Nonlinear Identifiability and Critical Transitions in a Model of School Bullying Dynamics

Abstract

In this paper, we investigate nonlinear identifiability and critical transitions in a compartmental model of school bullying dynamics through three inverse problems: parameter estimation, reconstruction of initial conditions, and their joint identification from observational data. The problems are formulated as nonlinear least-squares tasks and solved using forward simulations combined with multi-start optimization. Numerical results show accurate recovery in the noise-free case and reasonable robustness under noise. However, identifiability strongly depends on the dynamical regime. In particular, near the critical threshold $R_0 \approx 1$, the system exhibits dynamical indistinguishability, where different parameter configurations produce similar observable trajectories. This reflects a loss of practical identifiability associated with critical transitions. The study highlights both the effectiveness and the intrinsic limitations of inverse problem approaches in nonlinear dynamical systems.