Global stability properties of a class of renewal epidemic models with variable susceptibility

The paper formulates the variable-susceptibility renewal model (equation (3)) and phase space Ω, proves precompactness under A ∈ BV+ and τ̄ < ∞, and establishes global stability via explicit Lyapunov functionals: U for the disease-free equilibrium when R0 ≤ 1, and W for the endemic equilibrium when R0 > 1 (Theorems 1–2) . The candidate solution proves the same results under the same hypotheses (finite Σ, A ∈ BV+, τ̄ < ∞), but uses a window-maximum argument for R0 ≤ 1 and a different Jensen/g-based Lyapunov construction for R0 > 1. Aside from a minor heuristic bound on S(t,σ) used to obtain a uniform F bound, the model’s reasoning aligns with the paper’s results and assumptions. Hence both are correct, with substantially different arguments for the infection-free case and stylistically different (but compatible) Lyapunov constructions for the endemic case.