Emergent dynamical features in
behaviour-incidence models of vaccinating
decisions
Samit Bhattacharyya and Chris T. Bauch
Abstract
Vaccination is a cornerstone of infectious disease prevention. However, individ-
ual vaccinating behaviour does not always result in population-level vaccine cov-
erage patterns that are optimal for protecting public health. For example, vaccine
coverage may fall below the elimination threshold due to nonvaccinators who “free-
ride” on the herd immunity provided by vaccinators. Routine vaccination programs
for many pediatric infectious diseases now have an almost world-wide coverage,
but vaccine scares fuelled by such behaviours threaten eradication goals. This free-
riding behaviour can be seen as a manifestation of policy resistance, where hu-
mans respond to an intervention in such a way that tends to undermine the in-
tervention. However, policy resistance is only one such example of the types of
dynamics that emerge from the interaction between vaccinating behaviour and dis-
ease incidence or prevalence. Here we explore four types of emergent dynamics of
behaviour-incidence systems: policy resistance, policy reinforcement, outcome in-
elasticity, and outcome variability. We discuss examples of each of these dynamics
in the behaviour-incidence modelling literature, and suggest potential implications
for vaccination policy.
1 Introduction
Despite widespread controversies among the public, vaccination has proved to be
one of the most successful infectious disease interventions ever, and remains one
Samit Bhattacharyya
Departments of Mathematics and Biology, School of Medicine, University of Utah, Salt Lake City,
UT 84108, e-mail: samit@math.utah.edu
Chris T. Bauch
Department of Mathematics and Statistics, University of Guelph, 50 Stone Road, Guelph, ON
N1G2W1, e-mail: cbauch@uoguelph.ca
1
2 Samit Bhattacharyya and Chris T. Bauch
of the greatest public health achievements in the twenty-first century. Vaccination
against major infectious diseases, and the complete or near-complete eradication of
some diseases (such as smallpox and polio) has completely changed the demogra-
phy of many developed and developing countries worldwide [13].
The most common strategy is universal mass vaccination (UMV), which requires
covering as much of the population as possible, either through large-scale periodic
campaigns or through regular school-based programs. However, ring vaccination
has also been applied. Ring vaccination involves identifying infectious index cases
and vaccinating their close contacts to prevent them from being infected [37, 29].
Ring vaccination may perform better than UMV when the outbreak is localized and
infected individuals or their exposed contacts can be rapidly identified. Ring vac-
cination has been applied to outbreak control for hepatitis A [17], foot-and-mouth
disease in cattle [32, 41] and smallpox [21].
Large-scale vaccination programs confers population-wide benefits. Vaccination
not only prevents infection in vaccinated individuals–it also protects the unvac-
cinated through a “herd immunity” effect that slows down the circulation of a
pathogen in the entire population [1]. Herd immunity operates through disrupting
the chain of transmission between individuals. The greater the proportion of vacci-
nated individuals, the smaller the probability that a susceptible individual will come
into contact with an infectious individual and thereby become infected.
The epidemiology of many well-known vaccine preventable diseases is subject to
the effects of human belief and awareness of disease or vaccine [15, 22]. Human be-
haviour plays an important role in determining whether target vaccination coverage
can be reached in a given population. Common childhood diseases such as measles
or pertussis are timely examples [38]. While there has long been enthusiastic de-
bate in some higher-income countries about the relative merits and implications
of mandatory versus voluntary vaccination [16], it is clear that voluntary vaccina-
tion policies sometimes fail. Measles-Mumps-Rubella (MMR) vaccination in Great
Britain in the 1990s is one such example of failure due to a vaccine “scare” [31].
Vaccination coverage for seasonal influenza also remains suboptimal in many
countries, including the USA and Canada. Sub-optimal coverage has been observed
even among health care workers (HCWs). For example, a recent study indicates
that vaccination coverage for seasonal influenza among HCWs in Canada remains
below 50% [33]. Influenza vaccination coverage among children and individuals
at high-risk was also very low and significantly below the target level. This is a
worldwide phenomenon, with suboptimal influenza vaccine coverage among HCWs
having also been identified in Middle East countries (United Arab Emirates (UAE),
Kuwait and Oman), due to doubts about vaccine efficacy, lack of information about
the importance of immunization, and concerns about vaccine side effects [28]. A low
perceived risk of becoming infected, whether justified by historically low infection
rates or not, can contribute as much as inflated perception of vaccine risk does:
studies identify perceived lack of infection risk as a factor in non-uptake of influenza
vaccine [35]. If reduced infection rates due to previous vaccinations lead to reduced
perception of infection risks, then herd immunity can, ironically, lead to reduced
infection risk perception and thus reduced vaccine uptake.
Emergent dynamical features in behaviour-incidence models 3
An emerging fear of vaccine complications can combine with the temptation to
rely on herd immunity provided by those who have already vaccinated to impel
individuals to exempt themselves or their children from vaccination. Thus, herd im-
munity introduces a social dilemma in voluntary vaccination policy that amounts
to “free-riding”, or a “Tragedy of the Commons” [30, 22]. Equivalently, this dy-
namic implies a feedback loop between vaccinating decisions and disease dynam-
ics: individual vaccinating choices influence disease prevalence, but the level of dis-
ease prevalence in turn influences how many individuals choose to seek vaccination
[7, 9, 11] (Figure 1).
Classical game theory provides a useful tool to analyze and predict the outcomes
of strategic interactions [44, 43, 20, 18], including those arising from the interaction
between disease dynamics and human vaccinating behaviour [14, 5]. According to
a game theoretical perspective, individuals make a rational decision in weighing up
the costs and benefits related to vaccination against the cost of risking infection,
making assumptions about how much herd immunity will be provided by others in
the population. Although further empirical study is warranted regarding how well
game theory captures vaccinating behaviour and risk perception, game theoretic
modeling of individual vaccinating decisions is growing. This models often pre-
dict that rational self-interest leads to a Nash equilibrium vaccine coverage that is
suboptimal for the population, being below the level required to eliminate the infec-
tion [6, 5]. However, a variety of other non-game-theoretical approaches have also
been adopted to capture the interplay between disease dynamics and vaccinating
behaviour, and they often yield similar predictions [2, 7, 19].
The nonlinear feedback loop that springs from interaction between individual
vaccinating behaviour and disease dynamics can create interesting dynamical con-
sequences, including policy resistance, policy reinforcement, outcome inelasticity,
and outcome variability. In the next few sections, we will define these terms and dis-
cuss how they arise in models of the interplay between vaccinating behaviour and
disease dynamics.
2 Policy resistance
The most common implication of herd immunity for behaviour-incidence dynamics
is policy resistance, which is “the tendency for interventions to be defeated by the
systems response to the intervention itself” [40]. The vaccine coverage necessary to
achieve perfect herd immunity and thus elimination varies greatly from one infec-
tious disease to the next, but generally ranges from 80%-95% for common pediatric
infectious diseases [1]. If this level of vaccination is attained, those who refuse to
be vaccinated are nonetheless protected through the strong likelihood that they will
never be infected. As a result, there is a temptation not to seek vaccination, when
vaccine coverage is very high. If this temptation translates into individual action
not to seek vaccination, vaccine coverage will drop below the elimination threshold.
Hence, as a result of herd immunity and the nonlinear interplay between disease
4 Samit Bhattacharyya and Chris T. Bauch
dynamics and vaccinating behaviour, voluntary vaccination is subject to policy re-
sistance.
Policy resistance is often cast as a conflict between the Nash equilibrium vac-
cine coverage and the social optimum vaccine coverage. The social optimum can
be defined as the vaccine coverage such that the total population burden from either
vaccination or infection across all individuals is minimized, and in these models the
goal of public health is often conceived as being to reach the socially optimal cover-
age level (although typically, this definition ignores issues of equity). In contrast to
the socially optimal coverage, the Nash equilibrium driven by rational self-interest
in most models leads to a vaccine coverage that is different (often lower) than the
social optimum vaccine coverage (Figure 2).
Some models indicate that this free-riding manifestation of policy resistance can
emerge relatively quickly upon introduction of a new immunization program, and
that it can result in considerable instabilities in vaccine coverage [19, 42, 8]. For ex-
ample, new generation vaccines for childhood immunization programs are launched
in the United States every few years [3], and the success of the immunization pro-
gram depends to some extent on how the population will respond to it, which may
only partly be a function of demonstrated safety and efficacy of the vaccine. A re-
cent game theoretical model describes how populations respond to a new pediatric
infectious disease vaccine implemented through a universal mass vaccination pro-
gram administered at a specified age every year [8]. This model predicts that, due to
initially high infection prevalence, vaccine coverage remains reasonably high imme-
diately after introduction, but can succumb to free-riding on herd immunity within
4-5 birth cohorts (years). These drops occurs sooner (within 2-3 birth cohorts) when
the disease risk is low or vaccine efficacy is low. Moreover, due to instabilities in
behaviour-incidence dynamics, vaccine coverage can vary considerably from one
birth cohort to the next. The model also enables calculating the smallest vaccine
risk tolerable for each birth cohort so that an individual make a rational decision
of considering vaccination; this information may be useful for designing phase-III
trials and phase-IV safety studies for vaccines [3].
Somewhat similar dynamics have been explored in the context of influenza vac-
cination [42]. Influenza management is one of the most significant current concerns
for public health policy makers. While it is recommended that nearly 80% of in-
dividuals should get annual influenza vaccine, it is estimated that only 40-50% ac-
tually do so [23]. In response to the need to revaccinate for influenza every year,
universal influenza vaccines conferring long-term immunity are being developed,
and it is hoped that this might increase vaccine coverage [46]. However, a game the-
oretic model linking human cognition and memory for universal influenza immu-
nization and influenza epidemiology forces us to consider the potential for policy
resistance against universal influenza vaccines. This model predicts that a universal
vaccine which provides short-term protection will on average increase the vaccine
coverage, compared to the standard seasonal vaccine: short-term protection main-
tains risk communication of influenza among populations, resulting in stable vac-
cine coverage, which in turn creates small groups of free-riders and thus frequent
but small-size epidemics. In contrast, a universal vaccine that provides longer-term
Emergent dynamical features in behaviour-incidence models 5
protection may be counter-productive in some respects. Long-term protection cre-
ates large groups of free-riders who accept vaccination only after a severe epidemic
occurs. Because of long-term immunity, individuals mostly free-ride, or accept vac-
cination only once in a large time-frame, and this results in drop of vaccine coverage
after many years, in turn causing infrequent but very severe epidemics.
The imbalance between perceived and real risk and its negative effect on vac-
cine coverage is also reflected by several other examples of research. For example,
population surveys have been used to parameterize game theoretical vaccinating be-
haviour models for influenza and human papillomavirus (HPV) vaccination [27, 4].
These models confirm that rational individual vaccinating decision-making would
not allow populations to reach vaccine coverage levels that minimize disease preva-
lence in the population.
Another manifestation of policy resistance is individuals who vaccinate, but only
after a period of delay. Delaying behaviour has been observed in some real-world
immunization programs, and the game theoretical aspects of such behaviour have
been explored for the case of pediatric infectious diseases [11] and pandemic in-
fluenza [12]. Using a game theoretic model of vaccination the authors have shown
that relatively low disease incidence causes individuals to delay vaccination, for a
year or two in the case of school-based programs for pediatric infectious diseases, or
many weeks in the case of pandemic influenza. Naturally, delaying behaviour also
hinders disease control and can cause subsequent incidence spikes.
3 Policy reinforcement
Models of vaccinating behaviour can also exhibit policy reinforcement, which can
be defined as the tendency for interventions to be boosted by the system’s response
to the intervention. Instead of the negative feedback loop of policy resistance, where
an increase in vaccine coverage tends to create a disincentive for further vaccination
activity, the feedback loop in the case of policy reinforcement is positive, where
increased vaccination activity stimulates still further vaccination activity.
One situation in which policy reinforcement can occur in such models is dur-
ing the transient period when a new vaccine has been introduced and there is some
social-learning process, whereby individuals adopt a vaccinator strategy only if they
have learned that behaviour from someone else [7, 39, 20, 26, 9]. In that scenario,
disease is initially widespread and there is little herd immunity, and so it is optimal
for individuals to get vaccinated. At the same time, if individuals “sample” other
individuals at some rate and only switch to being a vaccinator when they sample
someone who is a vaccinator, then an increase in the abundance of vaccinators will
lead to more instances of vaccinators being “sampled”, and hence more opportuni-
ties for new vaccinators to be created. As a result, there is a virtuous cycle of in-
creasing vaccine coverage, at least until herd immunity creates a disincentive large
enough to outweigh the effect of increasing numbers of vaccinators.
![Fig. 1 Feedback loop arising from interactions between vaccinating behavior and disease incidence. Figure taken from [10].](/figures/fig-1-feedback-loop-arising-from-interactions-between-1kg860ta.webp)
![Fig. 4 Outcome inelasticity in pandemic influenza vaccination. (a) Nash equilibrium vaccine coverage and (b) resulting influenza incidence, at different R0 values and different values of the parameter a governing perceived risk evolution during the pandemic, over each week of the outbreak. Figure taken from [12].](/figures/fig-4-outcome-inelasticity-in-pandemic-influenza-vaccination-2z481qrd.webp)
![Fig. 5 Outcome variability in voluntary ring vaccination. The distribution across many stochastic realizations of the number vaccinated (top) and the number of secondary infections (bottom) in 100 neighbours of an index case, for imitation strength κ = 0.7. Figure adapted from [45]. For details see this reference.](/figures/fig-5-outcome-variability-in-voluntary-ring-vaccination-the-wfkgy2up.webp)
![Fig. 3 Policy enhancement in chickenpox vaccination. Nash equilibria versus social optimum for a range of cost of vaccinations in (a) USA and (b) Israel. Depending on the vaccination cost, there are three Nash equilibria indicated by triangles, squares and filled circles. Solid line (without symbols) indicates the social optima as a function of cost of vaccination. Figure taken from [34].](/figures/fig-3-policy-enhancement-in-chickenpox-vaccination-nash-2xvx60cv.webp)
![Fig. 2 Policy resistance in vaccination for paediatric infectious diseases. Vaccine coverage p∗ at the Nash equilibrium versus relative risk r, the ratio between risk of vaccination and risk of infection, for various values of R0 . Dashed horizontal lines demarcate the critical coverage level that eliminates the disease from the population. Figure taken from [5].](/figures/fig-2-policy-resistance-in-vaccination-for-paediatric-1o1njdx0.webp)