Abstract.

This paper proposes a new compartmental framework for describing early childhood risk development (ages 16–36 months) by integrating environmental screen exposure and composite demographic/contextual factors with early intervention mechanisms. The population is partitioned into four interacting classes: low-risk S(t), moderate-risk RM(t), high-risk RH(t), and children enrolled in early intervention I(t). Screen exposure is incorporated through a bounded normalized function T˜(t) ∈ [0, 1] with saturation beyond 4 hours/day, while additional covariates are aggregated into a composite risk pressure. To capture persistence and long-term accumulation effects, the classical model is extended to the Caputo–Fabrizio fractional setting, introducing a non-singular memory kernel. Existence and uniqueness of solutions are established via the Sumudu transform and an iterative scheme, and local stability conditions are derived through linearization and a Caputo–Fabrizio characteristic equation. Numerical simulations further demonstrate that stronger memory effects with smaller fractional order slow transitions between risk classes and delay convergence to equilibrium, supporting the relevance of fractional dynamics for modeling history-dependent behavioral exposure.

Keywords:

autism risk modeling, compartmental model, Caputo–Fabrizio fractional derivative, memory effect, Sumudu transform, screen exposure.