Incorporating changeable attitudes toward vaccination into an SIR infectious disease model

The paper’s Proposition 1–3 results are: (i) DFE threshold Rv = (ζξ/(ζξ+vωη))R0 with stability iff Rv<1, equivalently ω(0)>ωcr := rh̄Σ̄(0)ζ(0)/ρ(0) (ρ(Y)=Σ̄(Y)(v+rh)+R0hv) ; (ii) in the ε→0 limit, endemic equilibria occur exactly when ω(Y)=ω∗(R0Y) with p(y)=Σ̄(rh̄−y)/(ρ+R0hy), ω∗(y)=χ(y)p(y), and ω∗(0)=ωcr, ω∗(rh̄)=0, implying Y∗<Ymax=(1−R0−1)h̄; a sufficient monotonicity condition is (1−hζ)R0<Σ̄ζ+1 ; (iii) EDE linear stability in the ε→0 limit is equivalent to three Routh–Hurwitz-type inequalities (16)–(18), which for constant Σ reduce to the two-sided bound R0ω∗′(R0Y)<ω′(Y)<R0Υ(R0Y) with Υ(y)=w1/(v+rh) . The candidate solution derives precisely these formulas using the same next-generation matrix argument for (i), the same ε→0 fast–slow reduction and elimination for (ii), and a cubic characteristic polynomial with Routh–Hurwitz for (iii), including the same grouped parameters and explicit identities (e.g., ξ=h̄Σ̄−hω, ω∗(y)=χ(y)p(y), and the constant-Σ identity R0ω∗′(y)=−R0Σ̄w2/(ρ+R0hy)). Minor notation differences (e.g., occurrences of y versus ȳ in w1,w2) do not affect the substance. Net: the proofs are essentially the same and correct, with the candidate giving a slightly more explicit algebraic pathway in a few steps.