Global stability of SAIRS epidemic models

The paper rigorously proves global asymptotic stability of the endemic equilibrium for the SAIRS-with-vaccination model under R0 > 1 with the parameter condition βA < δI, using a Lu–Lu geometric Bendixson framework on the 4D system and the third additive compound, together with a carefully constructed diagonal transformation and a time-averaged Lozinski measure estimate (Theorem 14) . By contrast, the candidate solution attempts a Li–Muldowney approach on the 3D reduction but contains algebraic errors in the 2nd additive compound J^[2], unjustified symmetrization claims for off-diagonals, and a false inequality (S βI ≤ λ), all of which undermine the correctness of the proof. Although the candidate’s conclusion matches the paper’s, the derivation is not sound.