SIAM News Blog

Understanding Vaccination Dilemma through the Lens of Evolutionary Game Theory

By Xingru Chen and Feng Fu

Voluntary mass vaccination is a fundamental measure in achieving widespread herd immunity and preventing significant disease outbreaks. In recent years, however, the world has seen a resurgence of vaccine-preventable childhood diseases, such as pertussis (whooping cough) and measles. The measles outbreak at Disneyland in 2015 particularly caught nationwide media attention. Moreover, measles has recently reached the very verge of an endemic disease in France. These incidences of outbreak indicate that a lack of vaccine compliance is preventing achievement of the vaccination levels needed to maintain herd immunity. Those who intentionally skip vaccinations are prone to the risks of disease infection, and can trigger severe epidemic outbreaks that are rather costly to control.

Game theory is a powerful paradigm for the mathematical analysis of strategic interactions among individuals with competing interests. Evolutionary game theory—pioneered by John Maynard Smith and others—is the application of game theory to biology, with a focus on population dynamics of evolving strategies. An individual’s vaccination contributes to herd immunity, meaning these who forgo vaccination can be indirectly protected by the presence of herd immunity. The problem of vaccine compliance is thus often represented as a public-goods dilemma. A misalignment between individual self-interest and population interest can yield the “free rider” problem in vaccine uptake, thereby causing suboptimal vaccination coverage.

In light of this, we have combined evolutionary game theory models of vaccination behavior with an epidemiological process. We focused on exploring the role of social imitation, or peer influence, in individual vaccination decisions and its consequential impact on population health outcomes. This opens a new door for applications of evolutionary game theory to help understand public health behavior — in particular, the social dilemma of vaccination.

For example, we calculate the evolutionarily stable strategy (ESS) of vaccination, which represents the optimum strategy of individuals who cannot improve their own payoffs by unilaterally deviating from this equilibrium. The ESS is given by


where \(c\) is the relative cost of vaccination to infection \(0<c<1\), and \(R_0\) denotes the basic reproductive ratio of the disease — the average number of secondary infections caused by one single infected individual in a susceptible population. \(R_0\) is an important epidemiological parameter that quantifies the severity of the disease and its potential impact on the population. For measles, \(R_0\) is in range 11-18. For influenza, \(R_0\) is typically 2-3.

Only at \(c = 0\) does the ESS of vaccination coincide with the social optimum of vaccination coverage, \(x_h=1- 1/R_0\). \(x_h\) also stands for the herd immunity threshold, namely the minimum vaccination coverage needed to achieve herd immunity so the disease cannot spread. When \(c>0\), a coverage gap inevitably occurs between individual and social optimum, thus resulting in the social dilemma of voluntary vaccination.

Figure 1. Double-edged effect of social networks on vaccination behavior. The spread of vaccine compliance, or the opposite vaccine avoidance, via social networks is sensitive to the perceived risks or costs of vaccination. Node size is proportional to its degree, blue denotes vaccinated, yellow denotes unvaccinated but healthy, and red denotes unvaccinated and infected.
Understanding the impact of social networks on public health behavior and vaccination choices, for example, is of particular interest. A vaccine’s success can become its own demise. Once the incidence of vaccine-preventable common childhood diseases becomes rare, parents who are unfamiliar with the diseases pay more attention to concerns regarding the risks of vaccination rather than the disease itself. This leads to vaccine scares or skepticisms that can hinder vaccination efforts. As shown in Figure 1, we study vaccination dynamics in scale-free networks known to be highly heterogeneous. Our work indicates that social networks can have a double-edged effect on vaccination. Positive peer influence, which considers the cost of vaccination as reasonably small compared to the cost of infection, can greatly promote vaccination. However, in the presence of exaggerated perception of vaccination costs or risks, the vaccination rate can plummet abruptly from high levels to pervasive vaccine refusals. Another interesting finding is that social hubs tend to be more likely to vaccinate when compared to periphery nodes with just a few neighbors. 

Furthermore, our work-in-progress shows that vaccination effectiveness can cause bistability in vaccination behavior. Following a sharp decline in vaccine uptake, the vaccination rate recovery may depend not only on the extent of mitigating the perceived cost of vaccination and improving the vaccine efficacy, but also on its past trajectory. This phenomenon resembles the hysteresis effect, traditionally associated with the magnetic properties of materials. Our research shows that hysteresis can appear as an unprecedented roadblock for the recovery of vaccination uptake, thereby helping explain the vaccine compliance problem’s persistence. 

Over the years, researchers have proposed behavioral epidemiology as a means of integrating the study of epidemiology with an understanding of health decisions made by individual actors responding to infection risks. Nevertheless, a feedback loop exists between health behaviors and the spread of an epidemic: individuals may take preventative measures, such as vaccination or reduced contact with others, in response to perceived risks. These responses in turn modify the spread of infection. The interplay between between changing opinions of vaccination and epidemic spreading on social networks constitutes a “dueling contagion” process. It is of fundamental significance for public health providers to achieve a comprehensive understanding of the rich dynamics generated by this sort of dueling contagion.

To conclude, vaccination dilemma is just one example of real-world human cooperation problems, which require special mechanisms to avoid the tragedy of the commons. We should emphasize the role of altruistic behavior in promoting vaccination, as vaccination not only protects oneself but also contributes to the herd immunity that benefits others. After all, the efforts of one or two shepherds would not have sufficed in preventing our commons from being overgrazed. We can gain deep insights into the vaccine accomplice problem by leveraging evolutionary game theory models, which researchers have already used to explore human strategic behavior in social dilemma situations where people can cooperate for the common good.

The authors presented part of the related research at the 2017 SIAM Annual Meeting, held in Pittsburgh, Pa., this July.

Further Reading:
[1]. Bauch, C.T., & Galvani, A.P. (2013). Social factors in epidemiology. Science, 342(6154), 47-49.
[2]. Fu, F., Christakis, N.A., & Fowler, J.H. (2017). Dueling biological and social contagions. Scientific Reports, 7, 43634.
 [3]. Fu, F., Rosenbloom, D.I., Wang, L., & Nowak, M.A. (2011). Imitation dynamics of vaccination behaviour on social networks. Proceedings of the Royal Society of London B: Biological Sciences, 278(1702), 42-49.
[4]. Wadman, M., & You, J. (2017). The vaccine wars. Science, 356(6336), 364-365.
[5]. Wu, B., Fu, F., & Wang, L. (2011). Imperfect vaccine aggravates the long-standing dilemma of voluntary vaccination. PloS ONE, 6(6), e20577.

Xingru Chen is a second-year applied mathematics Ph.D. student at Dartmouth College. She is interested in the mathematical modeling of human strategic behavior with particular emphasis on real-world problems of great importance to society, such as vaccination behavior and antibiotic overuse.
Feng Fu is an assistant professor of mathematics and biomedical data science at Dartmouth College. His work combines mathematical modeling approaches with analyses of experimental and observational data to yield a better understanding of the evolution of cooperation, behavior-disease interactions, and cancer evolution. 
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