vaccine coverage/incidence data for _{fit}; dashed black line represents vaccine coverage/incidence data for _{fit} (data from years _{fit} were used to fit model and produce model extrapolation to _{fit}); dotted blue line represents the best fit of model to data for given value of _{fit}; thin red lines represent 50 Monte Carlo samples for a given value of _{fit}.

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Bootstrapping Results for Pertussis, _{fit} from 1975 to 1988. Solid black line represents vaccine coverage/incidence data for _{fit}; dashed black line represents vaccine coverage/incidence data for _{fit} (data from years _{fit} were used to fit model and produce model extrapolation to _{fit}); dotted blue line represents the best fit of model to data for given value of _{fit}; thin red lines represent 50 bootstrap samples for a given value of _{fit}.

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PSA for MMR, _{fit} values from 1997 to 2009. Solid black line represents vaccine coverage/incidence data for _{fit}; dashed black line represents vaccine coverage/incidence data for _{fit} (data from years _{fit} were used to fit model and produce model extrapolation to _{fit}); dotted blue line represents the best fit of model to data for given value of _{fit}; thin red lines represent 50 Monte Carlo samples for a given value of _{fit}.

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Bootstrapping Results for MMR, _{fit} from 1997 to 2009. Solid black line represents vaccine coverage/incidence data for _{fit}; dashed black line represents vaccine coverage/incidence data for _{fit} (data from years _{fit} were used to fit model and produce model extrapolation to _{fit}); dotted blue line represents the best fit of model to data for given value of _{fit}; thin red lines represent 50 bootstrap samples for a given value of _{fit}.

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Confidence interval of fitted parameters for all 5 risk evolution curves models for the behavioral model with social learning and feedback, derived from bootstrapping.

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Fitting results for behavioral model with social learning and feedback under 5 risk evolution curves.

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Fitting results for behavioral model with social learning but no feedback under 5 risk evolution curves.

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Fitting results for behavioral model with feedback but no social learning under 5 risk evolution curves.

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Fitting results for behavioral model with no feedback and no social learning under 5 risk evolution curves.

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Estimated parameter values from bootstrapping for behavior-incidence model for Pertussis. Values represent median (median −2 standard deviations, median +2 standard deviations) from 50 bootstrap samples.

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Estimated parameter values from bootstrapping for behavior-incidence model for MMR. Values represent median (median −2 standard deviations, median +2 standard deviations) from 50 bootstrap samples.

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Methods.

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The authors are grateful to Bryan Grenfell, Madhur Anand, and Timothy Reluga for interesting discussions. The authors also acknowledge the helpful comments of three anonymous reviewers.