Lyapunov Functions and Stability Analysis of Fractional-Order Systems

The paper proves (via new ABC/CF fractional-derivative inequalities and tailored Lyapunov functionals) that for a general SEIR model with Caputo/CF/ABC operators and incidence F(S,I) satisfying (H), the disease-free equilibrium is (globally) asymptotically stable for R0 ≤ 1 and the endemic equilibrium is (globally) asymptotically stable for R0 > 1; see the model definition and hypotheses (H), Theorem 4.1, and its proofs for V0 and V1, as well as the conclusions emphasizing global stability . By contrast, the candidate solution only establishes local stability via linearization and a quadratic Lyapunov argument, incorrectly asserts that the disease-free Jacobian is Hurwitz at the threshold R0 = 1 (in fact A3 = 0, so the characteristic polynomial has a zero root), and makes unproven assumptions (e.g., F(S,I) ≤ β I globally) to claim a scalar comparison. Hence the model’s proof is flawed and weaker than the paper’s result.