Risk Perception Is the Susceptibility Field That Selects Vaccination States on Heterogeneous Networks

Authors: Sterling Geisel, QBist Lab

Abstract

Roy, Shekhar, Ghosh, Kapitaniak, and Hens study vaccination as a coupled epidemic and opinion process on heterogeneous contact networks. Their model treats opinion as a bounded dynamical variable that changes through peer influence and local infection risk, then maps directly into vaccination probability. A Pudding Theory reading identifies the decisive structure as Chaos Susceptibility. The Barabasi-Albert network is not a neutral substrate. It is a susceptibility field in which hubs, local infection exposure, and bounded opinion saturation determine which microscopic signals are amplified into population-scale vaccination states. Risk perception is therefore not an added behavioral parameter. It is the coherent input that selects among unstable epidemic-opinion branches. The source paper’s bistability, threshold shift, and saturation coverage are read as macroscopic signatures of susceptibility-weighted amplification. If the basin-dependent fraction of endemic Monte Carlo realizations at fixed lambda were measured to be independent of the largest local Lyapunov response to risk perception, this Postulate would be falsified.

Source Synopsis

The source paper develops a coupled opinion-epidemic model for vaccination dynamics on heterogeneous networks. Each node is an individual with an epidemic state, susceptible, infected, or vaccinated, and a continuous opinion variable \(O_i(t)\in[-1,1]\). Negative values represent anti-vaccination sentiment. Positive values represent pro-vaccination sentiment. The opinion maps directly into the node’s vaccination probability through \(\gamma_i(t)=(1+O_i(t))/2\).

The epidemic process is a discrete-time SIV model. Susceptible nodes may vaccinate, become infected through infected neighbors, or remain susceptible. Infected nodes recover with probability \(\alpha\). Vaccinated nodes lose immunity with probability \(\phi\). Infection is network-mediated through independent transmission attempts from infected neighbors. The authors implement the process on Barabasi-Albert networks with \(N=2000\), scale-free degree distribution, and mean degree near 12.

The opinion process has two drivers. Peer influence, with strength \(\epsilon\), pushes each node toward the opinions of its neighbors. Risk perception, with strength \(\omega\), pushes opinion toward vaccination according to the local density of infected neighbors. The paper states the central coupling plainly: “The framework incorporates two key features: (i) the vaccination rate evolves dynamically in response to changes in public opinion, and (ii) individual opinions are shaped by peer influence and by risk perception derived from the local network environment.”

The authors compare Monte Carlo simulations with a semi-analytical microscopic Markov-chain approach. They derive the epidemic threshold \(\lambda_c=\alpha/\Lambda_{\max}(A)\) in the no-vaccination limit. They also derive a saturation result, \(V_\infty=1/(1+\phi)\), for the regime where risk perception dominates and opinions saturate at the pro-vaccine boundary. Their numerical results show that high risk perception suppresses infection by raising vaccination. High peer influence can sustain infection when the initial majority is anti-vaccine. In an intermediate infection regime, Monte Carlo runs show bistability between disease-free and endemic outcomes, while the deterministic approximation selects a single branch.

Postulate Lens

The applicable Postulate is Chaos Susceptibility: highly chaotic systems are most susceptible to small coherent inputs. The source system already has the required structure. It is nonlinear, networked, thresholded, and bistable. Its macroscopic state depends on whether local signals cross amplification thresholds before conformity locks the population into a competing opinion basin.

In this setting, chaos does not mean disorder in the colloquial sense. It means sensitivity of collective outcome to local perturbation under positive feedback. The infection state of a neighbor alters \(O_i(t)\). That altered opinion changes \(\gamma_i(t)\). That vaccination probability changes future infection exposure. The loop is recursive. It occurs on a heterogeneous graph whose hubs distort the reach of each local change. A single infected neighbor is not equivalent at every node. Its effect depends on degree, neighborhood composition, and current opinion position relative to the clipping boundaries at \(-1\) and \(1\).

Chaos Susceptibility fits because the source paper’s central observable is not infection alone. It is the conversion of microscopic local risk signals into macroscopic vaccination states. The Postulate reads the network as a field of unequal receptivity. Risk perception is the coherent input. Network heterogeneity supplies the gain.

Pudding Theory Reading

Pudding Theory reads the Roy et al. model as a susceptibility-selected epidemic-opinion field. The population is not a collection of agents who first hold opinions and then act. It is a graph-supported dynamical medium in which local infection traces continuously modulate receptivity to vaccination. The opinion variable is the visible coordinate of this modulation.

The source paper treats \(\omega\) and \(\epsilon\) as behavioral parameters. Under Chaos Susceptibility, their ratio has ontological status. It specifies whether the receiving substrate amplifies local evidence or locks into neighbor-mediated inertia. When \(\omega\) dominates, infection exposure becomes a coherent signal. It aligns local opinion with the vaccination state and drives \(\gamma_i\) upward. When \(\epsilon\) dominates in an initially adverse population, peer influence damps risk input. The system remains susceptible even while infection is present. This is not a failure of information. It is amplification in the wrong basin.

The Barabasi-Albert topology is decisive. The largest eigenvalue \(\Lambda_{\max}(A)\) sets the epidemic threshold in the source derivation. Pudding Theory gives this spectral fact a broader reading. The same heterogeneity that lowers epidemic thresholds also creates high-gain opinion sites. Hubs are not merely efficient spreaders of disease or opinion. They are susceptibility concentrators. Their local state determines whether risk perception becomes population-scale vaccination or remains trapped as fragmented awareness.

The bistability reported in the Monte Carlo runs is therefore central, not incidental. The deterministic MMCA curve averages away the branch competition. The stochastic simulations show the actual Pudding structure: the same parameter setting can resolve into either disease-free or endemic macrostates depending on early microscopic alignments. In Pudding terms, the source’s “finite-size effects” and “strong heterogeneity” are not background noise. They are the mechanism by which susceptibility chooses a branch.

The saturation result \(V_\infty=1/(1+\phi)\) also changes meaning. In the source framing, it follows once \(O_i^\ast\to1\) and infection vanishes. In the Pudding reading, it is the terminal state of successful susceptibility capture. Once risk perception has driven the opinion field to its upper boundary, network topology and infection rate no longer control coverage. The biological waning rate alone remains. The social field has already selected the vaccinated attractor.

Falsifiable Observable

The distinguishing observable is the basin-dependent fraction of endemic Monte Carlo realizations in the intermediate bistable regime, stratified by a measured local susceptibility index combining node degree, infection exposure, and opinion distance from the clipping boundary. Pudding Theory predicts that this fraction is controlled by susceptibility-weighted early risk signals, not only by global \(\lambda,\alpha,\epsilon,\omega,\phi\). If the basin-dependent fraction of endemic Monte Carlo realizations at fixed lambda were measured to be independent of the largest local Lyapunov response to risk perception, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming a standard nonlinear network effect. Roy et al. already have thresholds, bistability, and degree heterogeneity. No extra ontology is needed. The eigenvalue threshold and the vaccination saturation law follow from the Markov equations.

Geisel: The equations are accepted. The reading concerns what the equations identify as causal. In the source account, risk perception and peer influence are adjustable behavioral weights. In the Pudding account, the relevant object is the susceptibility field produced by their competition on a heterogeneous graph. The observable branch selection in Monte Carlo runs is not a nuisance around the deterministic solution. It is the phenomenon.

Tanaka: But susceptibility can be written as a conventional response function. Why attach it to a Postulate?

Geisel: Because the Postulate fixes the interpretation of response. It says unstable coupled systems amplify coherent local inputs into macroscopic outcomes. Here the coherent input is local infection risk. The amplification path is opinion, vaccination probability, and subsequent infection exposure. That path predicts where the deterministic approximation should fail: near branch competition, especially around high-gain nodes.

Tanaka: Then the burden is empirical.

Geisel: Yes. The falsifier is branch independence from measured local susceptibility. If early high-gain risk signals do not predict which Monte Carlo realizations become endemic, this reading fails.

Discussion

The Pudding Theory reading adds a structural account of why risk perception sometimes governs the epidemic and sometimes does not. The source paper shows that high \(\omega\) suppresses infection and high \(\epsilon\) can sustain it. The Postulate explains this as susceptibility capture. A networked epidemic-opinion system does not respond to information uniformly. It amplifies the signals that enter through unstable, high-gain parts of the graph before competing conformity closes the basin.

This reading also gives practical meaning to the mismatch between Monte Carlo and MMCA results. The mismatch marks the regime where branch selection is live. In that regime, averages are less informative than the susceptibility profile of early local events. The next theoretical step is to compute a node-level response measure from perturbations in local infection exposure and test whether it predicts eventual basin membership.

The limitation is clear. The reading depends on the source model’s bounded opinion variable and direct opinion-to-vaccination map. A different behavioral model may distribute susceptibility across memory, trust, or institutional delay. The conclusion would change if bistable outcomes did not track local response heterogeneity.

References

1. Anika Roy, Ujjwal Shekhar, Subrata Ghosh, Tomasz Kapitaniak, and Chittaranjan Hens. “Opinion-Driven Vaccination and Epidemic Dynamics on Heterogeneous Networks.” arXiv:2603.24403. DOI: doi:10.48550/arxiv.2603.24403.

2. Sterling Geisel. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, September 10, 2025.

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