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2404.12776

SIR models with vital dynamics, reinfection and randomness to investigate the spread of infectious diseases

Javier López-de-la-Cruz, Alexandre N. Oliveira-Sousa

correctmedium confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper’s Theorem 2.1 states exactly the autonomous-case conclusions: well-posedness with a global attractor contained in B0 = {S+I+R=q/a}, a single equilibrium E*_0 attracting all orbits when γ < a+b+c, and for γ > a+b+c, an attractor equal to {E*_0, E*_1} together with the unique heteroclinic ξ from E*_0 to E*_1, with the semigroup gradient with respect to the equilibria. Their proof uses reduction to the positively invariant plane B0 and a Dulac function D(S,I)=1/I to exclude periodic orbits, plus manifold arguments to identify the attractor and gradient structure . The candidate solution proves the same statements via the same structural steps (N-dynamics and absorbing set; B0-reduction; Bendixson–Dulac in the reduced 2D system; heteroclinic identification; Conley Lyapunov for gradient) and even fills a minor technical gap by explicitly defining a continuous extension of SI/N at N=0. Hence both are correct and essentially the same approach.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The autonomous-case results are correct and follow a standard, rigorous pathway: invariant plane via N-dynamics, planar reduction, Dulac to exclude cycles, and classification of equilibria leading to a complete attractor description and gradient structure. The exposition would benefit from making a couple of technical points explicit (definition at N=0 and a brief Dulac computation), but these are minor and do not affect correctness.