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, Text S1
). At some point, enough parameters are added to allow a “naked” risk evolution curve to outperform the corresponding behavior-incidence model; in the current analysis that point was reached with curve #5 with its five free parameters. However, our risk evolution curves were intended to represent phenomenologically a broad range of potential competing models, and in practice it may not even be possible to construct a mechanistic risk evolution model that can track the data as closely as curve #5 does. For example, we note that a simple SIR-type rumor propagation model could not replicate the approximately linear decline and recovery in vaccine coverage seen in the case of pertussis.
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 w should not matter. Otherwise, knowing which value of w to choose requires experimentation with the data and therefore forces use of large values of tfit, 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 t
, 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 (t
) and then rely on modeled incidence data for projections into the future (t
). However, there are technical difficulties arising from the switch at t
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 t
. 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 t
. 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 
. Our results suggest that vaccine scares or other forms of “free riding” could become more common as eradication goals are approached for more vaccine-preventable diseases. Behavior-incidence models may help mitigate the impact of vaccine scares, and assist in planning the global eradication endgame against some infectious diseases.