Theoretical mathematical modeling of compulsive social-media use for digital well-being: cumulative engagement and validation dynamics.

We develop a continuous-time mathematical framework that formalizes how different demographic groups engage with social-media platforms through validation-driven behavioural dynamics, with explicit relevance to digital well-being (DWB). This work is motivated by growing evidence that algorithmically mediated validation feedback (such as likes, comments, and notifications) can generate persistent reinforcement loops associated with compulsive checking behaviour and adverse mental health outcomes, yet lacks a rigorous dynamical formulation. Existing descriptions of likes, notifications, and algorithmic feedback loops remain largely qualitative; in contrast, we construct a measurable modelling structure that expresses these mechanisms as continuous-time processes influencing attention and engagement. Unlike purely exogenous approaches, the model incorporates endogenous feedback between cumulative engagement and platform-mediated validation exposure, allowing engagement to influence subsequent validation intensity through algorithmic amplification. Instantaneous validation intensity is represented as a demographic-specific function of deviations from baseline expectations, capturing responsiveness to normalized feedback environments. Cumulative engagement evolves through a nonlinear differential system with saturation constraints that model diminishing responsiveness and an exponential memory kernel that captures temporal decay of reinforcement. We further decompose validation intensity into like-driven and notification-driven components, formalizing distinct compulsive tendencies associated with visible metrics and intermittent alerts. The resulting nonlinear system permits equilibrium and stability analysis under demographic heterogeneity and feedback strength. Estimation is formulated within a continuous-time state-space framework using Kalman filtering. While the present study emphasizes theoretical development and simulation-based illustration, the framework is constructed to be empirically estimable from platform data. The model provides a mathematically precise representation of social-media reinforcement dynamics with implications for assessing and mitigating digital well-being risks across generations.