Mathematical Observations on a Delayed Smoking–Lung-Cancer Model: Positive Invariance, Stability, and Discrete Consistency

This technical note examines the mathematical formulation, positivity properties, stability arguments, and numerical discretizations presented in the article "Evaluating lung cancer control through a time-delay mathematical model with detailed sensitivity analysis" by Zeb et al., published in the Journal of the Nigerian Society of Physical Sciences, 8 (2026), Article 3539. The main findings are as follows: An explicit counterexample with a strictly positive initial susceptible population demonstrates that the nonnegative invariance assertion in Theorem 2.1 does not hold under its stated hypotheses. A corrected delay-dependent characteristic equation is derived. Despite a deficiency in the original stability proof, the reproduction threshold is shown to yield the correct local disease-free stability dichotomy. The Lyapunov derivative used in Theorem 2.5 is recalculated, revealing an algebraic discrepancy that invalidates the published global-stability argument as written. An explicit numerical evaluation demonstrates that the particular Lyapunov function proposed in Theorem 2.6 can have a strictly positive derivative, so its claimed monotonicity is not established. The published Euler, Runge–Kutta, and nonstandard finite-difference formulations are examined. As printed, these schemes discretize an instantaneous system rather than the stated delay differential equations. Additional conditions required for numerical positivity and boundedness arguments are identified. The note distinguishes established mathematical contradictions from deficiencies in proofs and numerical formulations. It does not claim that the underlying equilibria are necessarily unstable or that the complete epidemiological conclusions of the original article are false. The purpose of this contribution is to provide mathematically precise, constructive observations that may support clarification or correction of the original analysis. AI Assistance and Author Responsibility OpenAI ChatGPT assisted with English-language improvements, auxiliary numerical verification checks, and preparation of the preliminary mathematical review. The author assumes full responsibility for independently verifying, adopting, and approving all counterexamples, lemmas, propositions, theorems, proofs, and mathematical conclusions. Original article: https://doi.org/10.46481/jnsps.2026.3539 Correspondence: [email protected]