Analysis of an epidemic model with spontaneous human behavioral change

The paper’s Theorem 1 proves that, under the stated non-tangency and finite-alternation assumptions, the slow–fast Poletti model admits orbits Γε that converge to the concatenated singular orbit S, and the terminal point (Sε_f,0,1) → (Sf,0,1) as ε→0. The proof uses standard GSPT/entry–exit machinery, constructs the entry–exit integrals on x=0 and x=1, and concatenates slow SIR segments with fast x-jumps (Sections 4.4–4.6 and 6; see the model equations (3.1)–(3.3), slow–fast form (4.1)–(4.6), entry–exit integrals (4.18),(4.20), and Theorem 1) . The candidate solution replicates this structure: it identifies the critical manifold x∈{0,1} with the switch line I*=k/(mn−ma), defines Λ(I)=k−(mn−ma)I, tracks slow SIR flows while the entry–exit integral H(t)=∫Λ(I)ds remains signed, proves convergence of exit times/points via a variation-of-constants estimate for y=1−x (or z=x), shows fast jumps are vertical with o(1) change in (S,I), and uses a conserved-quantity argument to conclude Sε_f→Sf—matching the paper’s Proposition 5 and the breakdown in the proof of Theorem 1 (Section 6.5) . Differences are technical presentation (the paper leans on a general entry–exit theorem; the model sketches a direct VoC/Grönwall argument), but the logic and assumptions are the same.