Opinion-Dependent Vaccination Capacities

• Opinion-dependent vaccination capacities are models that merge social opinion dynamics with epidemiological processes to determine dynamic vaccination uptake limits.
• These frameworks integrate continuous, kinetic, and network-based approaches to capture discontinuous transitions and critical epidemic thresholds.
• Empirical calibrations and targeted interventions demonstrate how media, leader influence, and demographic factors dynamically steer vaccination behavior.

Opinion-dependent vaccination capacities quantify the maximum or actual rate of vaccination uptake in a population whose willingness is endogenously shaped by the social dynamics of opinion formation. In these coupled systems, vaccination capacity is not a fixed biological or logistic property, but emerges as a function of local or global opinion states—typically fluctuating according to social influence, individual risk perception, demographic heterogeneity, and often dynamic policy input. These frameworks reveal profound couplings between opinion propagation, network topology, epidemiological thresholds, and vaccination outcomes, frequently resulting in discontinuous phase transitions, multistable regimes, and counterintuitive policy responses.

1. Mathematical Foundations of Opinion-Driven Vaccination Capacity

Opinion-dependent vaccination capacity is formalized by coupling compartmental epidemic models (SIS, SIR, SEIR, and their vaccinated extensions) with dynamical systems for opinion formation. The basic idiom is that the vaccination rate for an agent or subpopulation is a monotone function of its current opinion, which itself evolves through both peer exchange and exogenous fields (such as disease prevalence or targeted persuasion). Prototypical formulations include:

• Continuous-opinion models: Each individual $i$ holds an opinion $o_i\in[-1,1]$, with the vaccination propensity defined as $\gamma_i=(1+o_i)/2$ (Pires et al., 2017).
• Kinetic opinion models: Agents are described by joint densities $f_J(w, t)$ over compartment $J$ and opinion $w$, with vaccination flow $K_V(f_S)(w, t)=f_S(w, t)\kappa_V(w)$, where propensity $\kappa_V(w)$ is typically increasing in $w$ (Bondesan et al., 2023).
• Threshold/switching models: When opinions evolve under discrete or majority-rule dynamics, system-level thresholds (such as the majority fraction of initial pro-vaccine agents) set sharp boundaries for long-term vaccination coverage (Pires et al., 2016).
• Networked and demographic extensions: When population structure is incorporated, the local vaccination capacity is a function $\kappa_i(o_i)$ (e.g., $\kappa_i(o_i)=\max(v_i, (1+2o_i)/(1-2o_i))$ on $o_i\in(-1/2, 1/2)$), and is further modulated by the social-signature and demographic responsiveness of each node (Casu et al., 6 Dec 2025).

Such formulations allow for explicit calculation of stationary coverages $V_\infty$, endemic states, and disease-free thresholds in terms of the distribution of opinions and parameters of social response.

2. Mechanisms Linking Opinion and Epidemic Dynamics

In all models, opinion dynamics serve as both a determinant of immediate vaccination decisions and a feedback pathway from epidemic prevalence to social response. Mechanisms include:

• Peer influence and memory: Opinions evolve by memory/self-retention, neighbor imitation (diffusive exchange), and external fields proportional to prevalence or controlled media (Pires et al., 2017, Leung et al., 2023).
• Majority-rule social pressure: Group-level interactions (e.g., majority in triplets) can induce global consensus on pro- or anti-vaccine stances, creating discontinuous jumps in effective vaccination capacity (Pires et al., 2016).
• Persuasion–compromise duality: The balance of persuasive versus compromising interactions determines the fraction of extremists eligible or motivated for vaccination (Alvarez-Zuzek et al., 2017).
• Payoff-driven and imitation-based uptake: Individual decisions may be best-response to ongoing payoff differentials (myopic rationalists) or result from imitating prior adopters (success-based learners), each generating distinct timescales and ceilings for vaccine uptake (Aghaeeyan et al., 2023).

Formally, these mechanisms generate nontrivial fixed points and collective behavior, often characterized by multistability and abrupt regime shifts.

3. Thresholds, Transitions, and Bifurcations

When vaccination rates are functions of social opinion, key epidemiological thresholds become endogenously determined and sensitive to opinion parameters:

• Critical coverage for eradication: In network-coupled models, the control condition for the healthy state is

$v_c = 1 - \frac{1}{\rho(G^{-1})\rho(B)}$

where $v_c$ is the minimal uniform vaccination needed for local stability, $G^{-1}$ is the diagonal recovery matrix, and $B$ the infection-rate adjacency (Leung et al., 2023). It is necessary to maintain opinions sufficient to drive all nodes $i$ to $v_i \geq v_c$.

• Discontinuous (first-order) transitions: For strong risk perception or social influence, the coupled system exhibits first-order transitions: e.g., $I_\infty$ and $V_\infty$ jump abruptly as parameters cross a threshold—unlike classical continuous (second-order) transitions in uncoupled SISV models (Pires et al., 2017).
• Majority-rule-induced bifurcation: In (Pires et al., 2016), the steady-state vaccine coverage $K(p, r)$ is zero for initial pro-vaccine fraction $p < 1/2$, and jumps strictly positive for $p > 1/2$, with explicit formulae for $K$ in both regimes.
• Phase diagrams in opinion/efficacy space: Critical curves $\omega^*(\beta; r)$ and $\beta^*(r; \omega)$ trace the relation between vaccine efficacy, social parameters, and epidemic extinction in multiplex models (Alvarez-Zuzek et al., 2017).

These results underscore the nontrivial relationship between social structure, opinion-modulation of capacity, and epidemic control.

4. Empirical Parameterization and Social Heterogeneity

Empirical calibration of opinion-dependent capacity models reveals substantial heterogeneity across populations:

• Type composition (rationalists vs. learners): Nationwide fits for COVID-19 vaccination in the US estimate that the α parameter—the fraction of myopic rationalists—ranges from 0.31 to 0.76, controlling both the speed and limit of vaccine adoption in each jurisdiction (Aghaeeyan et al., 2023).
• Demographic targeting: The SIS-Vo framework introduces $\bm d_i$ demographic vectors and exogenous input $\bm u$, modeling opinion responsiveness and suggesting that targeted persuasion aligned with demographic profiles can effectively raise subpopulation-specific $\kappa_i^*$ above critical values to secure disease eradication (Casu et al., 6 Dec 2025).
• Network topology and feedback: Both epidemic and opinion network structures (signed links, imitation layers, heterogeneities in peer influence) modulate local vaccination capacities and global thresholds (Leung et al., 2023, Casu et al., 6 Dec 2025).

This suggests that observed variation in coverage is not merely a function of supply or baseline attitudes, but of the dynamic interplay of heterogeneous social feedbacks and interventions.

5. Policy Levers: Media, Leaders, and Interventions

Several models provide explicit means for controlling or maximizing opinion-dependent vaccination capacities:

• Stabilizing opinion via media/actuator input: It is possible to design closed-loop strategies in which media nudges maintain opinion levels above critical values, thereby enforcing $v^* \geq v_c$ and ensuring local stability of the healthy state; discontinuing control leads to rapid reversion to endemic equilibria as opinions erode (Leung et al., 2023).
• Leader/follower dynamics: Explicit inclusion of leader distributions in the opinion Fokker–Planck operator enables quantification of their effect on mean opinion, showing that even small influence ($\lambda_L=0.2$–$0.3$) with strongly pro-vaccine leaders ($m_L=0.9$) accelerates and increases coverage (Bondesan et al., 2023).
• Microtargeting via demographic modulation: By tailoring $\bm u$ to increase $o_i^*$ among critical subgroups, public health messaging can selectively raise their vaccination capacity, providing a potential defense against echo chambers or misinformation that would otherwise localize low-uptake regions (Casu et al., 6 Dec 2025).

A plausible implication is that policy effectiveness depends as much on the social topology and alignment of messengers as on the intrinsic merits of the vaccine itself.

6. Counterintuitive Effects and Design Principles

The coupling between opinion, risk perception, and vaccination capacity induces a series of nonmonotonic or paradoxical phenomena:

• More effective vaccines can reduce long-term coverage: High efficacy ($\phi\to0$) rapidly suppresses prevalence, which weakens perceived risk ($w I$), reducing motivation for ongoing vaccination and causing $V_\infty$ to decline—a "victim of its own success" effect (Pires et al., 2017).
• High initial pro-vaccine sentiment may sustain endemicity: Raising the initial pro-vaccine fraction $D$ can both decrease the short-term epidemic peak and paradoxically sustain infection in the long run, as persistent prevalence maintains $w I$ and inhibits consensus (Pires et al., 2017).
• Bistability and hysteresis: Many systems admit bistable regimes or hysteresis, where small perturbations or transient changes in opinion can tip the population between eradication and endemic disease, with transitions that are irreversible without exogenous intervention (Pires et al., 2017, Pires et al., 2016).

These features highlight the failure of monotonic intuition and the need for dynamic, feedback-aware intervention strategies.

7. Comparison Across Model Structures

Several model classes have been employed to formalize opinion-dependent vaccination capacities. The table below summarizes key features of representative frameworks.

Model (arXiv)	Opinion Variable	Vaccination Capacity	Epidemiological Module
(Pires et al., 2017)	$o_i\in[-1,1]$	$\gamma_i=(1+o_i)/2$	SISV
(Pires et al., 2016)	$o_i=\pm1$ (majority)	$K(p, r)$ (discontinuous)	SISV
(Aghaeeyan et al., 2023)	$\alpha, 1-\alpha$ (type mix)	$\alpha$ dominates max rate	Uptake ODEs
(Bondesan et al., 2023)	$w\in[-1,1]$	$\kappa_V(w)$	SEIRV (kinetic)
(Casu et al., 6 Dec 2025)	$o_i\in(-1/2,1/2)$	$\kappa_i(o_i)=\frac{1+2o_i}{1-2o_i}$	SISV
(Leung et al., 2023)	$o_i\in[0,1]$	via $\eta_{ij}(o)$ imitation capacity	SIRSVo (layers)

Each class provides an explicit, quantitative mapping from the instantaneous (or stationary) opinion state to maximal and realized coverage, as well as control and measurement targets for external intervention.

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Opinion-dependent vaccination capacities thus constitute a unified paradigm for analyzing, predicting, and steering vaccine uptake in the presence of complex social dynamics. The mathematical, computational, and empirical frameworks reviewed not only yield explicit stability criteria and phase diagrams, but also establish the theoretical basis for real-time, demographically informed public health policy in the face of vaccine hesitancy, misinformation, and persistent heterogeneity.