Wave propagation for a discrete diffusive vaccination epidemic model with bilinear incidence

The paper proves that for the lattice vaccination model (1.3) there exist positive traveling waves for R0>1 and c≥c*, with c* characterized by the double-root condition for Δ(r,c), and that the waves connect E0 to the unique endemic equilibrium E*; see the abstract and goals, the wave system (2.4)–(2.6), the dispersion relation and Lemma 2.1, the explicit sub- and super-solution construction (Lemma 2.2), the Schauder fixed-point formulation (3.1) and compactness, the boundedness and persistence of I (Section 4), and the Lyapunov functional proving convergence (Theorem 5.1), as well as existence at c=c* by a limiting argument (Remark 5.1) . By contrast, the model’s outline relies on (i) an unsubstantiated “Harnack-type estimate” to ensure positivity of I behind the front (the paper instead proves lim inf I>0 via a ratio/log-derivative argument), and (ii) a Lyapunov functional lacking the nonlocal integral compensators crucial to control the discrete Laplacian; both points are essential in the paper’s proof and cannot be replaced by the stated shortcuts. Hence, the model’s proof is flawed/incomplete at critical steps.