Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator

The paper proves (i) a Poincaré–Bendixson-type alternative: for R0>1, every trajectory converges either to an endemic equilibrium or to a simple periodic orbit (Theorem 1), and (ii) under Assumption (A) (convex hysteresis loops) and a small loop-width parameter L, all solutions converge to an endemic equilibrium and periodic orbits are excluded (Theorem 2). Both results are rigorously established using a switched-systems viewpoint and Lyapunov functions tailored to the SIR branches and to the Preisach operator’s geometry . By contrast, the candidate solution leans on general monotone-cyclic negative-feedback theorems (Mallet-Paret–Smith) and monotone small-gain arguments, without verifying the strong monotonicity/semi-flow hypotheses for the infinite-dimensional closed loop with memory, and it asserts an explicit small-gain threshold L/β<1 (effectively L0=β) not supported by the paper’s analysis, which derives a problem-dependent bound L0 via detailed estimates. It also uses an unproven Lipschitz bound on the Preisach branches needed for the contraction claim. Hence, the paper’s results are correct, while the model’s proof is not justified as written .