Evolutionary vaccination dynamics under higher-order reinforcement pressure

TL;DR

This work studies how higher-order social interactions shape vaccination uptake and epidemic outcomes by coupling a triadic-learning behavioral layer to a classical SIR epidemic on a lattice. Using a two-layer spatial multiplex, the authors show that higher-order structure can create protective vaccination clusters, and that an optimal reinforcement level $α$ (around $0.5$ in some regimes) maximizes vaccination coverage while too little or too much reinforcement can hinder uptake. The analysis combines a homogeneous mean-field theory with extensive lattice simulations, revealing non-monotonic effects of $α$ on vaccination and highlighting distinct macroscopic patterns (rectangular clusters) that arise from HOIs. These findings bridge complex contagion and evolutionary game dynamics, offering insights for public health strategies that consider structured social influence in vaccination campaigns.

Abstract

Vaccination games in higher-order settings remain underexplored, despite their importance in shaping opinions and collective decisions. Here, we introduce a parsimonious behavioral-epidemiological model to evaluate how peer reinforcement pressure influences vaccination uptake. The framework consists of a two-layer multiplex: an epidemic layer governed by the SIR process on a square lattice, and a behavioral layer represented by a hypergraph of triadic interactions. Individuals update their vaccination strategy via imitation, modulated by a reinforcement parameter $α$ when peer support is present. We find that higher-order structure alone induces clusters of vaccinated individuals that act as protective barriers. Low but nonzero reinforcement ($α\approx 0.5$) maximizes coverage and suppresses outbreaks, while both negligible ($α\approx 0$) and moderate ($α> 0.1$) reinforcement reduce uptake, as excessive confirmation lowers adaptability and enables non-vaccinators to re-emerge. Our work bridges complex contagion theory with evolutionary game dynamics, offering insights into how contact structure and peer reinforcement jointly shape vaccination behavior.

Figures (8)

• Figure 1: Spatial structure and vaccination game scheme.(a) Schematic representation of the two-layer multiplex interaction structure. The top layer illustrates the behavioral network, modeled as a spatially embedded hyper-lattice. Each agent is embedded into four overlapping hyper-edges (triangles), each connecting the agent with two of its neighbors under periodic boundary conditions. The bottom layer depicts the disease transmission network, a regular square lattice through which infections propagate via pairwise contacts. (b) Overview of the seasonal dynamics. Each season consists of two stages: a vaccination decision-making phase, where agents revise their strategy based on payoffs and peer influence within a hyper-edge; and an epidemic spreading phase, modeled by a Susceptible-Infected-Recovered (SIR) process restricted to non-vaccinated individuals. These stages repeat across seasons until the system reaches a behavioral steady state. (c) Strategy updating when the observer and the reference agent share the same strategy, different from the focal agent's; in this case, the imitation probability follows a standard Fermi rule. (d) Strategy updating rule when the focal agent and the observer share the same strategy; here, the Fermi rule is rescaled by a factor $\alpha<1$, representing the reinforcement pressure parameter, reducing the probability of imitation.
• Figure 2: Macroscopic behavior in a well-mixed setting. Simulation results on a well-mixed system compared with analytical predictions from the homogeneous mean-field theory. Panels a) and c) show simulation outcomes on the well-mixed system, while panels b) and d) display the corresponding homogeneous mean-field theory predictions. Top panels depict vaccination coverage (VC) and bottom panels show the final epidemic size (FES).
• Figure 3: Macroscopic behavior of the higher-order vaccination game. Panel a: vaccination coverage, and Panel b: final epidemic size as functions of relative vaccination cost $c$ and reinforcement parameter $\alpha$ on a spatially embedded hyper-lattice. For fixed $c$, vaccination coverage peaks at an intermediate value of $\alpha$, indicating a non-monotonic relationship between higher-order influence and vaccine uptake in the structured lattice.
• Figure 4: Configuration evolutionary dynamics for $\alpha=0$. Panels a) to f) depict, respectively, the microscopic evolution of the system for $\alpha=0$ and $c=0.05$ at seasons $s=0$, $10$, $10^2$, $5\times 10^2$, $10^3$, and $10^4$. At the steady-state, the system self-organizes into a giant rectangular cluster of vaccinated individuals.
• Figure 5: Self-organized vaccination clusters at equilibrium. Panels (a), (b), and (c) depict, respectively, the equilibrium configuration for the relative cost $c=0.05$, $0.1$, and $0.15$, with fixed $\alpha=0$ always (i.e., no reinforcement bias).