Dynamic vaccination games and variational inequalities on time-dependent sets

This paper presents a model of a dynamic vaccination game in a population consisting of a collection of groups, each of which holds distinct perceptions of vaccinating versus non-vaccinating risks. Vaccination is regarded here as a game due to the fact that the payoff to each population group depends on the so-called perceived probability of getting infected given a certain level of the vaccine coverage in the population, a level that is generally obtained by the vaccinating decisions of other members of a population. The novelty of this model resides in the fact that it describes a repeated vaccination game (over a finite time horizon) of population groups whose sizes vary with time. In particular, the dynamic game is proven to have solutions using a parametric variational inequality approach often employed in optimization and network equilibrium problems. Moreover, the model does not make any assumptions upon the level of the vaccine coverage in the population, but rather computes this level as a final result. This model could then be used to compute possible vaccine coverage scenarios in a population, given information about its heterogeneity with respect to perceived vaccine risks. In support of the model, some theoretical results were advanced (presented in the appendix) to ensure that computation of optimal vaccination strategies can take place; this means, the theory states the existence, uniqueness and regularity (in our case piecewise continuity) of the solution curves representing the evolution of optimal vaccination strategies of each population group.     

Imitation dynamics predict vaccinating behaviour

There exists an interplay between vaccine coverage, disease prevalence and the vaccinating behaviour of individuals. Moreover, because of herd immunity, there is also a strategic interaction between individuals when they are deciding whether or not to vaccinate, because the probability that an individual becomes infected depends upon how many other individuals are vaccinated. To understand this potentially complex interplay, a game dynamic model is developed in which individuals adopt strategies according to an imitation dynamic (a learning process), and base vaccination decisions on disease prevalence and perceived risks of vaccines and disease. The model predicts that oscillations in vaccine uptake are more likely in populations where individuals imitate others more readily or where vaccinating behaviour is more sensitive to changes in disease prevalence. Oscillations are also more likely when the perceived risk of vaccines is high. The model reproduces salient features of the time evolution of vaccine uptake and disease prevalence during the whole-cell pertussis vaccine scare in England and Wales during the 1970s. This suggests that using game theoretical models to predict, and even manage, the population dynamics of vaccinating behaviour may be feasible.     

Dynamics of Vaccination Strategies via Projected Dynamical Systems

Previous game theoretical analyses of vaccinating behaviour have underscored the strategic interaction between individuals attempting to maximise their health states, in situations where an individual's health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here, we extend such analyses by applying the theories of variational inequalities (VI) and projected dynamical systems (PDS) to vaccination games. A PDS provides a dynamics that gives the conditions for existence, uniqueness and stability properties of Nash equilibria. In this paper, it is used to analyse the dynamics of vaccinating behaviour in a population consisting of distinct social groups, where each group has different perceptions of vaccine and disease risks. In particular, we study populations with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). We find that a population with a vaccine-inclined majority group and a vaccine-averse minority group exhibits higher average vaccine coverage than the corresponding homogeneous population, when the vaccine is perceived as being risky relative to the disease. Our model also reproduces a feature of real populations: In certain parameter regimes, it is possible to have a majority group adopting high vaccination rates and simultaneously a vaccine-averse minority group adopting low vaccination rates. Moreover, we find that minority groups will tend to exhibit more extreme changes in vaccinating behaviour for a given change in risk perception, in comparison to majority groups. These results emphasise the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by PDS and VI in mathematical epidemiology.     

Social Contact Networks and Disease Eradicability under Voluntary Vaccination

Certain theories suggest that it should be difficult or impossible to eradicate a vaccine-preventable disease under voluntary vaccination: Herd immunity implies that the individual incentive to vaccinate disappears at high coverage levels. Historically, there have been examples of declining coverage for vaccines, such as MMR vaccine and whole-cell pertussis vaccine, that are consistent with this theory. On the other hand, smallpox was globally eradicated by 1980 despite voluntary vaccination policies in many jurisdictions. Previous modeling studies of the interplay between disease dynamics and individual vaccinating behavior have assumed that infection is transmitted in a homogeneously mixing population. By comparison, here we simulate transmission of a vaccine-preventable SEIR infection through a random, static contact network. Individuals choose whether to vaccinate based on infection risks from neighbors, and based on vaccine risks. When neighborhood size is small, rational vaccinating behavior results in rapid containment of the infection through voluntary ring vaccination. As neighborhood size increases (while the average force of infection is held constant), a threshold is reached beyond which the infection can break through partially vaccinated rings, percolating through the whole population and resulting in considerable epidemic final sizes and a large number vaccinated. The former outcome represents convergence between individually and socially optimal outcomes, whereas the latter represents their divergence, as observed in most models of individual vaccinating behavior that assume homogeneous mixing. Similar effects are observed in an extended model using smallpox-specific natural history and transmissibility assumptions. This work illustrates the significant qualitative differences between behavior–infection dynamics in discrete contact-structured populations versus continuous unstructured populations. This work also shows how disease eradicability in populations where voluntary vaccination is the primary control mechanism may depend partly on whether the disease is transmissible only to a few close social contacts or to a larger subset of the population.     

A game dynamic model for delayer strategies in vaccinating behaviour for pediatric infectious diseases

Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice—rather than accessibility—is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake.     

Optimizing vaccine allocation at different points in time during an epidemic

Background

Pandemic influenza A(H1N1) 2009 began spreading around the globe in April of 2009 and vaccination started in October of 2009. In most countries, by the time vaccination started, the second wave of pandemic H1N1 2009 was already under way. With limited supplies of vaccine, we are left to question whether it may be a good strategy to vaccinate the high-transmission groups earlier in the epidemic, but it might be a better use of resources to protect instead the high-risk groups later in the epidemic. To answer this question, we develop a deterministic epidemic model with two age-groups (children and adults) and further subdivide each age group in low and high risk.

Methods and Findings

We compare optimal vaccination strategies started at various points in time in two different settings: a population in a developed country where children account for 24% of the population, and a population in a less developed country where children make up the majority of the population, 55%. For each of these populations, we minimize mortality or hospitalizations and we find an optimal vaccination strategy that gives the best vaccine allocation given a starting vaccination time and vaccine coverage level. We find that population structure is an important factor in determining the optimal vaccine distribution. Moreover, the optimal policy is dynamic as there is a switch in the optimal vaccination strategy at some time point just before the peak of the epidemic. For instance, with 25% vaccine coverage, it is better to protect the high-transmission groups before this point, but it is optimal to protect the most vulnerable groups afterward.

Conclusions

Choosing the optimal strategy before or early in the epidemic makes an important difference in minimizing the number of influenza infections, and consequently the number of influenza deaths or hospitalizations, but the optimal strategy makes little difference after the peak.     

Rapid Emergence of Free-Riding Behavior in New Pediatric Immunization Programs

Background

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.

Methodology/Principal Findings

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 for the first few birth cohorts under the new program. However, free-riding behavior emerges very quickly, with the Nash equilibrium vaccine coverage dropping significantly within 2-5 years after program initiation. Subsequently, a rich set of coupled dynamics between infection prevalence and vaccinating behaviors is possible, ranging from relatively stable (but reduced) coverage in later birth cohorts to wide fluctuations in vaccine coverage from one birth cohort to the next. Individual tolerance for vaccine risk also starts out at relatively high levels before dropping significantly within a few years.

Conclusions/Significance

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.     

Policy Resistance Undermines Superspreader Vaccination Strategies for Influenza

Theoretical models of infection spread on networks predict that targeting vaccination at individuals with a very large number of contacts (superspreaders) can reduce infection incidence by a significant margin. These models generally assume that superspreaders will always agree to be vaccinated. Hence, they cannot capture unintended consequences such as policy resistance, where the behavioral response induced by a new vaccine policy tends to reduce the expected benefits of the policy. Here, we couple a model of influenza transmission on an empirically-based contact network with a psychologically structured model of influenza vaccinating behavior, where individual vaccinating decisions depend on social learning and past experiences of perceived infections, vaccine complications and vaccine failures. We find that policy resistance almost completely undermines the effectiveness of superspreader strategies: the most commonly explored approaches that target a randomly chosen neighbor of an individual, or that preferentially choose neighbors with many contacts, provide at best a relative improvement over their non-targeted counterpart as compared to when behavioral feedbacks are ignored. Increased vaccine coverage in super spreaders is offset by decreased coverage in non-superspreaders, and superspreaders also have a higher rate of perceived vaccine failures on account of being infected more often. Including incentives for vaccination provides modest improvements in outcomes. We conclude that the design of influenza vaccine strategies involving widespread incentive use and/or targeting of superspreaders should account for policy resistance, and mitigate it whenever possible.     

Assessing the best time interval between doses in a two-dose vaccination regimen to reduce the number of deaths in an ongoing epidemic of SARS-CoV-2

The SARS-CoV-2 pandemic is a major concern all over the world and, as vaccines became available at the end of 2020, optimal vaccination strategies were subjected to intense investigation. Considering their critical role in reducing disease burden, the increasing demand outpacing production, and that most currently approved vaccines follow a two-dose regimen, the cost-effectiveness of delaying the second dose to increment the coverage of the population receiving the first dose is often debated. Finding the best solution is complex due to the trade-off between vaccinating more people with lower level of protection and guaranteeing higher protection to a fewer number of individuals. Here we present a novel extended age-structured SEIR mathematical model that includes a two-dose vaccination schedule with a between-doses delay modelled through delay differential equations and linear optimization of vaccination rates. By maintaining the minimum stock of vaccines under a given production rate, we evaluate the dose interval that minimizes the number of deaths. We found that the best strategy depends on an interplay between the vaccine production rate and the relative efficacy of the first dose. In the scenario of low first-dose efficacy, it is always better to vaccinate the second dose as soon as possible, while for high first-dose efficacy, the best strategy of time window depends on the production rate and also on second-dose efficacy provided by each type of vaccine. We also found that the rate of spread of the infection does not affect significantly the thresholds of the best window, but is an important factor in the absolute number of total deaths. These conclusions point to the need to carefully take into account both vaccine characteristics and roll-out speed to optimize the outcome of vaccination strategies.     

Uniqueness of Nash equilibrium in vaccination games

One crucial condition for the uniqueness of Nash equilibrium set in vaccination games is that the attack ratio monotonically decreases as the vaccine coverage level increasing. We consider several deterministic vaccination models in homogeneous mixing population and in heterogeneous mixing population. Based on the final size relations obtained from the deterministic epidemic models, we prove that the attack ratios can be expressed in terms of the vaccine coverage levels, and also prove that the attack ratios are decreasing functions of vaccine coverage levels. Some thresholds are presented, which depend on the vaccine efficacy. It is proved that for vaccination games in homogeneous mixing population, there is a unique Nash equilibrium for each game.     

Evolving public perceptions and stability in vaccine uptake

Recent vaccine scares and sudden spikes in vaccine demand remind us that the effectiveness of mass vaccination programs is governed by the public perception of vaccination. Previous work has shown that the tendency of individuals to optimize self-interest can lead to vaccination levels that are suboptimal for a community. We use game theory to relate population-level demand for vaccines to decision-making by individuals with varied beliefs about the costs of infection and vaccination. In contrast to previous work proposing that universal vaccination is impossible in a game theoretic context, we show that optimal individual behavior can vary between universal vaccination and no vaccination, depending on the relative costs and benefits to individuals. By coupling game models and epidemic models, we demonstrate that the pursuit of self-interest often leads to stable dynamics but can lead to oscillations in vaccine uptake over time. The instability is exacerbated in populations that are more homogeneous with respect to their perceptions of vaccine and infection risks. This research illustrates the importance of applying temporal models to an inherently temporal situation, namely, the time evolution of vaccine coverage in an informed population with a voluntary vaccination policy.     

Comparative Effectiveness of Different Strategies of Oral Cholera Vaccination in Bangladesh: A Modeling Study

Background

Killed, oral cholera vaccines have proven safe and effective, and several large-scale mass cholera vaccination efforts have demonstrated the feasibility of widespread deployment. This study uses a mathematical model of cholera transmission in Bangladesh to examine the effectiveness of potential vaccination strategies.

Methods & Findings

We developed an age-structured mathematical model of cholera transmission and calibrated it to reproduce the dynamics of cholera in Matlab, Bangladesh. We used the model to predict the effectiveness of different cholera vaccination strategies over a period of 20 years. We explored vaccination programs that targeted one of three increasingly focused age groups (the entire vaccine-eligible population of age one year and older, children of ages 1 to 14 years, or preschoolers of ages 1 to 4 years) and that could occur either as campaigns recurring every five years or as continuous ongoing vaccination efforts. Our modeling results suggest that vaccinating 70% of the population would avert 90% of cholera cases in the first year but that campaign and continuous vaccination strategies differ in effectiveness over 20 years. Maintaining 70% coverage of the population would be sufficient to prevent sustained transmission of endemic cholera in Matlab, while vaccinating periodically every five years is less effective. Selectively vaccinating children 1–14 years old would prevent the most cholera cases per vaccine administered in both campaign and continuous strategies.

Conclusions

We conclude that continuous mass vaccination would be more effective against endemic cholera than periodic campaigns. Vaccinating children averts more cases per dose than vaccinating all age groups, although vaccinating only children is unlikely to control endemic cholera in Bangladesh. Careful consideration must be made before generalizing these results to other regions.     

Evolutionary game theory and social learning can determine how vaccine scares unfold

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.     

COVID-19: vaccination problems

This minireview addresses problems of financing the vaccine development, regulatory questions, the ethics and efficacy of vaccine prioritization strategies and the coverage of variant viruses by current vaccines. Serious adverse effects observed with adenovirus vectored vaccines and mRNA vaccines in mass vaccination campaigns are reported. The ethical problems of continuing with placebo controlled vaccine trials and alternative clinical trial protocols are discussed as well as concrete vaccination issues such as the splitting of doses, the delaying of the second dose, the immunization with two different vaccine types and the need of vaccinating seropositive subjects. Strategies to increase vaccine acceptance in the population are shortly mentioned.     

Reactive vaccination in the presence of disease hotspots

Reactive vaccination has recently been adopted as an outbreak response tool for cholera and other infectious diseases. Owing to the global shortage of oral cholera vaccine, health officials must quickly decide who and where to distribute limited vaccine. Targeted vaccination in transmission hotspots (i.e. areas with high transmission efficiency) may be a potential approach to efficiently allocate vaccine, however its effectiveness will likely be context-dependent. We compared strategies for allocating vaccine across multiple areas with heterogeneous transmission efficiency. We constructed metapopulation models of a cholera-like disease and compared simulated epidemics where: vaccine is targeted at areas of high or low transmission efficiency, where vaccine is distributed across the population, and where no vaccine is used. We find that connectivity between populations, transmission efficiency, vaccination timing and the amount of vaccine available all shape the performance of different allocation strategies. In highly connected settings (e.g. cities) when vaccinating early in the epidemic, targeting limited vaccine at transmission hotspots is often optimal. Once vaccination is delayed, targeting the hotspot is rarely optimal, and strategies that either spread vaccine between areas or those targeted at non-hotspots will avert more cases. Although hotspots may be an intuitive outbreak control target, we show that, in many situations, the hotspot-epidemic proceeds so fast that hotspot-targeted reactive vaccination will prevent relatively few cases, and vaccination shared across areas where transmission can be sustained is often best.     

The impact of vaccine side effects on the natural history of immunization programmes: an imitation-game approach

When the incidence and prevalence of most common vaccine preventable childhood infectious diseases are constantly low, as is the case in many industrialized countries, the incidence of vaccine-associated side effects might become a key determinant in vaccine demand. We study an SIR transmission model with dynamic vaccine demand based on an imitation mechanism where the perceived risk of vaccination is modelled as a function of the incidence of vaccine side effects. The model shows some important differences compared to previous game dynamic models of vaccination, and allows noteworthy inferences as regards both the past and future lifetime of vaccination programmes. In particular it is suggested that a huge disproportion between the perceived risk of disease and vaccination is necessary in order to achieve high coverages. This disproportion is further increased in highly industrialised countries. Such considerations represent serious challenges for future vaccination programmes.     

Efficacy of vaccination against HPV infections to prevent cervical cancer in France: present assessment and pathways to improve vaccination policies

Background

Seventy percent of sexually active individuals will be infected with Human Papillomavirus (HPV) during their lifetime. These infections are incriminated for almost all cervical cancers. In France, 3,068 new cases of cervical cancer and 1,067 deaths from cervical cancer occurred in 2005. Two vaccines against HPV infections are currently available and vaccination policies aim to decrease the incidence of HPV infections and of cervical cancers. In France, vaccine coverage has been reported to be low.

Methods

We developed a dynamic model for the heterosexual transmission of Human Papillomavirus types 16 and 18, which are covered by available vaccines. A deterministic model was used with stratification on gender, age and sexual behavior. Immunity obtained from vaccination was taken into account. The model was calibrated using French data of cervical cancer incidence.

Results

In view of current vaccine coverage and screening, we expected a 32% and 83% reduction in the incidence of cervical cancers due to HPV 16/18, after 20 years and 50 years of vaccine introduction respectively. Vaccine coverage and screening rates were assumed to be constant. However, increasing vaccine coverage in women or vaccinating girls before 14 showed a better impact on cervical cancer incidence. On the other hand, performing vaccination in men improves the effect on cervical cancer incidence only moderately, compared to strategies in females only.

Conclusion

While current vaccination policies may significantly decrease cervical cancer incidence, other supplementary strategies in females could be considered in order to improve vaccination efficacy.     

The smallpox vaccine: a multidimensional model of choice

Following September 11, 2001, the U.S. government increased its efforts to prepare for future attacks, including those using dangerous biological agents such as smallpox. The smallpox vaccination program called for vaccinating military personnel and smallpox response teams, including healthcare workers and other first responders. The program of vaccinating healthcare workers was largely unsuccessful; few individuals volunteered to be vaccinated, highlighting the importance of understanding the factors that influence choice regarding this complex medical decision. This study examined stated choice and how it was associated with risk perceptions, knowledge, psychological distress, and general vaccine beliefs using a five-dimensional choice model. The model used multivariable modeling strategies in a sample of 256 undergraduate, graduate, and medical students. Sixty-three percent of the sample stated that they would elect to receive the smallpox vaccination. Multiple factors were related to stated choice in multivariable models, including perceived risk/worry, general vaccine beliefs, decisional conflict, and gender. However, the models were more successful at predicting acceptance of the vaccination than vaccine refusal. Although support was obtained for a multidimensional model of choice, several questions were raised by our results, including (a) whether refusal of smallpox vaccination can be more effectively characterized, possibly with additional questions; (b) whether the model translates to actual vaccination behavior; and (c) whether the model describes choice in more at-risk samples (e.g., first responders, healthcare workers). A multidimensional modeling approach should facilitate these and other studies of choice.     

Optimizing one-dose and two-dose cholera vaccine allocation in outbreak settings: A modeling study

BackgroundA global stockpile of oral cholera vaccine (OCV) was established in 2013 for use in outbreak response and are licensed as two-dose regimens. Vaccine availability, however, remains limited. Previous studies have found that a single dose of OCV may provide substantial protection against cholera.MethodsUsing a mathematical model with two age groups paired with optimization algorithms, we determine the optimal vaccination strategy with one and two doses of vaccine to minimize cumulative overall infections, symptomatic infections, and deaths. We explore counterfactual vaccination scenarios in three distinct settings: Maela, the largest refugee camp in Thailand, with high in- and out-migration; N’Djamena, Chad, a densely populated region; and Haiti, where departments are connected by rivers and roads.ResultsOver the short term under limited vaccine supply, the optimal strategies for all objectives prioritize one dose to the older age group (over five years old), irrespective of setting and level of vaccination coverage. As more vaccine becomes available, it is optimal to administer a second dose for long-term protection. With enough vaccine to cover the whole population with one dose, the optimal strategies can avert up to 30% to 90% of deaths and 36% to 92% of symptomatic infections across the three settings over one year. The one-dose optimal strategies can avert 1.2 to 1.8 times as many cases and deaths compared to the standard two-dose strategy.ConclusionsIn an outbreak setting, speedy vaccination campaigns with a single dose of OCV is likely to avert more cases and deaths than a two-dose pro-rata campaign under a limited vaccine supply.     

The influence of social norms on the dynamics of vaccinating behaviour for paediatric infectious diseases

Mathematical models that couple disease dynamics and vaccinating behaviour often assume that the incentive to vaccinate disappears if disease prevalence is zero. Hence, they predict that vaccine refusal should be the rule, and elimination should be difficult or impossible. In reality, countries with non-mandatory vaccination policies have usually been able to maintain elimination or very low incidence of paediatric infectious diseases for long periods of time. Here, we show that including injunctive social norms can reconcile such behaviour-incidence models to observations. Adding social norms to a coupled behaviour-incidence model enables the model to better explain pertussis vaccine uptake and disease dynamics in the UK from 1967 to 2010, in both the vaccine-scare years and the years of high vaccine coverage. The model also illustrates how a vaccine scare can perpetuate suboptimal vaccine coverage long after perceived risk has returned to baseline, pre-vaccine-scare levels. However, at other model parameter values, social norms can perpetuate depressed vaccine coverage during a vaccine scare well beyond the time when the population''s baseline vaccine risk perception returns to pre-scare levels. Social norms can strongly suppress vaccine uptake despite frequent outbreaks, as observed in some small communities. Significant portions of the parameter space also exhibit bistability, meaning long-term outcomes depend on the initial conditions. Depending on the context, social norms can either support or hinder immunization goals.