A SIRS-Type Dynamical Model for Generative-AI Data Pollution with Synthetic Seeding and Real-Data Preservation

Generative-AI systems are increasingly trained on datasets containing AI-generated content, creating recursive feedback that may cause progressive losses of tail mass, diversity, and effective variance in learned distributions–a phenomenon commonly associated with model collapse. Existing analyses are primarily empirical or based on discrete-generation mixture models, motivating the development of a continuous-time dynamical framework. We develop a SIRS-type compartmental model, in which a normalized training-data pool is divided into clean, AI-polluted, and flagged/recovered fractions, with synthetic seeding (σ) and real-data preservation (ρ). We establish well-posedness, positivity, boundedness, and positive invariance; derive the equilibria and an explicit contamination threshold governing the clean equilibrium in the unseeded system; and analyze local stability and global convergence using planar dynamical-systems methods. Numerical simulations and parameter sweeps illustrate the resulting dynamics. In addition, a proof-of-concept fit of the reduced contamination–seeding subsystem to independently reported web-content prevalence data reproduces the observed post-2022 growth trend (R2=0.977). Contamination (β) and synthetic seeding (σ) increase equilibrium pollution, whereas detection (γ) and real-data preservation (ρ) reduce it. Within the parameter ranges investigated, increasing the cleaning rate (α) can indirectly increase equilibrium pollution by returning more data to the clean compartment, where it is again exposed to contamination. These results identify continuous real-data preservation and effective detection as key mechanisms for limiting synthetic-data pollution in generative-AI training ecosystems.