Stochastic Dynamics of Health-Risk Information Seeking: Permutation Symmetry and Symmetry Breaking in a Probabilistic Dynamic RISP Framework

Public responses during health crises are shaped by interacting risk perceptions, affect, trust, information needs, overload, misinformation, and protective behavior. Existing applications of the Risk Information Seeking and Processing (RISP) model are largely static and therefore cannot represent stochastic multichannel exposure, delayed correction, or policy feedback. We develop the Stochastic Probabilistic Dynamic RISP (SP-D-RISP) model, which recasts RISP as a bounded stochastic state-space system. Its symmetry structure is explicit: the channel-allocation mechanism is equivariant under simultaneous relabeling of channels and their parameter blocks, while the multi-agent dynamics are invariant to agent relabeling under exchangeable sampling and a label-independent policy. Channel-specific effects, heterogeneous traits, rumor shocks, and interventions generate symmetry breaking. The model combines softmax–multinomial channel competition, discounted Bayesian trust updating, and policy-coupled state transitions. Projection guarantees feasible states by construction, whereas stronger stochastic stability is conditional on a coefficient-level small-gain criterion. For the stationary bounded-memory specification, this criterion is sufficient for Wasserstein contraction, uniqueness of the invariant distribution, and geometric forgetting of initial conditions. The criterion is formulated at the coefficient level and is kept distinct from finite-horizon simulation diagnostics. For the fully disclosed semi-synthetic coefficient vector, the scenario-specific gain matrices have spectral radii between 0.852765 and 0.857123; the worst-case column-sum norm is 0.983948. Thus, the fixed-policy kernels satisfy the stated contraction certificate. For deterministic time-varying paths, the calculation is used only as a common-path one-step certificate, and for the threshold-adaptive rule, it is used only mode by mode rather than as a stationary invariant-law claim. While concentration bounds and Monte Carlo inference quantify population and replication uncertainty, a semi-synthetic experiment with 2500 heterogeneous agents over 90 days examines trust and literacy heterogeneity, clarification delays, communication volume, and intervention portfolios. Within the calibrated SP-D-RISP scenarios, the simulations suggest that higher communication volume may reduce modeled protective behavior when overload effects dominate knowledge gains, delayed clarification may increase transient misinformation, and an integrated portfolio can yield a more favorable simulated outcome profile than the evaluated single-lever strategies.