Global stability of SIR model with heterogeneous transmission rate modeled by the Preisach operator

The paper proves global convergence to a single endemic equilibrium within a continuum of equilibria via a careful branch-by-branch Lyapunov construction and explicit smallness conditions depending on q0 = (R0^nat − R0^int) sup_Π q(α), culminating in Theorem 1 that every trajectory converges to an endemic state when R0^int ≥ 1 + ε for an explicit ε (see the statement of Theorem 1 and its proof sketch, including the roles of ε0 and κ, and the continuum of equilibria Eθ*). By contrast, the candidate solution’s ‘small-gain on slopes’ argument incorrectly assumes a unit bound on the Preisach slice-slope M (it should scale with sup_Π q), conflicting with the paper’s Lipschitz constant q0 and leading to conditions that ignore the necessary dependence on sup q; it also contains a flawed strip-invariance argument and an unjustified energy-supply bound.