Vaccination as a Game: Behavioural Dynamics, Network Effects, and Policy Levers—A Comprehensive Review

Abstract

Classical epidemic models treat vaccine uptake as an exogenous parameter, yet real-world coverage emerges from strategic choices made by individuals facing uncertain risks. During the last two decades, vaccination games, which combine epidemic dynamics with game theory, behavioural economics, and network science, have become a very important tool for analysing this problem. Here, we synthesise more than 80 theoretical, computational, and empirical studies to clarify how population structure, psychological perception, pathogen complexity, and policy incentives interact to determine vaccination equilibria and epidemic outcomes. Papers are organised along five methodological axes: (i) population topology (well-mixed, static and evolving networks, multilayer systems); (ii) decision heuristics (risk assessment, imitation, prospect theory, memory); (iii) additional processes (information diffusion, non-pharmacological interventions, treatment, quarantine); (iv) policy levers (subsidies, penalties, mandates, communication); and (v) pathogen complexity (multi-strain, zoonotic reservoirs). Common findings across these studies are that voluntary vaccination is almost always sub-optimal; feedback between incidence and behaviour can generate oscillatory outbreaks; local network correlations amplify free-riding but enable cost-effective targeted mandates; psychological distortions such as probability weighting and omission bias materially shift equilibria; and mixed interventions (e.g., quarantine + vaccination) create dual dilemmas that may offset one another. Moreover, empirical work surveys, laboratory games, and field data confirm peer influence and prosocial motives, yet comprehensive model validation remains rare. Bridging the gap between stylised theory and operational policy will require data-driven calibration, scalable multilayer solvers, and explicit modelling of economic and psychological heterogeneity. This review offers a structured roadmap for future research on adaptive vaccination strategies in an increasingly connected and information-rich world.

1. Introduction

1.1. Motivation and Context

Usually considered one of the first epidemiological models, Daniel Bernoulli’s 1766 work provided a quantitative approach to understanding the impact of smallpox inoculation on mortality, laying the foundation for modern mathematical epidemiology [1]. In other words, one of the earliest epidemic models has already addressed early immunisation (variolation).

Many of the models that would follow were based on the seminal SIR (Susceptible–Infected–Recovered) framework proposed by Kermack & McKendrick (1927) [2]. In this deterministic formulation, a closed population of size N is partitioned into three time-dependent compartments $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ that satisfy

$$\dot{S}=-\beta {\displaystyle \frac{SI}{N}},\dot{I}=\beta {\displaystyle \frac{SI}{N}}-\gamma I,\dot{R}=\gamma I,$$

where $\beta$ is the transmission rate and $\gamma$ the recovery rate. This model introduced the concept of an epidemic threshold and the basic reproduction number $R_0=\beta /\gamma$, establishing a foundation on which later deterministic extensions, including the SIRS models discussed below were built.

Thereafter, vaccination has remained central to epidemic modelling. In classical deterministic frameworks [3], vaccination is imposed exogenously and uniformly, implicitly assuming that everyone accepts the vaccine as soon as it becomes available. These models focus on reducing the basic reproduction number by lowering the susceptible fraction and hence ignore individual decision-making. Shulgin et al. [4] introduced pulse vaccination as a control strategy, yet individuals in that model still play no strategic role.

During the 1990s, Müller [5] and Francis [6] questioned the passive-uptake assumption. Müller’s SIRS (Susceptible–Infected–Recovered[Immune]–Susceptible) model allowed individuals to vaccinate only when their perceived infection risk outweighed vaccination cost, creating a gap between individually optimal and socially optimal coverage. Francis showed that whether vaccination markets are efficient depends critically on dynamic versus static treatment of externalities. These studies introduced economic and behavioural considerations into epidemiology. Until then, vaccination coverage was treated as an exogenous, uniformly applied parameter; what was missing was a framework in which individuals adapt their decisions in response to changing epidemic risks.

A turning-point came with Bauch et al. (2003, 2004) [7,8]. First, Bauch [7] demonstrated, using a smallpox scenario, that voluntary vaccination guided purely by self-interest falls short of the herd-immunity level. The follow-up paper formalised the vaccination game [8]. Here, individuals weigh vaccine risks against infection risks that decrease with increasing coverage, generating a “free-rider” dilemma. Subsequent work has confirmed that perceived vaccine risk, particularly after a “scare”, can keep uptake levels far below eradication thresholds even for highly transmissible pathogens such as measles and pertussis.

1.2. The Vaccination Decision as a Game

Vaccination games expose a paradigmatic conflict between individual and collective interests [8,9]. While public-health agencies seek high coverage, self-interested agents may free-ride on herd immunity, especially when vaccine hesitancy—fuelled by misinformation, safety concerns, or social norms—raises the perceived cost of immunisation [10,11]. Models incorporating game theory, strategy imitation, or explicit contact-network structure show that such feedbacks can generate fluctuating uptake, localised outbreaks, and endemic persistence [9,12,13]. Achieving herd immunity under voluntary uptake, therefore, requires understanding on how people gauge risks, imitate peers, and respond to policy.

Evolutionary-game frameworks capture adaptation over time: agents imitate successful strategies or update beliefs in response to incidence [14,15]. Network models reveal that local structure shapes incentives: clustering can amplify free-riding, whereas targeted incentives to hubs can suppress it [16,17]. Recent extensions address imperfect vaccines [18,19], dynamic perception [20,21], prospect-theoretic risk distortion [22,23], multilayer information-epidemic coupling [22], fractional behavioural memory [24], and multi-strain or zoonotic contexts [25,26]. Together, these approaches provide a wide methodology for designing interventions that anticipate behavioural feedback rather than assuming passive compliance.

Rooted in the adaptive-dynamics tradition of Maynard Smith, evolutionary game theory (EGT) offers a rigorous framework for situations in which the success of a strategy depends on how common it is. Over the past two decades, its reach has expanded well beyond theoretical biology, informing problems in economics, computer science, and the social sciences. Recent work illustrates this breadth: novel “infection–immunisation” dynamics for large strategy spaces in economics [27]; imitation-based models that reproduce real vaccine scares in England and Wales [28]; activity-driven contact networks that link individual mobility to vaccination decisions [29]; multilayer network games that couple choices made in distinct social contexts [30]; and COVID-19 models that embed stay-at-home compliance and shield-immunity policies in an EGT framework [31]. These examples demonstrate how EGT naturally captures the feedback loop between epidemic risk and individual incentives, making it an indispensable lens for viewing vaccination behaviour and the broader public-health dilemmas surveyed in this review.

1.3. Objectives and Structure of the Review

This review has three primary objectives:

(i)

Systematise the rapidly expanding literature on vaccination games, spanning mean-field, network, and multilayer models, and covering both pharmaceutical and non-pharmaceutical interventions.

(ii)

Compare methodological axes: population structure, decision heuristics, psychological realism, pathogen complexity, and mixed policy levers, highlighting how each axis alters the social dilemma.

(iii)

Identify analytical, computational, and empirical challenges and outline future research agendas for data-driven and policy-oriented modelling.

A comprehensive historical trajectory, from classical, non-behavioural mean-field models to feedback models on complex networks, is surveyed by Wang et al. [32]. Readers who require a broad statistical-physics background may consult that review, whereas the present paper narrows its scope to game-theoretic and behaviour-explicit approaches.

The review is structured as follows: Section 2 offers a concise game-theoretic primer on vaccination decisions (from basic SIR games through free-riding and Nash vs. social optima). Section 3 surveys recent extensions across three axes: psychological and behavioural influences, network-based effects, and policy-incentive mechanisms. Section 4 examines key challenges in empirical validation, dynamic feedback, and computational tractability. In Section 5, we draw our conclusions, highlighting emerging trends (including AI/ML integration) and outlining open questions for future research.

2. Short Introduction for Vaccination Games

2.1. Game Theory and Population Dynamics

Game theory provides a natural framework to model vaccination as a strategic choice among self-interested individuals. Consider a large, well-mixed population of constant size N in which each individual makes a one-shot vaccination decision at the start of an epidemic season. Let V denote the strategy “vaccinate” and $NV$ the strategy “not vaccinate”. If the vaccination coverage, that is, the fraction p of individuals who vaccinate, is p, then the expected pay-offs to a representative individual choosing $i\in \{V,NV\}$ are

$$U_V\left(p\right)=-c_v,U_{NV}\left(p\right)=-c_i\pi \left(p\right),$$

where

• $c_v>0$ is the perceived cost of vaccination;
• $c_i>0$ is the perceived cost of infection;
• $\pi \left(p\right)$ is the probability that an unvaccinated individual becomes infected when coverage is p. Here, $\pi \left(p\right)$ is strictly decreasing in the vaccination coverage p; that is, ${\pi }^{\prime }\left(p\right)={\displaystyle \frac{d\pi }{dp}}<0$.

In a Nash equilibrium, no individual can improve their expected payoff by unilaterally changing strategy; for two available actions, this occurs precisely when the payoffs are equal, i.e., when an individual is indifferent between vaccinating and not vaccinating. Hence, an interior Nash equilibrium coverage $p^{\mathrm{NE}}$ occurs when individuals are exactly indifferent between the two actions:

$$U_V\left(p^{\mathrm{NE}}\right)=U_{NV}\left(p^{\mathrm{NE}}\right)⟹-c_v=-c_i\pi \left(p^{\mathrm{NE}}\right)⟹\pi \left(p^{\mathrm{NE}}\right)={\displaystyle \frac{c_v}{c_i}}.$$

Since $\pi (·)$ is strictly decreasing, it admits an inverse function ${\pi }^{-1}$. Applying this inverse yields the equilibrium coverage:

$$p^{\mathrm{NE}}={\pi }^{-1}\left(c_v/c_i\right).$$

Because $\pi \left(p\right)$ decreases in p, a small ratio $c_v/c_i$ (cheap vaccination or severe infection) maps to a high $p^{\mathrm{NE}}$, whereas a large ratio produces low uptake.

The herd-immunity threshold is the minimum vaccination coverage, $p_{\mathrm{crit}}=1-1/R_0$, required to bring the effective reproduction number to $R_e=(1-p)R_0<1$; once this level is exceeded, each case generates, on average, fewer than one secondary infection and sustained transmission cannot occur. In most SIR-type settings, this Nash coverage lies below the herd-immunity threshold, illustrating the classic free-rider dilemma under voluntary vaccination.

Evolutionary models view vaccination strategies as traits with frequencies that evolve over time according to their relative success. A common formulation is the replicator dynamic, in which individuals randomly meet, compare pay-offs, and then imitate the more successful strategy at a rate proportional to the pay-off difference. $x\left(t\right)$ denotes the fraction of vaccinators in a well-mixed population at time t, and $u_V\left(x\right)$, $u_{NV}\left(x\right)$ denotes the average pay-offs to vaccinated and unvaccinated individuals when coverage is x. The replicator equation reads as follows:

$$\dot{x}=\kappa x(1-x)\left[u_V\left(x\right)-u_{NV}\left(x\right)\right],$$

where

• $\kappa >0$ is the learning or timitation rate, scaling how quickly individuals adjust their strategy.
• The term $x(1-x)$ equals the probability that a randomly selected pair consists of one vaccinator and one non-vaccinator in a well-mixed population; imitation can occur only in such mixed encounters, so the term naturally vanishes when either strategy goes extinct ($x=0$ or $x=1$).

Note that an individual who meets a peer using the alternative strategy imitates with probability proportional to the payoff gain; hence, if vaccinators currently earn more ($u_V\left(x\right)>u_{NV}\left(x\right)$), the fraction x of vaccinators increases, whereas it decreases when non-vaccinators fare better.

Fixed points of this dynamic occur at

$$x=0,x=1,\mathrm{and}\mathrm{any}\mathrm{interior}\mathrm{solution}{x}^{\ast }\mathrm{satisfying}{u}_V\left(x^{\ast }\right)=u_{NV}\left(x^{\ast }\right),$$

the latter corresponding to the Nash-equilibrium coverage $p^{\mathrm{NE}}$.

Coupling $\dot{x}$ to an underlying epidemic model—so that $u_{NV}$ depends on current disease prevalence—captures the classic behaviour–incidence feedback. As infections rise, vaccinating becomes more attractive ($u_V>u_{NV}$), driving x upward, which then suppresses transmission, eventually reversing the pay-off advantage and reducing coverage. Such boom-and-bust cycles were first demonstrated in an SIR+replicator framework by Bauch [9] and analysed via Hopf bifurcations by Reluga et al. [10]. These homogeneous replicator models provide a baseline against which extensions, including network structure, prospect-theoretic biases (systematic distortions in risk perception such as probability-weighting and loss aversion—see Section 3.1 for a primer on Prospect Theory), memory effects, and policy interventions, are measured.

Herd immunity represents a collective externality: as the fraction of vaccinated individuals increases, the risk of infection for all individuals, including the unvaccinated, decreases. The infection probability $\pi \left(p\right)$ is typically a decreasing function of the vaccination coverage p. In epidemiology, the basic reproduction number $R_0$ is the average number of secondary infections generated by a single primary case in a wholly susceptible population. With a vaccinated fraction p this quantity decreases to the effective reproduction number $R_e=(1-p)R_0$. Thus, herd immunity transforms vaccination from a purely medical decision into a strategic one, where individual incentives misalign with collective welfare.

Finally, under certain conditions, evolutionary stable strategies (ESSs) may emerge, corresponding to strategies that, if adopted by most of the population, cannot be invaded by alternative strategies [33,34]. While every evolutionary stable strategy (ESS) is a Nash equilibrium, the reverse is not true: an ESS must satisfy an additional stability condition—once it dominates the population, no rare mutant strategy can invade—making it a stricter refinement of Nash equilibrium. In vaccination contexts, these evolutionary dynamics help explain real-world phenomena such as the cyclical drop-and-rebound in pertussis vaccine coverage in England and Wales during 1974–1984, which coincided with three epidemic waves [35]; and the continued endemicity of seasonal influenza despite widespread vaccine availability, a pattern reproduced by behaviour–infection models that couple voluntary vaccination with network structure or adaptive decision-making [36,37].

2.2. Vaccination Game on the SIR Model

Bauch and Earn [8] embed a static (one-shot, at-birth) vaccination game inside a demographically balanced SIR framework, where the per-capita birth rate equals the per-capita death rate so that total population size remains constant. Let $p\in [0,1]$ denote the vaccine uptake level (assumed equal to the long-run vaccine coverage). Newborns enter the population at rate $\mu$; a proportion p is immunised immediately, and the remainder $1-p$ is susceptible. With transmission rate $\beta$ and recovery rate $\gamma$, the epidemiological subsystem is

$$\begin{array}{cc}\hfill \dot{\mathrm{S}}& =\mu \left(1-p\right)-\beta \mathrm{S}\mathrm{I}-\mu \mathrm{S},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \dot{\mathrm{I}}& =\beta \mathrm{S}\mathrm{I}-(\gamma +\mu )\mathrm{I},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{\mathrm{R}}& =\gamma \mathrm{I}+\mu p-\mu \mathrm{R},\hfill \end{array}$$

(3)

with $\mathrm{S}+\mathrm{I}+\mathrm{R}=1$. In the endemic equilibrium $(\dot{\mathrm{S}}=\dot{\mathrm{I}}=\dot{\mathrm{R}}=0)$, one finds

$${\mathrm{S}}^{\ast }={\displaystyle \frac{1}{R_0}},{\mathrm{I}}^{\ast }={\displaystyle \frac{\mu }{\gamma +\mu }}\left(1-{\displaystyle \frac{1}{R_0}}-p\right),{\mathrm{R}}^{\ast }=1-{\mathrm{S}}^{\ast }-{\mathrm{I}}^{\ast }.$$

(4)

with $R_0=\beta /(\gamma +\mu )$. The probability that an unvaccinated individual is ever infected, conditional on this endemic state, is

$$\phi \left(p\right)=1-{\displaystyle \frac{1}{R_0\left(1-p\right)}}.$$

(5)

Parents compare the perceived morbidity risk of vaccination, $r_v$, with that of infection, $r_i$. Setting $r:=r_v/r_i\in (0,1)$ as the relative risk, the expected payoff of choosing probability P to vaccinate, given population coverage p, is

$$E(P,p)=-rP-\left(1-P\right)\phi \left(p\right).$$

(6)

Writing $\partial E/\partial P=-r+\phi \left(p\right)$, every parent is indifferent between vaccinating and not when $r=\phi \left(p\right)$. Hence, the unique convergently stable Nash equilibrium (CSNE) vaccine uptake is

$$P^{\ast }=max\left\{0,1-{\displaystyle \frac{1}{R_0\left(1-r\right)}}\right\}.$$

(7)

Note that the equilibrium implies the following:

• If the perceived relative risk exceeds the threshold $r_0:=1-\frac{1}{R_0}$, then $P^{\ast }=0$ (pure free-rider equilibrium) and disease persists.
• If $r<r_0$, the equilibrium is mixed: only a fraction $P^{\ast }$ vaccinate, which is strictly lower than the eradication threshold $p_{\mathrm{crit}}=1-\frac{1}{R_0}$ obtained by setting $R_e=1$.

Equation (7) formalises why voluntary vaccination rarely attains herd-immunity levels: for realistic childhood diseases ($R_0\gtrsim 5$) even a modest perceived vaccine risk ($r\approx 0.1$) drives $P^{\ast }$ well below $p_{\mathrm{crit}}$. Moreover, after a vaccine scare (temporary increase in r), the nonlinear relation (7) predicts a rapid drop in uptake but a slow recovery, echoing historical patterns such as the pertussis scare in 1970s Britain. Thus, coupling a simple SIR model with a one-shot game—i.e., a game in which each parent makes a single, irrevocable vaccination decision at birth—captures essential trade-offs between individual risk–benefit calculus and collective disease control.

2.3. Voluntary Vaccination and Herd Immunity

Key research lines that deals with voluntary vaccination and herd immunity include the following.

(a) 

Heterogeneous perception and social structure.

Stratifying agents by beliefs widens the gap between Nash and social optima: the fraction of vaccine sceptics drives prevalence non-linearly [38], and age-dependent severity can even invert the usual ordering of $p^{★}$ and $p_{\mathrm{crit}}$ [39]. On cellular automata, clustered imitation produces quasi-periodic vaccination drives [13], while comparing payoffs to average strategy performance can temper free-riding on heterogeneous networks [40].

(b) 

Imperfect or multi-option protection.

Adding low-cost intermediate defences (mask-wearing, hand-washing) creates new equilibria. A continuous “sense-of-crisis” option—modelled as a real-valued level of inexpensive self-protection (e.g., increased hand-washing or reduced contacts) that individuals can tune between 0 and 1—on lattices can displace full vaccinators and raise epidemic size [41]. In a four-strategy mean-field model—where individuals may choose (i) no protection, (ii) vaccination only, (iii) an intermediate defence measure (IDM) such as hand-washing or masking, or (iv) both vaccination and IDM—combining vaccination with an intermediate measure may paradoxically worsen outcomes [19]. Social-learning frequency also matters: high update rates lower prevalence, whereas very slow learning reproduces a purely epidemiological baseline in which vaccination decisions remain fixed and only disease dynamics evolve [42].

(c) 

Pathogen complexity.

With two circulating strains, an imperfect, strain-biased vaccine can facilitate invasion of the mutant strain even while raising overall coverage [25]. When zoonotic reservoirs (i.e., non-human animal hosts that can sustain the pathogen) are present, voluntary human vaccination can eradicate infection only in a semi-endemic regime (infection persists in humans but not in the animal hosts); otherwise, reservoir control (targeted vaccination or culling of the reservoir population) is required [26]. Owner–pet games (evolutionary-game models in which cat owners weigh the costs and benefits of vaccinating their animals) for toxoplasmosis (the protozoan disease caused by Toxoplasma gondii) reveal sharp vaccination-cost thresholds owing to the long-lived environmental oocyst reservoir [43].

(d) 

Empirical and experimental evidence.

Laboratory games confirm strategic motives: in Chapman et al. [44], each participant is assigned a “young” or “elderly” role that mirrors influenza epidemiology (the young drive transmission, and the elderly bear the highest mortality risk). Two payout schemes are compared. Under an individual-payoff scheme, players keep the points they personally save by not vaccinating, so vaccination choices follow the self-interested Nash prediction (more elderly than young vaccinate). Under a group-payoff scheme, every player is paid according to the group total of points, which depends on how many elderly avoid infection; this utilitarian incentive induces more young than elderly to vaccinate, producing higher collective welfare. Real-time interaction experiments further show omission bias and selfish non-vaccination [45,46]. Lim and Zhang [47] demonstrate that if the individual infection risk is a concave function of local coverage—as for highly transmissible childhood diseases where risk falls steeply once moderate coverage is reached—voluntary play can surpass the herd-immunity threshold; with a linear risk curve (appropriate to low-$R_0$ pathogens), this tipping effect disappears.

The central point from these studies remains the same: rational self-interest, however updated, rarely achieves herd immunity without external incentives. Yet, the magnitude, and occasionally the sign, of the Nash-optimality gap depends sensitively on perception heterogeneity, network structure, alternative protective options, and pathogen ecology. Quantifying this gap remains essential for designing subsidies, mandates, and communication strategies that internalise the positive externality of vaccination.

2.4. Free-Riding and Social Dilemmas

Because vaccination confers a public reduction in force of infection but imposes a private cost, the game can be considered a public-goods dilemma [48]. A social planner who values every individual equally minimises the expected total loss

$$C_{\mathrm{soc}}\left(p\right)=p{c}_v+(1-p)c_i\pi \left(p\right),$$

yielding the utilitarian or social-optimal coverage $p^{\mathrm{SO}}=arg{min}_p{C}_{\mathrm{soc}}\left(p\right)$, which is also Pareto-efficient in this symmetric setting (no individual can be made better off without making another worse off). By contrast, a self-interested agent vaccinates only when the private cost is lower than the expected infection loss, $c_v<c_i\pi \left(p\right)$.

The Nash equilibrium coverage is strictly smaller than $p^{\mathrm{SO}}$ so long as $c_v>0$ and ${\pi }^{\prime }\left(p\right)<0$. Depending on the ratio $c_v/c_i$ and the curvature of $\pi \left(p\right)$, the resulting two-player normal form corresponds to a Prisoner’s Dilemma, Chicken or Stag-Hunt game. These classifications support quantitative metrics such as the dilemma strength—the ratio of the temptation and risk payoffs that characterises how strongly defection is favoured—and the social-efficiency deficit (SED), which measures the proportional welfare loss $({\langle \pi \rangle }_{\mathrm{SO}}-{\langle \pi \rangle }_{\mathrm{NE}})/{\langle \pi \rangle }_{\mathrm{SO}}$ [49]. SED is directly analogous to the Price of Anarchy, which is “the ratio between the worst possible Nash equilibrium and the social optimum, as a measure of the effectiveness of the system” [50].

Key research topics for free-riding and social dilemmas include the following.

(a) 

Network amplification and imitation.

Heterogeneous contact graphs create clusters of non-vaccinators that elevate outbreak risk; imitation dynamics generate boom–bust coverage cycles [13]. Aggregating neighbourhood information can damp free-riding when vaccination costs are low, but fails to do so at higher costs, allowing non-vaccinator clusters to persist [40]. On power-law networks, a degree threshold appears when individuals form their decisions according to prospect theory, that is, they overweight small probabilities and underweight large ones and are loss-averse, so that only nodes with degree exceeding a critical value choose to vaccinate (see Section 3.1 for a primer on Prospect Theory); hubs, therefore, become pivotal for breaking the dilemma [23].

(b) 

Empirical and experimental evidence.

Survey and Discrete Choice Experiment (DCE) data show that altruistic concern moves choices towards the social optimum [51]. Framing herd immunity as a collective benefit limits free-riding, whereas individual framing does the opposite [11]. Interactive games corroborate these findings: differential pay-outs for young and elderly reverse free-riding under group incentives [44]; real-time studies reveal that observing high coverage discourages further vaccination, direct evidence of free-riding [46]; the I–Vax paradigm demonstrates omission bias and prosocial heterogeneity [45]. I–Vax is a real-time laboratory game in which groups make repeated vaccinate vs. not-vaccinate choices while an explicit coverage-dependent infection risk generates the herd-immunity externality. When the individual infection risk is a concave function of local vaccination coverage—so that risk drops sharply once a neighbourhood reaches moderate coverage—voluntary play can push uptake past the herd-immunity threshold. In contrast, a linearised (approximately flat) risk curve lacks this tipping property, and voluntary uptake stalls below herd immunity [47].

(c) 

Policy levers and incentive design.

Partial subsidies may inadvertently encourage some former vaccinators to delay immunisation until aid becomes available, whereas well-targeted full subsidies or node-priority mandates achieve higher final coverage [12,52]. Economic relief that compensates households for lockdown losses can stabilise compliance; it serves as the non-pharmaceutical-intervention (NPI) analogue of subsidising vaccination [31]. Budget-allocation studies show that optimal spending profiles depend on perceived risk and migration (short-range commuting and long-term travel that exchange susceptible and infectious individuals between two metapopulations) [53]. Dual-dilemma models—coupling a proactive vaccination game with a retroactive antiviral-treatment game, each possessing its own free-rider problem—indicate that excessive drug use can offset the benefits of immunisation [54].

Theory and evidence converge on the conclusion that free-riding is structurally embedded in voluntary vaccination. Network heterogeneity, behavioural biases and pathogen ecology modulate—but rarely eliminate—the dilemma. Successful policy, therefore, requires aligning private payoffs with public benefits through subsidies, targeted mandates, prosocial framing and cost-sharing mechanisms.

2.5. Comparing Nash and Group-Optimal Vaccination Strategies

To evaluate how far voluntary behaviour can deviate from the collective best, we now compare the vaccination coverage that emerges under self-interest with the coverage that maximises social welfare. Quantifying this gap helps policy-makers judge when incentives or mandates are justified.

Let ${\langle \pi \rangle }_{\mathrm{NE}}$ denote the average social payoff obtained at the Nash equilibrium coverage $p^{\mathrm{NE}}$, and ${\langle \pi \rangle }_{\mathrm{SO}}$ the average social payoff at the welfare-maximising (and therefore Pareto-optimal) coverage $p^{\mathrm{SO}}$. Following Kabir [49], the welfare loss caused by free-riding is measured through the social-efficiency deficit

$$\mathrm{SED}={\displaystyle \frac{{\langle \pi \rangle }_{\mathrm{SO}}-{\langle \pi \rangle }_{\mathrm{NE}}}{{\langle \pi \rangle }_{\mathrm{SO}}}}=1-{\displaystyle \frac{{\langle \pi \rangle }_{\mathrm{NE}}}{{\langle \pi \rangle }_{\mathrm{SO}}}},$$

i.e., the fractional shortfall of realised welfare relative to what is achievable under coordinated play. SED is closely related to the price of anarchy (PoA) widely used in routing and congestion games, differing only by normalisation; both quantify the efficiency loss incurred when individuals act selfishly rather than cooperatively [50]. For a childhood infection with $R_0\approx 10$ and moderate perceived vaccine risk, typical values are $p^{\mathrm{SO}}\approx 0.95$, $p^{\mathrm{NE}}\approx 0.65$, and $\mathrm{SED}\approx 0.40$, implying a 40% loss in social welfare compared with the first-best solution.

Key Research Strands

(a) 

Existence and uniqueness in well-mixed populations.

If the attack ratio decreases monotonically with coverage, the vaccination game admits a unique Nash equilibrium [55]. Although convenient analytically, uniqueness leaves $p^{\mathrm{NE}}<1-1/R_0$, so laissez-faire rarely achieves elimination.

(b) 

Demographic heterogeneity.

Age structure can invert the usual ordering $p^{\mathrm{NE}}<p^{\mathrm{SO}}$. For varicella, moderate coverage shifts cases into riskier adult cohorts so that the socially optimal coverage actually falls below the Nash level, $p^{\mathrm{SO}}<p^{\mathrm{NE}}$ [39]. For poliomyelitis, by contrast, high transmissibility among infants keeps $p^{\mathrm{NE}}\ll p^{\mathrm{SO}}$, prompting many countries to introduce school-entry or infant-schedule mandates that require proof of vaccination for enrolment [56].

(c) 

Network structure and behavioural feedback.

On activity-driven temporal graphs, networks in which, at each timestep, a random subset of nodes becomes active and initiates a handful of short-lived contacts, only the most active individuals choose to vaccinate; in certain cost regimes, this decentralised outcome coincides with the Pareto optimum [57]. Local cost-sharing narrows the gap on heterogeneous networks [58], while node-priority mandates outperform untargeted compulsion (a blanket mandate applied uniformly to every node, regardless of its centrality) [52]. In small networks that are incomplete (missing many potential ties) or asymmetric (nodes have very different numbers of contacts even though the links are undirected), one can observe over-vaccination—the total number of doses exceeds the eradication requirement, but they are concentrated in low-influence nodes—highlighting that who vaccinates can matter as much as how many vaccinate [59].

(d) 

Quantitative indices of the dilemma.

Mapping vaccination games onto the four canonical $2\times 2$ dilemmas, Kabir uses the dilemma strength (DS) and shows that higher vaccine reliability lowers SED in Prisoner’s Dilemma classes but raises it in Chicken games [49]. Concretely, the expected payoffs to a vaccinator and a non-vaccinator at a given coverage are placed in the two-strategy matrix; this “lifts” the population model into a standard $2\times 2$ framework, so the resulting game is still analysed as a two-player dilemma while its payoff entries encode the aggregated epidemiological effects of a large population.

(e) 

Mixed interventions and wider social dilemmas.

When other measures interact with vaccination, the Nash-social gap changes again. Voluntary isolation widens $\mathsf{\Delta }p$ as perceived risk wanes [60]; alternating coercion and laissez-faire never reaches elimination in cellular automata [13]. Similar disparities appear in distancing-versus-vaccination choice [61] and in donor budget games [53].

Although context specific, the deficit $p^{\mathrm{SO}}-p^{\mathrm{NE}}$ seldom vanishes. Targeted subsidies, mandates for high-externality nodes, and accurate risk communication can shrink the gap, whereas demographic asymmetries, imperfect vaccines, and competing interventions can enlarge it. Robust policy therefore hinges on understanding how epidemiological, structural, and behavioural heterogeneity shape the Nash-social landscape.

3. Extensions and Recent Advances

The classical vaccination-game literature establishes a baseline in which perfectly rational, risk-neutral individuals interact in either well-mixed or static network settings. Over the past decade, however, research has moved far beyond this canonical framework. Three broad trends motivate a new tier of models. (i) Behavioural realism: laboratory and field evidence shows that people distort probabilities, care about others, follow peers and update beliefs from personal experience-features poorly captured by expected-utility theory. (ii) Structural realism: empirical contact data reveal temporally varying, multiplex networks with heterogeneity that reshapes both epidemic thresholds and strategic incentives. (iii) Policy diversification: contemporary public-health tools range from targeted subsidies and mandates to real-time risk communication, mixed with non-pharmaceutical measures such as distancing, quarantine and antiviral treatment. These developments have spawned a suite of “second-generation” vaccination games that couple psychological heuristics, dynamic networks, and multi-layer policy levers to pathogen spread.

This section surveys those advances in three parts. We first examine psychological and behavioural influences, focusing on prospect theory, memory, omission bias, and prosocial preferences. We then turn to Network effects, reviewing how heterogeneous, adaptive, and multilayer topologies alter strategic outcomes. Finally, we discuss policy interventions and incentive mechanisms—from subsidies and penalties to communication campaigns—and show how their success hinges on the behavioural and structural contexts introduced earlier. Each sub-section emphasises both modelling innovations and empirical findings, highlighting directions for data-driven, policy-relevant research.

3.1. Psychological and Behavioral Influences

3.1.1. Risk Perception and Prospect Theory

Classical vaccination games implicitly assume that individuals are risk-neutral expected-utility maximisers. Empirical and experimental evidence, however, shows systematic departures from this assumption: people distort probabilities, weigh losses more than gains, and are strongly affected by recent personal experiences. Prospect theory (PT) provides a coherent framework to embed these cognitive heuristics into epidemiological games.

Li and Li [34] integrate PT into a network SIR game by replacing an objective infection probability with a perceived probability. When vaccination costs are low, a lower rationality coefficient (i.e., stronger overweighting of small probabilities) increases equilibrium uptake and reduces both prevalence and social cost, because risk-averse individuals perceive infection as more likely than it is. Han and Li [57] generalise these findings to activity-driven temporal networks: they derive an explicit activity-rate threshold above which agents vaccinate and prove that the unique pure Nash equilibrium coincides with the Pareto optimum, although the final social cost still grows with the vaccination price and infection rate.

Li et al. [22] extend prospect-theoretic vaccination games to a two-layer setting in which epidemic dynamics considering a Susceptible–Infected–Recovered–Vaccinated (SIRV) model interact with information diffusion unaware–aware–unaware (UAU), forming the susceptible–infected recovered–vaccinated/unaware–aware–unaware model. The prospect-theory rationality coefficient $\gamma$ strongly shapes outcomes: for modest vaccination costs, lower $\gamma$ (stronger probability weighting) raises equilibrium uptake and suppresses prevalence, whereas for high costs, the same bias depresses coverage. Their analysis further shows that social heterogeneity in the epidemic layer shifts the epidemic threshold more than comparable heterogeneity in the information layer, highlighting the need for multilayer representations when prospect theory is applied to imperfect vaccines.

Working with a networked Susceptible–Infected–Susceptible (SIS) model, Hota and Sundaram show that prospect-theoretic probability weighting yields a unique degree threshold equilibrium: only nodes with a degree that exceeds $k^{★}$ vaccinate [23]. In practice, exact node degrees are unobserved, but high-activity individuals can be approximated via contact-diary surveys, Bluetooth proximity logs, or occupation-based proxies (e.g., healthcare workers, teachers, ride-share drivers). Empirical studies show that targeting such observable ‘hub’ groups achieves much of the theoretical benefit of degree-threshold vaccination even without full network information. Analytical bounds on $k^{★}$ reveal the same cost–bias trade-off seen in Li’s mean-field work [34]: for low vaccination cost, overweighting of small probabilities lowers the threshold and raises uptake, whereas for high cost, the bias underweights large infection probabilities, pushing $k^{★}$ to much higher degrees and reducing coverage. On power-law graphs, the ratio of behavioural to rational thresholds can grow without bound as the cost approaches the infection loss, highlighting how misperception compounds heterogeneity in network degree.

Beyond one-shot perceptions, vaccine choices evolve through subjective learning. Wells and Bauch [21] show that memory length for past infections versus vaccine failures governs the stability of coverage cycles: long memory stabilises behaviour-incidence dynamics, but the direction of its effect on average uptake depends on whether infection or adverse-event memories dominate. Shim et al. [38] partition the population into vaccine sceptics and believers with different perceived risks; they find that increases in either the proportion of sceptics or their perceived vaccine risk widen the gap between Nash and socially optimal coverage, illustrating the power of misperception cascades.

The propensity to free-ride is not solely cognitive but also contextual. Reluga and Galvani [62] show, via a Markov decision framework, that even small vaccine side-effect concerns create multiple equilibria that depend on the timing of information release. In a spatial cellular-automaton game, Schimit and Monteiro [13] demonstrate quasi-periodic vaccination waves driven by local risk cues and governmental campaigns, underscoring how boundedly rational decisions interact with neighbourhood structure. A systematic review of 178 behavioural change models confirms the trend towards embedding such realistic cognition and information heterogeneity, yet warns that only 15% of studies validate assumptions with data [63].

Taken together, these studies reveal that probability weighting, loss aversion, and memory effects can either hinder or help vaccination campaigns, depending on cost and network context. Policies that frame vaccination as avoidance of a salient loss emphasise recent infection risks and provide clear, timely safety information can harness prospect-theoretic biases to close the gap between individual and collective optima.

3.1.2. Altruism and Prosocial Behaviour

Accumulating empirical evidence indicates that other motives increase voluntary vaccination uptake beyond the level predicted by a purely self-interested Nash equilibrium. Shim et al. quantified altruistic concern about infecting family, friends, and colleagues and embedded it in an influenza vaccination game; even modest prosocial weight shifted the equilibrium close to the social optimum, markedly reducing morbidity and mortality [51]. Laboratory experiments by Betsch et al. confirmed that messaging which stresses the social benefit of herd immunity raises stated intentions, whereas highlighting the individual benefit invites free-riding [11].

Field data confirm these findings. During Israel’s 2013 silent poliovirus circulation, children already protected by inactivated vaccine nevertheless received an oral booster with the sole purpose of protecting others; game-theoretic reconstruction shows the observed 79% coverage is explicable only if a large share of parents acted prosocially [64]. In evolutionary simulations, imitation dynamics amplify vaccinator strategies when agents place positive value on group welfare [9], whereas neglecting such payoffs leads to quasi-periodic government “catch-up” campaigns without elimination [13].

Prosocial preferences extend beyond immunisation. Kabir and Tanimoto modelled compliance with COVID-19 economic shutdowns as a dual dilemma: individuals weigh personal income loss against communal benefit of reduced transmission. Compensation schemes that socialise these private losses stabilise cooperation, mirroring subsidies or risk-communication strategies that harness altruism in vaccination games [31].

Survey evidence also challenges the canonical assumption that higher coverage necessarily depresses individual demand. In a Bayesian D-efficient discrete-choice experiment involving 1919 Belgian adults, Verelst et al. found that both local and population coverage exert a positive linear effect on vaccine utility, whereas side-effects and accessibility remain the dominant drivers [65]. Approximately 40% of respondents were “acceptors” who vaccinate with little deliberation, further weakening free-riding motives. These findings emphasise that peer influence may raise, not lower, uptake-aligning with the network-based imitation results reviewed above.

These studies demonstrate that altruism and prosocial framing can narrow, and sometimes close, the gap between Nash and socially optimal coverage, underscoring the policy value of appeals to collective responsibility. For a broader synthesis of the empirical literature on prosocial motives, the recent review by Böhm and Betsch [66] summarises survey, laboratory, and field evidence, from pre-COVID studies to pandemic data—showing how collective-responsibility beliefs, empathy, and social-benefit framing shape vaccination intentions. They conclude that emphasising community protection is effective chiefly when individual vaccine costs are perceived as low and structural barriers are minimal, a boundary condition that fits well with the experimental and modelling results discussed here.

3.2. Network Effects on Vaccination Decisions

3.2.1. From Homogeneous Populations to Network-Based Vaccination Models

Traditional epidemiological models based on systems of ordinary differential equations (ODEs), such as the classical SIR framework, assume homogeneous mixing, where each individual in the population has the same probability of interacting with any other. This simplification facilitates analytical analysis but neglects important heterogeneities observed in real social structures. The study by [13] addresses this limitation by employing a probabilistic cellular automaton (PCA), in which each individual occupies a cell on a lattice and interacts primarily with local neighbours, mimicking a spatially structured population. Although not explicitly framed as a network model, this setup captures features typical of heterogeneous networks by allowing irregular local interactions.

A natural extension of this line of research involves representing populations as graphs, where nodes correspond to individuals and edges represent potential contacts. Network-based models enable a more realistic characterisation of disease transmission pathways, particularly when the network exhibits complex topological features such as degree heterogeneity, clustering, and modularity. In such models, the epidemiological process often follows the Susceptible–Infected–Recovered (SIR) dynamics adapted to the network context, and vaccination decisions are modelled as a game where each agent optimises their expected utility based on local or global information.

Mathematically, let $G=(V,E)$ be a contact network where V is the set of individuals and E is the set of edges. For each individual $i\in V$, the set of neighbours is denoted by ${\mathcal{N}}_i$. At each epidemic season, the infection spreads according to probabilities determined by the network topology and individuals’ immunisation states. Concurrently, individuals decide whether to vaccinate based on their payoff, which depends on parameters such as the cost of vaccination, the cost of infection, and potentially the behaviour of neighbours.

Behaviour-incidence feedback emerges most clearly when agents let past clinical experience shape present choices. Wells and Bauch [21] used this mechanism in a contact-network model where individuals update their perceived vaccine efficacy after each season, distinguishing between infections, vaccine failures, and even non-influenza ILI (influenza-like illness) episodes that can be misdiagnosed. Here, memory is captured by an exponential-decay parameter $m\in (0,1]$: for instance, an event that occurred d seasons ago enters the current pay-off with weight $exp(-md)$. Smaller m therefore means longer effective memory, because past experiences fade more slowly. Longer memory stabilises the coverage trajectory, yet the equilibrium level depends on whether recollections of infection or of vaccine failure dominate, representing a dichotomy that foreshadows real-world swings in uptake.

Complementing this epidemiological feedback, Han et al. [33] introduce explicit memory decay and conformity into an evolutionary game on Erdös–Rényi and Barabási–Albert networks. Their simulations reveal that strong tie strength and short memory can depress vaccination, while homogeneous networks, despite lacking hubs, achieve higher final coverage under imperfect vaccines.

Kuga and Tanimoto [67] propose a multi-agent simulation model that combines imperfect vaccination with alternative protective measures and test it on Erdős–Rényi and scale-free networks. Network topology proves decisive: heterogeneity in scale-free graphs facilitates outbreaks because hub nodes act as super-spreaders, even when mean vaccination coverage is moderate. Extending this line of work, Tang et al. [58] add global and local “internal-support” mechanisms that subsidise vaccinators (local-support schemes are discussed in detail in Section 3.2.2); their simulations show that, even without any external intervention, local cost-sharing can substantially increase vaccination uptake in complex networks.

Meng et al. [52] couple a node-targeted mandatory phase with a voluntary vaccination game on the USAir transportation network, the Facebook friendship graph, and synthetic Barabási–Albert (BA) scale-free networks. Using eight standard node-importance metrics— degree, betweenness, closeness, eigenvector, PageRank, k-core, structural-hole, and WTOPSIS (abbreviation for weighted technique for order preference by similarity to an ideal solution) centralities—they seed immunity in the highest-ranked nodes before the voluntary stage begins. This targeted seeding reduces the final epidemic size and boosts subsequent voluntary uptake compared with random or wholly voluntary schemes, illustrating how structural heterogeneity can be leveraged to overcome the free-rider dilemma on complex graphs.

Before coupling behaviour to network transmission, one may first ask how vaccination strategies perform when agents do not adapt. Wang et al. review uniform, degree-targeted, and acquaintance vaccination on static, adaptive, and multiplex graphs, deriving analytic thresholds via percolation and spectral criteria [32]. These baselines are valuable for benchmarking the additional effects of strategic decision-making discussed in the remainder of this section.

Iwamura, Tanimoto, and Fukuda extend network games by introducing a third “self-protection” strategy—mask-wearing, hand-washing, and similar low-cost, imperfect measures—and even a continuous strategy space interpolating between full vaccination and complete defection [41]. Counter-intuitively, their evolutionary analysis on both square lattices and Barabási–Albert networks shows that intermediate measures neither raise social welfare nor curb epidemic size; in some parameter regimes, they drive the population towards lower overall immunity by displacing full vaccinators. The result underscores that adding inexpensive but only partially effective options can intensify the free-rider dilemma, especially on heterogeneous graphs where hubs magnify contagion.

A related modification of activity-driven graphs introduces a closeness parameter g, representing the probability that an activated individual reconnects with a previously contacted neighbour rather than a random stranger. Han and Sun show that increasing g simultaneously suppresses epidemic prevalence and reduces vaccination uptake, because tighter social circles diminish perceived risk [29]. For intermediate g, the final recovered density exhibits a non-monotone response to the transmissibility ratio: it first rises as infection spreads more easily, then falls once heightened awareness drives a surfeit of free-riders. The study highlights a network-induced paradox in which interventions that restrict mixing, or simply strong social homophily, may backfire by lowering the incentive to vaccinate even as they curb transmission.

Wei, Lin, and Wu explore how social rewiring shapes the vaccination dilemma on an SIS/SIR network [68]. After each epidemic season, an agent either imitates a neighbour’s strategy or, with probability $\omega$ (rewiring rate), cuts a cross-strategy link, preferentially breaking vaccinator-healthy edges at rate $k_{VH}$ and vaccinator-infected edges at rate $k_{VI}$. Mean-field analysis plus simulations show that vaccination persists at an interior equilibrium fraction $x^{\ast }$ whenever the basic reproduction number $R_0$ satisfies

$$R_0>1+{\displaystyle \frac{k_{VI}}{k_{VH}}}{\displaystyle \frac{tanh\left(\beta P\right)}{tanh\left[\beta (C-P)\right]}},$$

where $\beta$ is the imitation (selection) intensity, P the private cost of vaccination, and C the cost of infection. Rewiring thus acts as a social multiplier on $R_0$, raising or lowering the effective epidemic threshold according to the bias $k_{VI}/k_{VH}$. Selection strength has opposite effects across cost regimes: for $C>2P$, greater rationality ($\beta \uparrow$) depresses coverage, whereas for $C<2P$, it boosts it. The study confirms that tie severing can weigh as heavily as epidemiological parameters in determining long-run vaccination levels.

A complementary approach is offered by Li et al. [22], who couple a prospect-theoretic vaccination game with a multilayer SIRV–UAU network. Their results indicate that local social reinforcement and a higher baseline probability of protection both increase vaccination coverage, while even a small reduction in the infection cost borne by vaccinated individuals can shift the population towards the vaccinator equilibrium. The study highlights how psychological perception and network structure interact to determine epidemic thresholds and equilibrium behaviours.

In summary, the shift from homogeneous ODE-based models to structured network representations offers a richer and more realistic framework for studying vaccination dynamics. These models capture the interplay between topology, strategic behaviour, and epidemiological outcomes, and point to the importance of local incentives and memory effects in shaping public health strategies.

3.2.2. Social Impact and Decision Correlations

Vaccination behaviour rarely emerges from isolated cost–benefit reasoning; instead, individuals look to their peers and the wider information environment. Some works shows that such social impact can either amplify or suppress uptake depending on network structure, cost parameters, and policy context.

Ichinose and Kurisaku coupled an SIR process with an evolutionary game in which agents compare the number, not the payoff, of vaccinating and non-vaccinating neighbours. On scale-free contact graphs, this popularity-based imitation promotes uptake when vaccination is cheap, because once hubs vaccinate, the majority signal cascades through the network; yet the same mechanism decreases uptake when costs are high. In contrast, on degree-homogeneous graphs, social impact typically reduces coverage, exposing a hidden risk for communities with locally uniform contact patterns [17]. These findings generalise earlier time-series work by Reluga et al., who showed that homogeneous perceptions and imitation can generate stable or oscillatory vaccination cycles depending on parameter variance within the population [10].

Internal cost-sharing can reinforce peer influence. Tang et al. introduced a local support mechanism in which every individual contributes a small amount that is redistributed only among vaccinated neighbours. On heterogeneous networks, the scheme maintains high coverage even when the private vaccination cost nearly equals the infection cost, and remains effective when only a subset of nodes participate; by contrast, a global pooling scheme offers weaker incentives and lower steady-state uptake [58]. Because the subsidy is channelled through existing social ties, it naturally aligns economic incentives with the correlation structure of decisions, increasing the probability that neighbouring nodes share the vaccinator strategy.

Large-scale data support the model predictions that decision correlations cluster spatially and digitally. Johnson et al. mapped nearly ${10}^8$ Facebook users and found that anti-vaccination groups form many small, tightly interlinked clusters that penetrate undecided communities, whereas pro-vaccination clusters remain peripheral. Their model predicts that, without intervention, negative sentiment will dominate within a decade, highlighting the role of meso-scale network architecture in steering collective outcomes [69].

A complementary line of work models social influence via social impact theory. Xia and Liu introduce a dual-perspective framework in which each agent minimises the monetary cost of vaccination while simultaneously conforming, with probability $p_{\mathsf{conf}}$, to the dominant opinion in her local interaction network [70]. Simulations on a real high-school contact graph show that high conformity boosts coverage when the vaccine is inexpensive but reduces coverage when the cost is high, because followers adopt the majority non-vaccination opinion. When agents behave as pure social followers ($p_{\mathsf{conf}}\to 1$), uptake converges to a level determined solely by the initial willingness in the network, independent of the actual price. The study thus highlights the non-linear manner in which peer conformity and cost combine to shape steady-state vaccine levels.

Information itself can act as a non-pharmaceutical intervention. Kabir, Kuga, and Tanimoto embed an SIR vaccination game in a multiplex “aware–unaware” framework where information diffuses on a separate layer and toggles individuals between aware and unaware states [71]. Three update rules are examined: individual-based risk assessment, strategy-based imitation, and direct commitment. Phase diagrams in the plane of relative vaccination cost and vaccine effectiveness show that faster information spread consistently lowers the final epidemic size even when vaccination remains costly, because aware individuals adopt intermediate self-protection (mask-wearing, hand-washing) that reduces their infection rate. Yet the choice of social-learning rule matters: society-based imitation achieves the highest average social payoff, whereas direct commitment can yield lower prevalence but at the expense of higher cost. The study demonstrates that awareness diffusion and vaccination strategy interact in complex ways, reinforcing the need to model information contagion alongside biological contagion.

Government incentives interact with social influence in subtle ways. Zhang et al. compare partial- and free-subsidy schemes in an evolutionary game: partial subsidies raise coverage smoothly as individual rationality increases, whereas under a free-subsidy policy, a randomly chosen subset of individuals (“donees”) receive vaccination at zero personal cost. Because this benefit is visible to their peers, the donees act as role-models: their cost-free uptake signals that vaccination is both safe and affordable, encouraging imitation in the local neighbourhood and boosting overall coverage. This role-model amplification persists until the intrinsic vaccine cost becomes so high that even social imitation cannot overcome free-riding, at which point coverage falls back below the level attainable with a partial subsidy [12]. Thus, the efficacy of any intervention hinges on how it reshapes peer signals and decision correlations across the contact network.

3.3. Policy Interventions and Incentive Mechanisms

3.3.1. Subsidies, Penalties, and Mandates

Because voluntary uptake typically stalls at the individually optimal level $p^{\mathrm{NE}}<p^{\mathrm{SO}}$, many jurisdictions deploy economic or regulatory levers to realign incentives. Game-theoretic and optimal-control studies reveal several consistent design principles.

Using an evolutionary SIR game, Zhang et al. compared two schemes: (i) a partial-subsidy that reimburses a fixed fraction of every vaccination and (ii) a free-subsidy that randomly vaccinates a number of individuals at no cost until a budget is exhausted [12]. In well-mixed and networked populations, the partial scheme raises coverage monotonically with the selection strength (agents’ sensitivity to payoff differences), whereas the free scheme displays a non-monotone response: role-model effects initially boost uptake but are eventually offset by free-riding once the budget is large. Targeted partial subsidies, therefore, achieve higher vaccinated individuals per unit expenditure when vaccine cost is moderate to high.

When individuals can choose either vaccination or social distancing, an interior Nash mixture arises only for a narrow band of relative costs $c_v/c_d$ (vaccination versus distancing). Choi & Shim show that if $c_v<c_d$ most agents vaccinate, whereas if $c_v>c_d$ distancing dominates; a regulator can therefore steer behaviour by modest price differentials rather than large mandates [61]. Similar trade-offs appear in SIR/SVIR control problems where vaccination and treatment are jointly optimised. Pontryagin analysis finds that, for realistic parameter ranges, a composite strategy (vaccination during the growth phase, treatment afterwards) yields the smallest discounted cost functional [72].

Incentives need not be tied to the act of vaccination. Kabir & Tanimoto model lockdown compliance as a binary game with payoff $-c_q$ for staying home and $-c_i$ for risking infection; emergency-relief funds that lower $c_q$ shorten the epidemic and the associated macro-economic loss, even when “shield immunity” later emerges [31]. Analogously, Amaral et al. show that reducing the perceived quarantine cost suppresses the amplitude of recurrent infection waves in voluntary isolation dynamics [60].

Alam, Kabir, and Tanimoto extend incentive analysis beyond purely financial levers by embedding forced non-pharmaceutical interventions, quarantine, and isolation into an SVEIR vaccination game [73]. In their mean-field model, individuals choose whether to vaccinate before the season begins, after which public-health authorities may quarantine exposed persons ($q_1$) and/or isolate symptomatic cases ($q_2$). Numerical phase diagrams show that either measure lowers the vaccination level needed to suppress prevalence, but their relative efficacy depends on epidemiological conditions: isolation outperforms quarantine when the progression rate from exposure to infection is high, whereas quarantine suffices at low progression rates. A joint policy (moderate $q_1$ and $q_2$) achieves the smallest final epidemic size at large $R_0$, highlighting the value of coordinating compulsory controls with voluntary vaccination rather than relying on either instrument alone.

A complementary “mixed” design is proposed by Meng et al. [52], who vaccinate a fixed proportion of high-centrality nodes mandatorily before allowing the remainder of the population to decide voluntarily via an evolutionary game. Simulations on empirical (USAir, Facebook) and synthetic (BA) networks show that such node-priority mandates out-perform untargeted compulsion or subsidies of equal size, achieving smaller epidemics and higher average social payoff. The study suggests that structural targeting can substitute for broad coercion when vaccine supply or political capital is limited.

Kabir and Tanimoto formalise a dual-dilemma framework in which proactive vaccination and retroactive antiviral treatment operate on different time scales, each with its own free-rider problem [54]. In their SITR/V (Susceptible–Infected–Treated–Recovered–Vaccinated) model, the treatment game unfolds day-by-day, while vaccination decisions update seasonally. Excessive treatment accelerates the emergence of a drug-resistant strain, raising the control reproduction number $R_r$ and widening the social-efficiency deficit. Phase diagrams show that, even with highly effective vaccines, over-prescribing antivirals can eliminate the gains from vaccination unless prescribing norms are restrained. The study highlights that pharmaceutical interventions applied after infection can create a second social dilemma that feeds back into vaccination uptake, complicating the design of coherent prevention policies.

Most incentive studies take the available budget as given, but Deka and Bhattacharyya integrate budget sizing and distribution with evolutionary vaccination behaviour [53]. In a single well-mixed population, they show that the optimal flow of public funds and the budget-release sensitivity are highly non-linear in the perceived vaccine risk: when scare-driven risk rises, minimal infection cost is achieved by raising the maximum budget yet slowing its release, because rapid spending triggers free-riding once prevalence drops. Extending the model to two countries linked by migration, they find that a third-party donor should prioritise the richer country whenever the poorer country already exhibits high prevalence (and therefore high self-motivated uptake), a counter-intuitive result driven by the higher per-case economic loss in the rich country. The framework thus connects behavioural game dynamics with classical optimal-control questions of how much and where to spend scarce public-health resources.

For vaccines with value that varies across demographic groups, subsidised insurance or school-entry mandates can shift coverage toward $p^{\mathrm{SO}}$. Basu et al. integrate survey-based utility weights into an HPV transmission model and estimate that a federal subsidy covering patients’ direct expenses would raise uptake from the Nash equilibrium ($\approx 40\%$) to the social optimum ($\approx 75\%$) [74]. When sub-populations disagree about vaccine safety, however, raising $c_v$ through penalties can prove counter-productive: Shim et al. show that higher costs widen the gap between sceptics and believers, lowering aggregate coverage unless accompanied by risk-communication campaigns [38].

3.3.2. Public Communication Strategies

Clear, timely, and audience-specific communication is a key pillar of any incentive programme because information mediates how individuals evaluate vaccine risks and benefits. Laboratory evidence shows that framing messages around collective benefits curbs free-riding: when Betsch et al. provided a one-sentence explanation of herd immunity, vaccination intent rose, provided the wording stressed the social value of immunisation; highlighting only the individual benefit, by contrast, depressed uptake as participants strategically relied on others to vaccinate [11]. Field experience corroborates these laboratory findings. In Israel’s 2013 silent polio outbreak, campaign slogans (“Just two drops and the family is protected”) emphasised prosocial motives; survey-calibrated modelling shows that this framing, rather than fear of personal disease, explained the rapid achievement of 79% oral-vaccine coverage among already-protected children [64].

Communication can also reduce hesitancy by correcting false views of relative risk. Shim et al. divide the population into believers and skeptics. Their game model shows that even a small fall in the vaccine risk perceived by skeptics, which targeted education can achieve, removes much of the gap between the Nash and socially optimal coverage [38]. Along similar lines, Reluga and Galvani’s population-game framework finds that clear facts on waning immunity and imperfect efficacy nudge individual choices towards the community best, turning policy resistance into cooperation [62].

Timing is critical. During the 2009 H1N1 pandemic, Bhattacharyya and Bauch showed that early “wait-and-see” behaviour creates a feed-forward loop: high initial concern about adverse events suppresses early uptake and sustains elevated perceived risk, delaying the epidemic peak and prolonging sub-optimal coverage [20]. Their results imply that intensive risk communication is most effective at the very start of an outbreak.

Systematic evidence confirms these patterns. A review of 178 behavioural-change models found that studies incorporating multiple information channels or explicitly modelling information thresholds reproduce empirical uptake patterns more faithfully than models without adaptive communication [63]. Mean-field analyses likewise show that rumours or “vaccine scares” can lock populations into pessimistic equilibria unless countered by transparent messaging, the mechanism identified by the mean-field game treatment of the 2009 H1N1 campaign in France, where individuals stopped vaccinating once fearful narratives dominated [75]. On spatial networks, local signals matter: Schimit and Monteiro’s cellular-automaton model shows that when susceptibles vaccinate only after observing infections in their neighbourhood, coverage oscillates and never reaches elimination, necessitating quasi-periodic public campaigns [13].

4. Challenges in Modelling Vaccination Behaviour

Game-theoretic vaccination models have progressed from proof-of-concept replicas of the classical SIR framework to highly detailed multilayer systems that incorporate cognition, information flow, economic incentives, and evolving contact networks. Yet this greater realism exposes three persistent obstacles that limit the policy relevance and predictive power of the field. First, most studies still calibrate psychological parameters a priori, with only sparse use of surveys, experiments, or digital traces for validation. Secondly, once behaviour and transmission co-evolve, the combined system often exhibits rich nonlinear feedback oscillations, multi-stable regimes, or abrupt transitions, that are difficult to analyse and, in practice, still harder to monitor or stabilise. Third, the very mechanisms introduced to capture heterogeneity, such as large state spaces, temporal networks, and fractional memory, also make the resulting models computationally demanding and analytically intractable.

In this section, we examine each of these bottlenecks in turn. Section 4.1 reviews the current state of empirical grounding, comparing survey calibration, field data, and new digital streams. Section 4.2 summarises the dynamical consequences of two-way behaviour–incidence coupling, highlighting conditions under which voluntary uptake cycles or collapses. Finally, Section 4.3 discusses the computational and analytical hurdles that arise when models include complex contact structures, long memory or high-dimensional strategy sets, and surveys recent attempts, both numerical and theoretical, to overcome these difficulties.

4.1. Model Validation and Empirical Data

Despite increasingly sophisticated theory, most vaccination-behaviour models still rely on assumed behavioural parameters and are seldom confronted with data. In a systematic review of 178 behavioural change models published between 2010 and 2015, only 15% used any real-world evidence for parameterisation or validation; fewer still employed multiple information sources to triangulate behavioural inputs [63]. This evidentiary gap limits the credibility of forecasts and the design of intervention policies that depend on quantitative thresholds (e.g., subsidy levels or messaging intensity).

One avenue is to embed stated-preference data directly into the game. Basu, Chapman, and Galvani collected nationally representative survey responses on perceived risks of HPV, genital warts, and vaccine side-effects, then propagated those perceptions through an epidemiological game-theoretic model [74]. The calibrated Nash equilibrium predicted female vaccination uptake of roughly 32%, far below the social optimum of 67%, highlighting both the usefulness of behavioural data and the policy leverage obtainable by addressing misperceptions. Yet such surveys are expensive, temporally sparse, and prone to social-desirability bias.

A larger and more recent study is the discrete-choice experiment by Verelst et al. [65]. Nearly two thousand Belgian adults ranked six vaccine features, such as peer coverage and side-effects, and the authors converted these rankings into utility weights. Modellers can insert these data straight into evolutionary vaccination games, showing how stated-preference surveys link theory with real behaviour.

Behavioural games with direct links to operational decisions remain rare. Murray’s area-management game for Scottish salmon farms is an exception: the payoff matrix was parameterised with industry figures on testing costs, disease prevalence, and farm clustering [76]. Although the application is aquacultural rather than human vaccination, it illustrates how transactional records can substitute for surveys when behaviour leaves a measurable economic trace.

By contrast, many influential studies are still proof-of-concept. Zhang et al. compared partial and free-subsidy policies in a vaccination game on synthetic networks, but all behavioural parameters (cost, imitation strength, subsidy fraction) were arbitrary, selected for dynamical interest rather than empirical realism [12]. Such models generate qualitative insight but offer limited guidance on whether, say, a USD 10 copayment reduction will materially affect coverage in a real city.

The lack of empirical calibration is not just oversight. Key behavioural inputs, such as imitation rate, risk threshold, and trust, are hard to measure and vary by context. Digital traces from search queries, social media posts, and mobility records add new data but come with sampling bias and privacy limits. A mixed strategy that blends these digital sources with focused surveys and outbreak reports could narrow the validation gap noted by Verelst et al. [63]. Until that happens, the results from behavioural vaccination games should be viewed with caution, and authors should highlight sensitivity checks and clearly state parameter uncertainty.

Wang et al. [32] list many digital data sources, such as participatory apps, social media feeds, Bluetooth contact logs, and mobile phone records, and note the strengths and biases of each. Their review lays the groundwork, yet weaving these detailed behavioural data into vaccination games remains an open task.

4.2. Dynamic Feedback in Behaviour and Epidemics

When the transmission dynamics of an infection and the adaptive choices of individuals are modelled jointly, the resulting two-way feedback can destabilise equilibria and generate persistent oscillations in both vaccine uptake and disease incidence. In the seminal coupling of an SIR model with an imitation dynamic, Bauch showed that vaccine coverage evolves according to a replicator equation that depends on the current prevalence and on perceived vaccine risks [9]. Whenever individuals react strongly to fluctuations in the current prevalence or imitate neighbours with high frequency, the endemic equilibrium loses stability via a Hopf bifurcation, producing relaxation cycles that reproduce the boom-and-bust pattern observed during the pertussis scare in England and Wales (1974–1984).

Reluga, Bauch, and Galvani subsequently generalised these results with a population-game framework in which payoff heterogeneity is allowed [10]. Their analysis shows that (i) polymorphic perceptions of vaccine risk stabilise the feedback loop, whereas (ii) homogeneous perceptions facilitate large-amplitude oscillations. Extending this approach, Reluga and Galvani applied Markov-decision techniques and path-integral methods to vaccination games with waning immunity, demonstrating that delayed booster schedules can also induce cyclical behaviour if the waiting time is treated as a strategic variable [62].

Memory and social conformity modulate these dynamics. Han and Sun embedded payoff memory and celebrity-driven conformity into a network-based game and found that short memory (large decay rate) enhances vaccination responsiveness, thereby decreasing oscillations, whereas long-term memory locks populations into low-coverage, high-incidence cycles [33]. Tang et al. introduced internal support mechanisms that subsidise vaccinators either globally or locally; although the subsidies raise the mean coverage, the feedback loop between epidemic size and vaccine demand remains, and oscillations persist whenever the support parameter is small [58].

The same positive feedback appears in non-pharmaceutical contexts. Kabir and Tanimoto coupled an SEIR model with an evolutionary game in which individuals weigh the infection risk against the economic burden of staying at home. A rising case count increases voluntary lockdown compliance, which suppresses transmission, lowers perceived risk, and eventually triggers premature relaxation, producing recurrent infection waves analogous to the vaccination cycles described above [31].

A recent contribution by Wei and Zhuang generalises these ideas to non-pharmaceutical interventions. They build a two-group asymmetric evolutionary game in which susceptible and infected individuals decide whether to adopt an NPI such as mask wearing, with group-specific costs and efficacies [77]. Replicator analysis shows that the system converges only to pure evolutionary stable strategies (either universal adoption or universal non-adoption), and sensitivity tests identify parameter regions where small cost reductions flip the population from zero to full compliance. Calibrating the model with U.S. mask-wearing data confirms its predictive realism and highlights how heterogeneity in psychological burden (infectors’ guilt) can raise overall NPI uptake. The study extends feedback-driven dynamics beyond vaccination and distancing to everyday protective behaviours.

Taken together, these studies reveal a generic dynamic mechanism: whenever individual behaviour responds endogenously to real-time epidemiological states, the coupled system can oscillate unless some form of damping (behavioural heterogeneity, memory decay, or sustained subsidy) is present. Understanding and quantifying this feedback is therefore critical for designing interventions that avoid the “boom-and-bust” pattern of vaccine uptake.

4.3. Computational and Analytical Challenges

The shift from well-mixed models to explicitly structured/temporal networks has expanded the explanatory power of vaccination games, yet it has also multiplied their computational burden and reduced analytical tractability. Activity-driven frameworks, for instance, require one to track the joint distribution of epidemiological state, activity rate, and strategy across time; even after mean-field closure, this yields a high-dimensional, non-linear ODE system with stability that must be investigated numerically [57]. When degree heterogeneity, imperfect vaccines, and intermediate defence measures are combined, the state space grows further and researchers often resort to large-scale multi-agent simulations to validate pair approximation results [67].

Introducing more realistic decision heuristics compounds the difficulty. Iwamura and Tanimoto’s “sense-of-crisis” update rule depends on each node’s local infection history; the resulting Markov chain cannot be analysed by standard pairwise comparison arguments and must instead be explored through extensive lattice simulations [78]. Memory-based rules such as those used by Wells and Bauch likewise require storing and updating an infection and vaccination ledger for every individual on the contact network, driving the computational cost of a single season into the millions of operations for modest network sizes [21].

A promising attempt to improve both efficiency and convergence properties is the Infection and Immunisation Dynamics (InfImmDyn) proposed by Bulò and Bomze [27]. Starting from Nash’s original “mass-action” idea, InfImmDyn alternates discrete “infection” steps, introducing a small share of a better-reply mutant, with “immunisation” phases that let the population move just far enough to block further invasion. In partnership games, the authors show that the average payoff always rises, the process settles on a Nash equilibrium, and, most importantly, it separates strategies in finite time, something interior-point rules such as the replicator or best-response dynamics cannot guarantee. Because each iteration requires only a handful of payoff comparisons, InfImmDyn offers a computationally attractive equilibrium selector for large strategy spaces, suggesting a path forward for scalable vaccination-game solvers.

A second line of complexity arises when behavioural memory is modelled with fractional-order derivatives. Ullah, Higazy, and Kabir embed evolutionary game dynamics in an SVIR framework of Caputo order $0<\epsilon \le 1$, thereby capturing the non-local “long memory” effect often observed in health behaviour [24]. Analytical results and Adams–Bashforth–Moulton simulations show that increasing the fractional order (longer memory) raises vaccination uptake and lowers prevalence, whereas a cheaper, more reliable vaccine produces similar benefits even when natural immunity is allowed to wane. The study demonstrates that fractional calculus not only widens the stability region of the epidemic system but also amplifies the behavioural feedback, at the cost of substantially heavier numerical solvers and parameter-estimation challenges.

Analytical progress is possible, but only with heavy mathematical machinery. Reluga and Galvani showed that coupling Markov decision processes with population-scale dynamics yields closed-form expressions for Nash equilibria, yet solving the resulting path-integral equations still demands specialised numerical quadrature and is currently feasible only for very small strategy spaces [62]. Hybrid epidemic-game models that embed evolutionary dynamics in quarantine decisions face similar hurdles: capturing recurring waves requires stochastic simulations to complement linear-stability analysis, since small noise can move the system between coexisting attractors [60].

These studies highlight a fundamental trade-off: models rich enough to mirror empirical contact patterns and human psychology are rarely amenable to purely analytical treatment, yet simulations that capture all relevant dimensions quickly become computationally intensive. Further discussion of analytical toolkit-pair approximations, next-generation operators on networks, and percolation theory, can be found in the study of Wang et al. [32], which also surveys recent spectral methods for vaccination threshold estimation on large, heterogeneous graphs. Advancing the field will therefore depend on systematic model-reduction techniques, efficient stochastic solvers, and the judicious integration of empirical data to constrain parameter spaces before simulation.

5. Conclusions and Future Perspectives

In this concluding section, we first summarise the central themes that have emerged from our survey of vaccination-behaviour games, showing how strategic decision-making, network structure, and psychological realism jointly shape immunisation outcomes. We then outline several frontiers for future work, from tighter empirical validation and multi-layer feedback models to adaptive incentive design and AI-driven payoff calibration. Our aim is to chart a path toward vaccination-game frameworks that are not only mathematically rigorous but also grounded in real-world data and capable of guiding dynamic, context-sensitive public-health interventions.

5.1. Summary of Key Insights

This review charts the evolution of vaccination games from Bernoulli’s actuarial calculus to multilayer prospect-theoretic network models that interact with quarantine, antiviral treatment, and real-time information diffusion. Three cross-cutting insights emerge.

1. 

Behaviour-incidence feedback. Endogenising behaviour turns coverage into a state variable that rises and falls with prevalence. From homogeneous replicator models [9] to fractional-memory SVIR games [24], the same pattern recurs: when imitation or risk perception is strong, the coupled system oscillates. Such boom–bust dynamics explain the pertussis scare in 1970s Britain and the stop and go demand observed during COVID-19 lockdowns [31].

2. 

Strategic heterogeneity and network structure. Behavioural thresholds hinge on who meets whom. In activity- driven networks only highly active nodes vaccinate, and local cost-sharing can narrow the Nash-social gap [57,58]. Degree-targeted mandates outperform random ones [52]; by contrast, small asymmetric networks may deliver “too many” vaccinations in the wrong places [59]. Pathogen complexity (multi-strain [25] or zoonotic reservoirs [26]) and prospect-theory weighting [22,23] further modulate these thresholds.

3. 

Memory, perception, and prosocial motives. Probability weighting, loss aversion, and omission bias can either depress or amplify uptake depending on cost [34,45]; long memory for past infection stabilises dynamics, whereas memory for vaccine failure destabilises [21]. Altruistic or social-benefit framing consistently raises demand [11,51,64]. Discrete-choice experiments reveal that coverage by peers often increases rather than decreases utility, contradicting simple free-riding assumptions [65].

Table 1. Modelling aspects and the papers that address them.

Aspect	References (Numeric Keys)
Analytical approximation
Mean-field ODE	[3,8,19,24,25,26,42,43,47,49,53,54,55,72,73,79,80]
Population structure
Cellular automata/lattices	[13,41]
Static or generic networks	[16,17,22,23,29,41,52,59,68,69,70]
Erdos–Rényi graphs	[15,23,33,69,81]
Scale-free/heterogeneous graphs	[15,17,29,55,67,69,81]
Super-spreader hubs	[17,67,69,81]
Decision-making basis
Neighbourhood prevalence	[15,16,17,22,23,33,41,68,69,70]
Population-wide prevalence	[8,19,25,26,42,43,47,55,79,80]
Perceived vaccine risk	[8,11,19,22,23,31,42,45,47,55,68,69,75,79]
Perceived disease risk	[8,15,25,26,42,47,69,79,80]
Direct imitation	[15,16,17,23,33,41,42,68]
Additional processes
Payoff optimisation	[8,49,52,53,55,62,68,72]
Economic cost analysis	[8,47,49,53,54,69,72,79]
Subsidies	[12,52,53,61]
Penalties	[8,61,75]
Mandates/priority	[26,52,56]
Communication/framing	[11,44,64,65,75]
NPIs (distancing, masks)	[31,60,73,77,80]
Vaccination + treatment	[54,73]
Game-theoretic specifics
Voluntary vaccination	[8,42,44,45,46,47,59,68]
Herd-immunity analysis	[26,43,44,47,55]
Prospect theory/memory	[21,22,23,24,45,80]
Altruism	[11,44,51,64,66]
Pathogen complexity
Multi-strain	[25,54]
Zoonotic reservoirs	[26,43]
Psychological realism
Prospect-theory weighting	[22,23]
Fractional memory	[24]
Omission bias	[45,46]
Hybrid interventions
Mandatory + voluntary	[52,73]
Vaccination + treatment	[54]

provides a compact map between these insights and the 80+ studies surveyed.

Overall, the evidence confirms that vaccination programmes cannot be judged on biological efficacy or static cost-benefit criteria alone. Success hinges on anticipating how heterogeneous, memory-laden, and network-embedded agents will respond to evolving risks and incentives.

5.2. Emerging Trends in Vaccination Game Research

Recent scholarship has begun to bridge long-standing gaps between theoretical formalisms and empirically observed behaviour by drawing on three complementary directions.

(i)

Explicit psychological modelling. Early vaccination games treated agents as risk-neutral utility maximisers; contemporary work instead embeds bounded rationality via cognitive biases from behavioural economics. Integrating prospect theory, Li et al. show that probability weighting and loss aversion can raise equilibrium coverage when vaccination costs are modest but may depress uptake in high-cost regimes, reproducing the context-dependent elasticity seen in survey data [34]. These perception-based extensions mark a decisive move beyond homogeneous “expected-utility’’ assumptions.

(ii)

Network-aware dynamical frameworks. Classical vaccination games on static, well-mixed populations are giving way to temporally resolved, heterogeneous network models that capture both contact dynamics and information flow. Han and Li derive closed-form activity-rate thresholds in activity-driven graphs, showing that pure Nash equilibria coincide with Pareto optima only under particular cost–infection regimes [57]. At the opposite extreme of scale, Johnson et al. map over 100 million Facebook users into pro-, anti-, and undecided clusters, revealing how anti-vaccination content penetrates core communities, while pro-vaccine voices remain peripheral [69]. These developments are surveyed in Wang et al. [81], who trace the shift from well-mixed to multilayer, behaviour-feedback models, and in their multilayer-network colloquium, which provides a methodological taxonomy spanning spectral criteria, percolation, pair approximations, and inter-layer coupling [30]. A further frontier is the use of higher-order networks—hypergraphs and simplicial complexes that encode group rather than pairwise contacts—which can fundamentally alter epidemic thresholds and strategic incentives; Majhi, Perc, and Ghosh offer a comprehensive overview of these methods and their dynamical consequences [82].

(iii)

Toward data-driven validation. A systematic review by Verelst et al. finds that only about 15 % of behavioural–epidemic models published between 2010–2015 used real-world data for calibration, but notes a rapid uptake of social media and survey streams in more recent studies [63]. Amaral et al. exemplify this shift: their voluntary-quarantine game links risk perception to SIR dynamics and reproduces the multi-wave COVID-19 patterns observed in Brazil [60]. Likewise, mean-field-game analyses of the 2009–2010 H1N1 campaign in France quantify how pessimistic vaccine-risk perceptions curtailed uptake despite ample supply [75]. These efforts illustrate a growing emphasis on validating strategic models against behavioural and epidemiological time series.

5.3. AI and Machine Learning for Strategic Vaccination Game Models

Recent work has considered artificial intelligence (AI) and machine learning (ML) to shed light on vaccine decision processes, typically by mining social-media posts for sentiment and hesitancy signals. For example, sentiment and stance analyses on Twitter—using classifiers such as SVM, Random Forest, LSTM and ensemble stacks—have revealed how public mood fluctuates with vaccine rollout and news cycles [83,84,85]. Data-augmentation techniques further improve these models’ robustness when labelled data are scarce [86]. Although insightful, these studies remain largely semantic: they detect whether a tweet is positive, neutral, or negative, but seldom link those measurements to real-world vaccination outcomes.

Bridging that gap, Bar-Lev et al. [87] uniquely combine Facebook discussion threads with children’s immunisation records held by Israel’s health maintenance organisations (HMOs—integrated health-insurance providers that run nationwide electronic registries). They show that spikes in negative or highly engaged posts correlate with lower on-time coverage for Hepatitis B, Rotavirus and DTP. This fusion of online sentiment with objective HMO data offers a template for setting payoffs in game-theoretic models, mapping negativity to a higher perceived vaccination cost or to an increased infection probability.

Similarly, Shaham et al. [88] bypass social media entirely, using electronic medical records from 250,000 Israeli patients and an XGBoost classifier to predict individual influenza-vaccination choices. A key finding is the strong predictive power of household members’ prior uptake, a clear behavioural externality that game models represent via peer-dependence in payoff functions.

Yuan et al. [89] apply a mixed-methods design to contrast parental decision routes for seasonal influenza (habitual, rule-based) versus COVID-19 vaccines (deliberative, cue-driven). Their ML feature-importance analysis quantifies how situational anxiety or emotional memory weigh into vaccination payoffs, suggesting that multi-dimensional game models incorporate both fast (social-impact) and slow (habit/memory) decision channels.

On a regional scale, Liu et al. [90] leverage a causality-aware BertMCNN (Bert multi-channel convolutional neural network) model to link brand-specific vaccine hesitancy on Twitter with local epidemic prevalence. Their results underline how dynamic infection intensity should modulate the benefit term in vaccination games (i.e., the $\pi \left(p\right)$ function).

Beyond sentiment and intention, Jaffry et al. [91] use ML on VAERS (Vaccine Adverse Event Reporting System) to test associations between Guillain–Barré syndrome and COVID-19 vaccines. Although they find no significant increase in population incidence, their self-controlled case-series and supervised models refine estimates of perceived infection cost $c_i$ and vaccine-risk ratio.

These AI/ML studies demonstrate both the promise and the current limitations of data-driven vaccination models. To fully leverage these digital signals in vaccination-game frameworks, future work must carry out the following steps:

• Map semantic outputs to game payoffs. Translate sentiment scores, topic intensities or risk-prediction probabilities into quantitative adjustments of vaccination cost $c_v$ and infection probability $\pi \left(p\right)$.
• Fuse multi-source data. Combine social media, electronic health records, and epidemiological time series to calibrate both behavioural payoffs and dynamic feedback loops (e.g., $\pi \left(p\right)$ evolving with local incidence).
• Simulate strategic interventions. Embed AI-inferred payoffs in multi-agent or networked game simulations to test how targeted communication, subsidies, or mandates shift equilibria and eradicate free-riding.

By closing the loop between real-time perception analytics and formal game-theoretic models, AI and ML can drive the next generation of strategic vaccination interventions that proactively align individual incentives with public-health goals.

5.4. Open Questions and Future Directions

Despite substantial theoretical progress, several critical gaps limit the translation of vaccination-game models into actionable public-health tools.

(i)

Empirical grounding and validation. A systematic review found that only about 15% of behavioural models published between 2010 and 2015 used any real-world data for calibration or validation, and even fewer employed multiple data streams or prospective testing [63]. Survey-based calibration has been attempted—for example, Basu et al. calibrated an HPV game with nationally representative risk perceptions [74], while Verelst et al. used a large discrete-choice experiment to estimate utilities for six vaccine attributes in Belgian adults [65]. Such studies remain rare, expensive, and subject to bias. Operational field data are scarcer still: Murray’s salmon-farm area-management game uses industry cost records [76], and Johnson et al. provide population-scale sentiment maps from Facebook clustering [69], yet neither directly parameterises dynamic vaccination games. Laboratory experiments (e.g., Betsch et al. [11], Shim et al. [38]) furnish behavioural anchors, but linking these insights to real epidemics remains challenging. Integrating social-media analytics, mobility logs, or electronic health records with game-theoretic models—and validating predictions prospectively—will be essential for moving from proof of concept to policy tool.

(ii)

Multi-layer feedbacks. Coupled behaviour–disease models show that small changes—such as memory length for past infections versus vaccine failures [21], symptomatic misclassification, or economic fatigue—can flip systems from stable coverage to large oscillations [31]. Reluga et al. demonstrated that payoff heterogeneity can stabilise or destabilise boom–bust cycles depending on belief variance [10], while path-integral methods reveal delay-induced oscillations in booster-timing games [62]. Environmental feedbacks in replicator–ecology couplings generate “oscillating tragedy-of-the-commons’’ dynamics [92]; network-based memory decay or internal subsidies can damp or perpetuate waves [33,58]. Extending these frameworks to real-time, data-driven settings—potentially via machine-learning hybrids that ingest live epidemiological and behavioural streams—remains a pressing frontier.

(iii)

Endogenous incentives. Most incentive studies still assume exogenous subsidies or penalties. Yet, internal support mechanisms, where agents pool small contributions for local redistribution, can outperform global schemes in heterogeneous networks [58]. Partial- versus free-subsidy designs yield non-trivial uptake responses [12], and budget-allocation games reveal counter-intuitive donor strategies that depend on migration and economic loss [53]. Hybrid mandates—vaccinating a core of high-centrality nodes before voluntary stages—achieve higher social payoff than untargeted compulsion [52]; forced quarantine/isolation adds another layer of strategic interplay [73]. Designing truly adaptive incentives that evolve with perceived risk, network topology, and supply constraints remains an open research avenue.

(iv)

Policy realism and heterogeneity. Demographic structure, imperfect vaccines, and multi-pathogen interactions can radically alter the Nash–social-optimum gap. Age-structured models show that for varicella, moderate vaccination may worsen outcomes by shifting cases into older cohorts—reversing the usual inequality $p^{\mathrm{NE}}<p^{\mathrm{SO}}$ [39]; two-age poliomyelitis frameworks underscore the need for mandates in high-transmission infant classes [56]. Multi-strain analyses reveal that asymmetric cross-protection can facilitate mutant invasion despite high coverage [25], and zoonotic-reservoir games highlight cost thresholds beyond which human vaccination alone cannot eradicate disease [26]. Extending such heterogeneous, multi-pathogen models—e.g., to RSV–COVID co-circulation or combined vaccination-distancing choices—could yield more nuanced policy portfolios.

(v)

Behaviour under macro-economic stress. The COVID-19 pandemic exposed how financial constraints shape compliance. Models coupling voluntary lockdown games with SEIR dynamics show that emergency relief—reducing the private cost of isolation—shortens economically costly epidemic waves, but calibration requires granular data on household liquidity and risk perception [31]. Integrating macro-economic modules (e.g., job-loss risk, income support) with vaccination games—and validating them against real economic and epidemiological indicators—remains largely uncharted territory.

Addressing these challenges will require integrated efforts across epidemiology, behavioural science, economics, and data science to build models that are empirically anchored, policy-relevant, and responsive to the socio-ecological complexity of real vaccination programmes. Such advances are essential for guiding effective interventions against both existing and emerging pathogens.