Relapse feedback and a counterexample to the disease-free stability threshold in a seven-compartment depression model

This technical note examines the threshold calculation in Mbanguta, Aphane, Shah and Abdeljawad, “A Compartmental Model for the Epidemiology of Depression: Formulation, Dynamics and Implications” (ES General 13, 2026, 2408; DOI: 10.30919/g2408). Retaining the recovery–relapse feedback gives a corrected next-generation number. With the published baseline parameters except beta = 0.9, equation (31) gives 0.913219 < 1, while the full transition system gives 1.945702 > 1 and the disease-free Jacobian has a positive real eigenvalue in (0.0588, 0.0589). A positive endemic equilibrium is constructed with all seven ODE residuals exactly zero in rational arithmetic. For beta = 1.48, the corrected model threshold is approximately 3.199599 rather than 1.501738. The upload includes the PDF, editable LaTeX source, a Python standard-library verification script, and exact-check results. The note addresses the mathematical threshold and disease-free stability assertion; it makes no empirical prediction of real-world elimination and no claim that endemic global stability is false. Correspondence: zeraoulia@univ-dbkm.dz. AI assistance: OpenAI ChatGPT assisted with mathematical drafting and algebraic and computational verification. The derivations and exact rational calculations provide the evidence for the claims.