Stochastic Reproduction Number and Ergodicity of a Saturated-Incidence SIHR Rumor Model with Optimal Control

This paper develops a stochastic susceptible–infected–hesitating–removed (SIHR) rumor model with Hill-type saturated incidence and compartment-specific Brownian perturbations. The model focuses on when environmental noise changes a rumor’s deterministic fate and how long-run behavior is organized as an invariant law rather than a fixed level. The hesitating class is interpreted as a post-spreading verification state: users have already engaged with the rumor but temporarily suspend further forwarding before removal. A corrected stochastic local reproduction number is derived from the logarithmic growth rate of the linearized spreader equation along the stochastic rumor-free boundary. Its Itô correction includes the second moment of the susceptible boundary process and therefore depends on the common attrition noise. Global existence and positivity are established from linear-growth and positive-invariance arguments, and the deterministic skeleton is shown to be well-posed, positive, bounded, and globally attracted to the endemic equilibrium when R0>1. Under explicit dissipativity and comparison assumptions, a Foster–Lyapunov argument with the full diffusion covariance matrix yields positive Harris recurrence, a unique invariant probability measure, and an ergodic time-average theorem. A media-debunking control problem is formulated through the stochastic maximum principle, while the implemented numerical schedule is stated as a drift-level open-loop approximation and then evaluated under the full stochastic dynamics. Numerical experiments, including 95% confidence bands, strong-saturation tests, stationary-density diagnostics, sensitivity surfaces, and control comparisons, support the revised threshold, recurrence, and intervention interpretations.