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

The authors have declared that no competing interests exist.

Mathematical models have formalized how free-rider effects can threaten the stability of high vaccine coverage levels under established voluntary vaccination programs. However, little research has addressed the question of when free-riding begins to develop when a new vaccine is first introduced in a population.

Here, we combine a game theoretical model of vaccinating behavior with an age-structured compartmental model to analyze rational vaccinating behavior in the first years of a universal immunization program, where a new vaccine is free to all children of a specified age. The model captures how successive birth cohorts face different epidemiological landscapes that have been shaped by the vaccinating decisions of previous birth cohorts, resulting in a strategic interaction between individuals in different birth cohorts. The model predicts a Nash equilibrium coverage level of

These results suggest that even relatively new immunization programs can be vulnerable to drops in vaccine coverage caused by vaccine scares and exacerbated by herd immunity effects, necessitating vigilance from the start.

Voluntary vaccination programs–where individuals are free to choose whether to vaccinate–can be victims of their own success. As vaccine-derived herd immunity builds, the probability of being infected declines and eventually goes to zero. However, the perceived risk of vaccination remains constant. Hence, rational individuals may cease to vaccinate once herd immunity is sufficiently strong. This effect has been variously described in terms of a Free-Rider problem, the Prisoner's Dilemma, or the Tragedy of the Commons. It can also be seen as an example of policy resistance, which has been defined as “the tendency for interventions to be defeated by the systems response to the intervention itself”

More broadly, the free-rider problem in vaccination is one possible outcome of the nonlinear interplay between infection prevalence and individual vaccinating behavior. This interplay can be captured by a model consisting of a submodel of disease dynamics and a submodel of individual vaccinating behavior. Such integrated frameworks have been used previously to model phenomena such as free-riding behavior under voluntary vaccination, vaccine scares, and related phenomena

Some of these approaches use game theory, which predicts what strategies an individual should adopt when their payoff depends on what strategies others adopt. Game theory requires solving for the Nash equilibrium, which in population games is a strategy such that no sufficiently small group of individuals can achieve a higher payoff by adopting a different strategy

Previous models focus on vaccinating behavior in a population without age structure where an immunization program has been in place for a long time, and often ignore age structure with a few exceptions

This question is very topical given the large number of new generation vaccines that have been introduced in recent years, with many others to be introduced in the future

A parent's decision to vaccinate a child is often not a simple comparison of the risk/benefit of vaccinating versus not vaccinating, but involves a complex interplay of values with scientific facts. Before trying to understand the effects of this complex interaction, examining how the decisions of a hypothetical perfectly “rational” person who simply weighs the risks and benefits is warranted. By so doing, we can provide a baseline against which to compare “real world” decision making. In addition such a model could provide a framework for decisions about safety study timing and size.

At present decisions about how large to make vaccine safety studies at different phases of vaccine development and use, in the absence of a specific safety concern, are made using rules of thumb derived from expert judgment, experience, and engagement with the public

Addressing these issues requires a model that is consistent with the context in which individuals make vaccination choices under a universal immunization program. For example, the framework must be able to capture cohort structure, and in particular how each successive birth cohort faces a different landscape of infection risks depending on the vaccinating decisions that have been made by previous birth cohorts. As a game, this would describe a strategic interaction through time between individuals born in different birth cohorts.

In this paper we develop a game theoretical model of the time evolution of vaccine uptake following the introduction of a newly licensed vaccine at a specified age through a universal immunization program. Vaccination is assumed to be voluntary and free. Disease dynamics in the first years of the program are modeled using an age-structured compartmental transmission model. Our objective is to investigate the evolution of vaccinating behavior when a new vaccine is first introduced. In the following section we describe the behavioral and disease dynamic submodels. In the

The players of the game are children in birth cohort

We denote the vector of strategies played by all cohorts by

where

Each individual in a given cohort receives a baseline payoff

where

To define a Nash Equilibrium for this game, consider a population where a fraction

as

We computed the quantities

Susceptible individuals in age class

Birth cohort

Model parameters were chosen to reflect measles epidemiology. Transmission rates were estimated using a contact surface approach

The Nash equilibrium for the game described by Eqns. (1)–(4) under the SEIRV model dynamics was determined numerically using a simulated annealing algorithm

The process was iterated 1000 times. Every 30 iterations, the vaccine coverage in each cohort was adjusted up or down randomly to prevent convergence to local minima. (In this context, a local minimum represents a solution where the payoff to vaccinate is not as close to the payoff not to vaccinate as another solution elsewhere in parameter space, and therefore globally is not the best Nash equilibrium candidate.) After 1000 iterations, it was confirmed that each cohort

We determined Nash equilibria for a range of values of the vaccine penalty

The first birth cohort for which the vaccine becomes available (

This strategic interaction through time between individuals born in different cohorts can result in a rich variety of behavior (

Solid line represents coverage, and dashed line represents cases.

The model dynamics across the eight scenarios can be visualized by surface plots of the Nash equilibrium coverage

Blue >80% coverage, green 60-80%, yellow 40–60%, orange 20–40%, red 0%–20% coverage.

When disease risk is high, the effect of reduced vaccine efficacy is to increase vaccine coverage (

An aspect of behavior-prevalence dynamics that has previously been neglected is the role of case importation. This is the only source of disease burden once local chains of transmission have been interrupted through herd immunity, therefore, inclusion of case importation should potentially modify predicted vaccinating behavior once herd immunity has been built through immunization programs. In the scenarios of high disease risk in the present model, including case importation stabilizes patterns of vaccine coverage, creating greater consistency in predicted behavior for various values of

An important question in the design of phase III vaccine trials and phase IV safety studies is: how small a vaccine risk must be ruled out before the vaccine is considered acceptable for licensing? From the standpoint of an individual's decisions, this question becomes: how small does the vaccine risk have to be for a rational person to decide it is in their interest to vaccinate? The answer to this will depend on the birth cohort, since different birth cohorts face different epidemiological landscapes that have been shaped by the vaccination decisions of older birth cohorts.

To address this question, we define

Triangles: 95% coverage, circles: 50% coverage.

Lowering disease risk lowers the average tolerance for vaccine risk across birth cohorts, which is not surprising (

Increasing vaccine efficacy also has some surprising and paradoxical effects. In the scenarios of high disease risk, increasing vaccine efficacy decreases tolerance for vaccine risk in most birth cohorts. This occurs because use of a higher efficacy vaccine makes it possible to achieve very low incidence through herd immunity, and the resulting low force of infection reduces the tolerance for vaccine risks. By comparison, in the scenarios of lower disease risk, increasing vaccine efficacy results in widely variable vaccine risk tolerance across cohorts.

There are few qualitative differences in

The social optimum was defined as the vaccine coverage

Here we developed a game theoretical model of vaccinating behavior that is closely tailored to the way that many vaccines are implemented in real populations, through free, universal programs that offer vaccine to all children at a recommended age. Using a compartmental measles model to capture the response of disease dynamics to vaccination choices, we modeled the strategic interaction between individuals in the same birth cohort as well as individuals in different birth cohorts. We studied the resulting transient solutions of behavior-prevalence dynamics in the first years after a new vaccine has been introduced into the population. This framework built in the realistic cohort structure that is an integral part of many publicly-financed, school-based immunization programs, and thus the framework captures how successive birth cohorts face different epidemiological conditions, especially in the first years of an immunization program. This situation is highly relevant for many new generation vaccines that have been recently introduced (such as new generation rotavirus vaccines) or that will eventually be introduced. The design of this approach was based on a desire to address issues that arise in the design of vaccine licensing protocols, where one must decide how large a vaccine risk is acceptable, either from the viewpoint of a decision-maker or from the viewpoint of a rational individual under a voluntary vaccination policy.

Free-riding behavior under voluntary vaccination policies is widely expected for long-established immunization programs where significant levels of herd immunity derived from immunization have been built up. This potential instability in long-established immunization programs has been captured by many previous models

More generally, we found that vaccine efficacy, disease risk, and vaccine risk can interact to result in a rich and often surprising variety of possible vaccine coverage and outbreak patterns at epidemiologically plausible parameters values. These patterns range from relatively high, stable vaccine coverage across all cohorts with occasional outbreaks (for relatively low disease risk) to extremely variable vaccine coverage from one cohort to the next with regular outbreaks (for relatively high disease risk).

We also found that the tolerance for vaccine risk depends on the birth cohort. Earlier birth cohorts are subject to higher levels of infection prevalence and hence are more willing to tolerate higher risks and vaccinate at higher levels of coverage. However, the tolerance for risk drops dramatically at some point within the first five birth cohorts. This drop occurs sooner for lower disease risk and/or lower vaccine efficacy. For the case of high efficacy vaccine and high disease risk, this drop does not occur until the fifth birth cohort vaccinated, under the

These findings indicate that sample size calculations for phase III vaccine trials and database size determination for phase IV safety studies should consider that risk tolerance can change during a vaccine rollout campaign. The simulation results suggest that for vaccine-disease systems where disease risk and vaccine efficacy are sufficiently high, there may be a window during the early years of an immunization program where individuals are willing to tolerate greater uncertainty in vaccine risk. All other things being equal, one might therefore argue that shifting more safety data collection to the time after the vaccine is introduced into use is a rational policy. As a result, large phase III trials may not be needed, although a large-scale safety study may be needed post-licensure because tolerated risk may drop significantly after a few years. The present analysis demonstrates how modeling this interaction between vaccine risk tolerance, vaccine acceptance, and infection rates in the presence of vaccination can provide a baseline to quantitatively evaluate current licensing and safety monitoring strategies, as well as a method of computing required sample sizes for clinical trials and safety study database size. The practicality of such an approach would need to be assessed. For instance, enrolled patients are sometimes lost to follow-up and/or can be very poor at self-diagnosing adverse events. Such factors can complicate measurement of vaccine risk and thus required sample size.

Our analysis assumed individuals have perfect rationality. Normally, this may be considered to be a limitation of classical game theoretical models. However, for the question of informing vaccine licensing protocol, rationality is actually a useful assumption. For the purposes of vaccine licensing, regulators wish to determine the required size of phase III vaccine safety trials. An important criterion in determining the required size of phase III trials is the need to rule out vaccine adverse events up to a certain acceptable probabilityin other words, there is a need to determine largest acceptable vaccine risk. The largest acceptable vaccine risk could be determined by several criteria. However, one criterion is the level of vaccine risk that a self-interested but rational person would be willing to tolerate, given information on disease risks and epidemiology. By comparison, how actual irrational individuals will react to a new vaccine is arguably not a good basis for deciding whether or not a vaccine should be licensed, because licensing decisions should be based on scientific data.

On the other hand, for other types of questions, such as predicting actual behaviour upon introduction of a new vaccine, the assumption of perfect rationality could be a significant limitation. Game theoretical models that assume perfectly rationality tend to predict more irregular behaviour (such as oscillations) for certain parameter regimes than are observed in reality. Hence, if more realistic human behavior is introduced into this framework, there may be greater stability of vaccine coverage across cohorts. This model was also developed for measles-like infection and vaccine. Infections other than measles may exhibit more regular patterns or may exhibit less extreme oscillations, since measles transmission is significantly nonlinear. The wide range of dynamical behavior seen in this model emphasizes the need to understand the impact of our model assumptions on the predicted dynamics. Future work could investigate how relaxing these assumptions influences predictions under this framework. Other future applications of this framework might involve case studies for specific disease/vaccine systems. One such example might be to explore whether the general acceptance of MMR vaccine despite known adverse effects such as idiopathic thrombocytopenic purpura (ITP)

As noted above, this model suggested that free-riding behavior can emerge relatively soon after a new immunization program for measles-like infection has been introduced. This is true even before vaccine-derived herd immunity has been built up in a large proportion of the overall population. This prediction may hold for a variety of disease systems. Mixing between age classes–both in terms of social contacts and in terms of sexual contacts–tends to be highly assortative

Additional information on model equations, parameterization, and supplemental results.

(2.62 MB PDF)

The authors are grateful to three anonymous reviewers for comments.