Fractional-order safe mental-health corridor modelling with Matignon spectral analysis of post-pandemic...

Mental health disorders, particularly anxiety, fatigue, and depression, exhibit complex dynamical features including delayed onset, memory-dependent progression, cyclical relapse patterns, and heterogeneous population outcomes that transcend conventional integer-order modeling frameworks. This study develops a novel fractional-order HAFCU model partitioning the population into five interacting compartments: Healthy (H), Anxious (A), Fatigued (F), Critical (C), and Under Recovery (U), incorporating an explicit relapse pathway from recovery back to anxiety to represent the recurrent nature of psychological disorders observed clinically. Employing the Atangana-Baleanu fractional derivative in the Caputo sense with its non-singular Mittag-Leffler kernel, the model uniquely captures fading memory effects wherein past traumatic experiences and cumulative stress continuously influence current psychological transitions- a phenomenon well-documented in psychiatric epidemiology but previously unaddressed in compartmental frameworks. The basic reproduction number, derived via the NGM approach as R₀=max{θ₁H^∗κ₁+μ₁,θ₂H^∗κ₂+μ₂} with computed value R₀=0.625, establishes the threshold below which psychological distress naturally attenuates without sustained intervention. Rigorous mathematical analysis proves existence, uniqueness, positivity, boundedness, and both local and global stability via Lyapunov method, while a novel Matignon spectral stability theorem adapted for ABC-kernel systems provides eigenvalue-based criteria for local asymptotic stability under memory-driven dynamics. Key innovations include: (i) the first mathematically defined safe mental-health corridor- a positively invariant region where clinical depression and combined psychological load remain below empirically motivated thresholds informed by WHO global estimates; (ii) a predictor-corrector numerical scheme specifically tailored to the ABC kernel with proven convergence of order 1+υ; and (iii) the introduction of the Lorenz curve and Gini coefficient (G=0.287) as quantitative metrics for inequality in psychological burden distribution across compartments- a methodological breakthrough enabling empirical assessment of mental health disparities. Numerical simulations reveal convergence to equilibrium at t=100 days with compartmental distribution H=0.1517, A=0.0947, F=0.1466, C=0.1566, and U=0.4504, confirming that memory effects delay transitions and smooth trajectories while preserving global stability. Sensitivity analysis identifies anxiety induction (θ₁) and fatigue induction (θ₂) as the most influential parameters for distress propagation (sensitivity indices +1.000), and optimal control characterization provides explicit expressions for relapse prevention (u₁^∗) and recovery enhancement (u₂^∗) strategies. Validated according to WHO (2022) post-pandemic mental health statistics documenting a 25% global increase in anxiety and depression, the model successfully reproduces three fundamental empirical patterns: (i) the steep rise in psychological distress during crisis periods, (ii) delayed recovery trajectories extending beyond 100 weeks, and (iii) heterogeneous outcomes reflected in the Lorenz curve disparity. This integrated framework advances fractional calculus theory in psychiatric epidemiology while providing a mathematically grounded decision-support instrument for mental health policy, intervention timing, and resource allocation across diverse population cohorts.