The persistent spread of rumors and misinformation on social media poses severe challenges to public safety. Traditional interventions overlook the idea that repeated debunking induces cognitive fatigue, thus causing diminishing efficiency over time. This paper proposes a novel SHIR-F (Susceptible-Hesitant-Infected-Recovered with Fatigue) dynamic model that incorporates a cognitive fatigue mechanism. By building upon the classical compartmental structure, the model introduces a fatigue state variable and characterizes the diminishing marginal effect of interventions through an exponentially decaying effective debunking efficiency, thereby achieving a quantitative description of public cognitive response mechanisms in information propagation. Regarding the control design, this work constructs a stage-weighted optimal control objective that balances the infection scale against the intervention costs, and develops a closed-loop feedback control strategy parameterized by a dual-branch neural network. This architecture achieves the end-to-end numerical approximation of continuous-time control problems through the collaborative operation of peak and regular branches combined with a fixed-weight fusion mechanism, thus circumventing the computational difficulties associated with solving high-dimensional adjoint equations in traditional Pontryagin methods. A rigorous theoretical analysis proves the positive invariance, global existence, and uniqueness of system solutions, derives explicit expressions for the basic reproduction number, and reveals the monotonic modulation relationship between the fatigue levels and propagation thresholds. Numerical experiments demonstrate that the proposed strategy reduces the infection peaks by 37.4% and the total costs by 48.8%, while ensuring bounded control inputs and fatigue variables, thus exhibiting favorable numerical robustness under parameter perturbation conditions.

We establish global well-posedness for a two-species competitive reaction-diffusion system in bounded two-dimensional domains under Neumann boundary conditions. The system incorporates spatially heterogeneous growth continuous over the closed domain and time-dependent bounded asymmetric intervention. A sharp threshold condition relating the target species' promotion strength to its natural decay rate is necessary and sufficient to maintain the carrying-capacity constraint globally in time, with the two species' total density not exceeding 1. Meanwhile, the unit square with both species' densities ranging between 0 and 1 remains positively invariant, regardless of the target species' promotion strength. Solutions are classical and unique, uniformly bounded, and Lipschitz-continuous with respect to initial data. Numerical simulations show that asymptotically applied interventions that satisfy the threshold condition result in a near-complete eradication of the competitor species at full efficiency. The promoted species establishes a heterogeneous spatial distribution shaped by growth heterogeneity. This framework extends classical diffusive Lotka-Volterra models and provides rigorous theoretical support for ecological management and misinformation suppression strategies.