SIR Model with Vaccination: Bifurcation Analysis

The paper establishes the location of the double-zero point at (R0,p)=(2,A^2/(4m)), derives the SN, T, H and B curves, and argues (by interpolation) for a heteroclinic-cycle curve; it explicitly avoids checking the Bogdanov–Takens nondegeneracy conditions and does not rigorously prove the global connection, so parts of the argument are incomplete. See Theorem A for the curves and DZ location, and the authors’ decision to bypass BT nondegeneracy; the heteroclinic curve is obtained numerically and is a cycle between the axis equilibria Ep0 and Ep1, not involving Ep2 (as stated in Section 6.6) . The candidate solution correctly reproduces the invariant calculations, the formulas for SN, T, H, and B, and even verifies the DZ nondegeneracy via det(∂(τ,Δ)/∂(R0,p))≠0. However, it misidentifies the global connection: the heteroclinic cycle in the paper is between Ep0 and Ep1, whereas the candidate claims a connection involving Ep2. The candidate also appeals to generic BT unfolding (tangency of the global-connection curve to SN) without a proof tailored to this model; the paper does not prove that tangency either. Hence, both are incomplete: the paper on BT-genericity and the heteroclinic proof, and the model on the global-connection identification.