Stochastic Dynamics of Obesity Transmission in a SRO Model: Extinction, Persistence, and Ergodicity Analysis

Obesity is a major public health concern influenced not only by individual lifestyle choices but also by social interactions and environmental conditions. However, deterministic obesity models may not adequately capture the uncertainty associated with behavioral changes and environmental fluctuations, which can substantially affect long-term obesity dynamics. This study aims to investigate the stochastic dynamics of obesity transmission and establish mathematical conditions governing the extinction and persistence of obesity-related behaviors. To this end, we formulate a stochastic Susceptible–Risk–Obese (SRO) model incorporating multiplicative Gaussian white noise. The global existence, uniqueness, and positivity of solutions are established to ensure the mathematical consistency and biological feasibility of the model. Using stochastic Lyapunov techniques and martingale theory, we derive sufficient conditions for the extinction of obesity-related behaviors in terms of a stochastic reproduction index. We further employ the theory of ergodic diffusion processes to establish the existence of a unique stationary distribution under appropriate persistence conditions, characterizing the long-term stochastic persistence of the system. Numerical simulations based on the Euler–Maruyama method support the theoretical results and illustrate how environmental noise influences obesity dynamics. In particular, the numerical examples demonstrate distinct extinction and persistence regimes consistent with the derived threshold criteria, while intervention simulations indicate that reducing social transmission rates can suppress the obese population. The novelty of this study lies in its unified analysis of solution properties, extinction conditions, stochastic persistence, and ergodicity within a single stochastic SRO framework. These findings provide a mathematical basis for understanding how environmental uncertainty and social transmission jointly shape obesity dynamics and may inform the evaluation of interventions targeting obesity-related behaviors.