The transcritical Bogdanov Takens bifurcation with boundary due to the risk perception on a recruitment epidemiological model

The paper establishes the model, disease-free equilibrium E0=(T/(1+μ),0,0), R0=β/(μ+τ), and local stability via explicit eigenvalues λ1=−(1+μ), λ2=(μ+τ)(R0−1), λ3=−(μ+γ), which matches the candidate’s Step 1 and supports a transcritical-with-boundary exchange at R0=1 (E0 on I=U=0) . It derives the endemic equilibrium relations I1=S1(R0−1)+U1(R0(1−η)−1) and S1=C0U1 with the closed forms for C0, D0 and U1, consistent with the candidate’s Step 2; it asserts existence for R0>1 (Theorem 2), though without a careful positivity proof of the U1 denominator that the candidate provides . For the transcritical Bogdanov–Takens (tBT) analysis, the paper defines the unfolding parameters δ1=R0−1, δ2=μ+γ and identifies a double-zero eigenvalue at E0 (Lemma 1) but then explicitly states that the analytic proof of a tBT with boundary is beyond scope and presents only numerical evidence of a small cycle and near-homoclinic behavior (Figures 4–5) . In contrast, the candidate performs an explicit center manifold reduction at δ1=δ2=0 (β=τ, μ=γ=0), obtains a two-dimensional reduced system with nonvanishing mixed quadratic terms, shows the transcritical degeneracy (no u^2 term in v̇) and gives the Hopf curve scaling δ1=κH δ2^2 and a nearby homoclinic curve, completing the missing analytic part. Hence, the paper’s qualitative claims are largely correct up to the tBT proof, which it does not provide, whereas the model’s solution supplies a consistent and standard analytic derivation of the tBT with boundary configuration.