Advancing Mathematical Epidemic Modeling via synergies with Chemical Reaction Network Theory and Lagrange-Hamilton Geometry

The paper states exactly the two bounds as Theorem 1 (RJ ≤ inf_F RF in the instability domain; RJ ≥ sup_F RF in the stability domain) and defines RJ via a Descartes-type Jacobian factorization. However, its proof makes the logically invalid jump “RJ>1 ⇒ DFE unstable ⇒ RF>1; thus RJ ≤ RF,” which does not justify a numerical inequality from a same-side-of-1 implication. The candidate solution provides the missing and correct argument via a one-parameter scaling of new-infection terms, together with the Descartes property and the standard DFE-stability/NGM equivalence, yielding the desired inequalities. Hence the result is right but the paper’s proof is incomplete; the model’s proof is correct and fills the gap. See the paper’s definitions and Theorem 1 in the Jacobian factorization bound section and the Descartes-lemma therein ; background on the DFE/NGM threshold and non-uniqueness issues appears earlier in the essay , and their algorithmic F−V recipe is in §2.4 .