Two Exact Counterexamples to a Proposed Threshold for Periodic SIS Epidemic Models

This technical note examines the periodic epidemic threshold assertions in Section 6 of the article by Omaima Slimani, Bouchra Aylaj, and Saber Elaydi, “On the global stability of the discrete-time epidemic models: A new approach,” Mathematical Biosciences and Engineering, 23(3) (2026), 702–721, DOI: 10.3934/mbe.2026027. We identify a mathematical discrepancy between the product of phasewise reproduction ratios used in the published article and the Floquet multiplier governing the local stability of the disease-free periodic solution. Two explicit period-two counterexamples are constructed within the exponential-incidence SIS model considered by the authors. In the first example, the published reproduction quantity satisfies (R_0=4/9<1), whereas the actual Floquet multiplier equals (63/40>1). Moreover, a strictly positive, nonconstant periodic solution exists, contradicting the asserted disease-free global-convergence conclusion. In the second example, the published reproduction quantity satisfies (R_0=3/2>1), while the Floquet multiplier equals (117/125<1). Nevertheless, infection converges to zero for every initial condition with positive total population. This contradicts the corresponding unconditional endemic-persistence conclusion. Additional observations concern the inconsistent concavity requirement in Assumption 3.1, an omitted term in the derivative of the infection map, and the application of the chain rule in the periodic composition. The counterexamples concern the periodic results of Section 6 and do not invalidate the autonomous Beverton–Holt or Ricker stability conclusions. To facilitate independent verification, supplementary materials include the LaTeX source, exact-rational computational checks, reproducibility instructions, verification logs, and file-integrity checksums. AI Assistance Disclosure AI-assisted tools were used for English-language improvements, auxiliary numerical and algebraic checks, and critical review of draft arguments. The author assumes full responsibility for constructing and validating the counterexamples, identifying gaps in the proofs, and ensuring the correctness of all lemmas, propositions, theorems, and mathematical arguments. The mathematical conclusions are supported by explicit reasoning and reproducible calculations rather than the authority of AI-generated output. Publication Status This document is an independent revised technical note and does not constitute an official journal corrigendum.