Behavioral model

In stage one, we formulated a social learning process based on the imitation dynamic of evolutionary game theory [18]. An individual samples others in the population at a constant rate. If the sampled person is playing a different strategy and is receiving a higher payoff, the individual switches to that strategy with a probability proportional to the expected gain in payoff. The payoff gain depends on the difference between the penalty for being vaccinated and the penalty for risking infection. In our model, individuals can choose to vaccinate, or not to vaccinate (“vaccinator” versus “non-vaccinator” strategies). The infection penalty is the perceived probability of being infected—which we assumed is simply proportional to the current disease incidence—times the perceived cost of being infected. In stage one, we simply took disease incidence directly from the data in Figure 1 instead of an incidence model, resulting in a “behavioral model” rather than a full behavior-incidence model. The resulting equations for the behavioral model are(1)where x is the proportion of vaccinators in the population at time t, s is the sampling rate, θ is the proportionality constant influencing the probability of switching strategies according to the expected gain in payoff, c_v is the penalty to vaccinate, c_i is the penalty for becoming infected, L is the number of case notifications at time t (taken from the data in Figure 1), and m is a proportionality constant governing the perceived probability of being infected (we note that for m and θ sufficiently small the relevant probabilities are always less than 1). The expression (−c_v+c_imL) is the payoff gain for switching strategies and its sign determines whether vaccinator or nonvaccinator is the favored switch. Equation (1) is derived in the Supporting Information (Text S1). Equation (1) can be further distilled to(2)where κ = sθc_im and ω = c_v/mc_i. This is the form of the model we use in the analysis. The parameter ω has absorbed c_v which, unlike other parameters, evolves over time as the perceived vaccination penalty changes during the vaccine scare.

Risk evolution curves

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 Figure 1. Hence, we formulated five risk evolution curves that govern how the perceived vaccination penalty could rise and fall during the scare. The function ω = ω(t) denotes the risk evolution curve describing time evolution of the vaccine penalty. ω(t) is constant at ω_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 (Figure S1).

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:(3)(b) feedback but no social learning:(4)where ρ is a proportionality constant, and (c) neither social learning nor feedback:(5)These equations are derived in Supporting Information (Text S1).

We used the AICc—a modified Akaike Information Criterion [19]–[22]—to evaluate the parsimony of the four models under all five risk evolution curves for each model, yielding 20 candidates altogether. Information criteria have a strong rooting in information theory, and favor models that explain the data as well as possible with as few parameters as possible. The model with the most negative AICc score is the one with the greatest parsimony, suggesting that it is likely capturing crucial determinants of the observed dynamics. We obtained confidence intervals using a non-parametric bootstrapping method. Additional details on model fitting and bootstrapping appear in the Supporting Information (Text S1).

Behavior-Incidence model

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 [23]–[25]. In the SIR model, individuals are either Susceptible, Infectious, or Recovered (immune). Susceptible individuals are infected at some rate and thereby moved to the Infectious compartment. From the Infectious compartment they recover at some rate and enter the Recovered compartment. Susceptible individuals who are efficaciously vaccinated are also moved to the Recovered compartment. Individuals are born into the Susceptible compartment at some rate, and leave the population due to death at some rate. For measles, the transmission rate was also made to vary seasonally [23]–[25]. For the behavioral component of the model, instead of making the perceived probability of being infected depend on the disease incidence data (L), it now depends on the disease prevalence generated by the SIR model (I). In turn, a proportion of infants are vaccinated according to the abundance of vaccinator strategists in the population at a given time (x), completing the feedback loop. The equations for the resulting behavior-incidence model are:(6)where μ is the birth/death rate per capita, ε is the vaccine efficacy, β is the transmission rate, τ is the case importation rate, and γ is the recovery rate. For measles, a delay was also introduced between changes in incidence and changes in vaccine coverage, to capture phenomenologically the fact many parents have opted to delay immunization rather than avoid it altogether. As a result, for measles the x equation becomes(7)where δ is the delay, in years. We opted to introduce a fixed delay in the equations rather than explicitly incorporate delayer strategies in order to keep the number of parameters relatively low.

The design of the parsimony analysis for the behavior-incidence model was similar to that of the behavioral model (see Supporting Information, Text S1). We note that the goodness of fit of the behavior-incidence model to disease incidence data does not, and cannot, contribute to the AICc score because in this respect there is no way to make a fair comparison between the behavior-incidence model (which is capable of predicting incidence) and the reduced models (two of which are not capable of predicting incidence, by definition).

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 Figure 1 (t≤t_fit), to see whether it can predict later data on vaccine coverage and disease incidence (t_fit>t). We fitted disease incidence and vaccine coverage simultaneously, by minimizing a weighted sum of the residual sum of squares (RSS) for vaccine coverage and the RSS for disease incidence. We also conducted a probabilistic sensitivity analysis (PSA) to assess the sensitivity of these predictions to parameter uncertainty. PSA defines plausible intervals for crucial model parameters and initial conditions. For each model realization, samples are drawn from statistical distributions based on those intervals and the model is fitted using those parameter values. Over many such realizations it is possible to see how sensitive the model predictions are to variations in the input parameters. A bootstrapping analysis was also performed to further test model sensitivity to input parameter uncertainties and to acquire confidence intervals. Details of fitting, PSA and bootstrapping appear in the Supporting Information (Text S1).

Stage one

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 (Figure 2). In comparison, the two reduced models with feedback but no social learning, and social learning but no feedback, produced poor fits and were much less parsimonious (Supporting Information Figure S2). The third reduced model with neither social learning nor feedback also did worse in terms of fit and parsimony, except under risk evolution curve #5 (Figure 2). Thus, on the whole, adding social learning and feedback significantly improved model parsimony and fit. Results were very similar for MMR, with the behavioral model doing better in all cases except for the reduced model with neither social learning nor feedback under curve #5 (Supporting Information Figure S3). Confidence intervals and best-fitting parameter values appear in Supporting Information Tables S1, S2, S3, S4, S5. We discuss the significance of the reduced model with neither social learning nor feedback under curve #5 in the Discussion section.

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Figure 2. Parsimony analysis of behavioral model for the pertussis vaccine scare: vaccine coverage data (solid black line) and best-fitting model (dashed blue line) under behavioral model with both social learning and feedback (a, c, e, g, i) and reduced behavioral model with neither (b, d, f, h, j), under risk evolution curves #1–#5 (left-hand column).

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

https://doi.org/10.1371/journal.pcbi.1002452.g002

Stage two

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 Figure S4). In contrast, for the case of MMR, the behavior-incidence model under curve #1 becomes the most parsimonious of all 20 candidates. Interestingly, its best-fitting solution for vaccine coverage is almost indistinguishable from the data for most of the vaccine scare (Figure 3; see Supporting Information Figures S5 for full results).

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Figure 3. Parsimony analysis of behavior-incidence model and the reduced model with neither social learning nor feedback under risk evolution curves #1–#5 (left-hand column), for the MMR vaccine scare: best fitting model (red) and vaccine coverage data (black).

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

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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 (Figure 3), we can see the effects of adding social learning and feedback to an underlying model of risk perception evolution: social learning delays the trough in vaccine coverage (an effect that is also observed in the data, where vaccine coverage bottoms out many years after the hypothesized link between MMR vaccine and autism was published), and feedback allows the model to capture the undulations in vaccine coverage observed in the data, which appear to be caused by entrainment of vaccine coverage dynamics with disease dynamics.

The results for the reduced model with feedback but no social learning are also telling (Supporting Information Figure S4): although the overall trend in vaccine coverage is tracked approximately, the predicted vaccine coverage is too irregular because without the inertial effects of an imitation dynamic, vaccine coverage responds too rapidly to slight changes in infection prevalence and the coupled behavior-incidence dynamics become unstable.

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 Figures S6, S7, S8, S9, S10, S11, S12, S13, S14, S15, S16, S17, we show that the model fits to these arbitrary time series are worse than its fit to the empirical vaccine coverage data in Figure 3, and thus the model may not be underdetermined.

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 (t≤t_fit) to see how well it predicts the second part (t>t_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 (t_fit = 1973; Figure 4a, b). This situation remains unchanged through 1977 (t_fit = 1977, Figure 4c, d). However, in 1978, the first large incidence peak occurs, resulting in an abrupt increase in predictive power: now, the best-fitting solution predicts both future vaccine coverage and disease dynamics fairly well up until 1988, and the sampled realizations of the PSA converge around future data points (t_fit = 1978; Figure 4e, f). Hence, the 1978 incidence peak acts to provide information that collapses model uncertainty, enabling reasonably accurate long-term predictions. This occurs despite the fact that—based on information available in 1978—it would not have been clear whether vaccine coverage had actually bottomed out or how quickly vaccine coverage would rebound.

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Figure 4. Predictive analysis of behavior-incidence model for the pertussis vaccine scare: predictions up until 1988 were made using data up until t_fit = 1975 (a, b); 1977 (c, d), 1978 (e, f) and 1988 (g, h) for both vaccine coverage (a, c, e, g) and case notifications (b, d, f, h).

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

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The model also qualitatively captures the subtle undulations in vaccine coverage between 1982 and 1987 that are superimposed on the longer-term trend (Figure 4e): in both model and data, two incidence peaks occur during this time period, each followed shortly thereafter by a surge in vaccine coverage. However, the amplitude of the surges is larger in the model than in the data, and the first surge is predicted to occur a year before it actually happened. From the incidence plot (Figure 4f) we see that the model predicts the first incidence peak a year too soon, which is what causes the model to predict the first vaccine coverage a year too soon as well. This suggests that using a slightly more sophisticated transmission model might result in better alignment of predicted and observed vaccine coverage surges. These simulations highlight the fact that vaccine coverage in the data surges at exactly the time it should, if vaccine coverage were partly driven by disease dynamics. We also note that the ability of the model to track subtle undulations is responsible for much of the model's AICc score, especially in the case of MMR (Figure 3). From 1978 onward, model predictions are gradually refined and the vaccine coverage undulations become better aligned (Figure 4g, h; see Supporting Information Figure S18 for all t_fit values). However, even when t_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 Figure 4, the modeled vaccine coverage is close to a steady state. The modeled vaccine coverage returns to this steady state after the scare is finished. This pattern is also observed in the vaccine coverage data. However, given that the whole cell pertussis vaccine was replaced with an acellular vaccine in the early 1990s, vaccine coverage data from this time period cannot be used to validate the model.

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 Figure S19). Using the whole time series to fit the model (t_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 Table S6.

Predicting behavior-incidence dynamics during the MMR vaccine scare is more challenging. Vaccine coverage declined less. Measles did not become endemic until 2008 [1], so there is a lower volume of lab-confirmed cases with which to parameterize the model, and no large epidemic outbreaks until later. As a result, measles dynamics are highly stochastic until 2008, meaning that deterministic models such as the SIR model are less suited to describing this phase of the vaccine scare. Perhaps as a result of this, the model does not develop good predictive power until 2005 (Figure 5; see Supporting Information Figure S20 for all t_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 Figure S21).

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Figure 5. Predictive analysis of behavior-incidence model for the MMR vaccine scare: predictions up until 2009 were made using data up until t_fit = 2000 (a, b); 2004 (c, d), 2005 (e, f) and 2009 (g, h) for both vaccine coverage (a, c, e, g) and case notifications (b, d, f, h).

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

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Using the whole time series to fit the model (t_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 Table S7.

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.