A Counterexample to Endemic Global Stability in a Discrete-Time SEIR Model with Saturating Incidence

This technical note critically examines the endemic global-stability conclusions in Saber N. Elaydi, “Global dynamics of a discrete-time SEIR model with saturating incidence,” Journal of Difference Equations and Applications 32 (2026), 1475–1491, DOI: 10.1080/10236198.2026.2697804. Under parameter values satisfying the published assumptions and with the basic reproduction number (\mathcal R_0>1), we construct an explicit counterexample demonstrating that the unique positive endemic equilibrium is locally unstable. An analytical determinant-sign argument, supported by exact rational bounds, proves that the restricted Jacobian possesses a real eigenvalue smaller than −1. Furthermore, a reproducible computer-assisted contraction certificate using rational interval arithmetic establishes the existence of a nonconstant period-two orbit for the same admissible parameter configuration. These results contradict the unconditional global asymptotic stability assertions in Theorems 6.1 and 6.2(ii) of the original article. The note also identifies the limitation in the proposed Volterra–Lyapunov descent argument. The findings do not challenge the basic reproduction-number formula or the existence and uniqueness of the endemic equilibrium. For transparency and reproducibility, a complete supplementary research package is provided, containing the mathematical manuscript, LaTeX source, exact-rational verification scripts, recorded computation results, and SHA-256 checksums. AI Disclosure Statement AI-assisted tools were used for English-language improvements and auxiliary numerical checks. The author assumes full responsibility for the construction and verification of the counterexamples, identification of gaps in the cited arguments, and correctness of all mathematical lemmas, theorems, propositions, and proofs. No AI output is treated as mathematical evidence in place of the explicit analytical and rational-arithmetic certificates. Note: This document is an independent technical note and does not constitute an official journal corrigendum.