Modeling the interplay between seasonal flu outcomes and individual vaccination decisions

The paper defines the vaccine-coverage map (Eq. 3.7) and reports its long-term behavior largely from numerical iteration. It correctly states that any cost r>0 prevents lasting herd-immunity fixed points and that, for r=0, [pcrit,1] is an invariant interval of fixed points with an additional fixed point p*=1/(2−s), and it presents period-2 oscillations for specific parameters (e.g., R0=1.4, r=0.55, s=0.9). However, the paper provides analytic criteria only in the r=0 case and relies on “numerical simulations suggest” for stability and convergence claims, and on numerics alone for the r>0 two-cycle. By contrast, the candidate solution gives clean, rigorous proofs: (i) no fixed point can exist in [pcrit,1] when r>0 (immediate from Eq. 3.7 with α≡0 on that interval), (ii) a contraction identity establishes the r=0 stability classification and convergence, and (iii) an intermediate value argument yields existence of a 2-cycle crossing the herd-immunity threshold, with a concrete example matching the paper’s Figure 3. Thus, the paper’s claims are correct but analytically incomplete, while the candidate solution supplies the missing proofs. See the map derivation and conclusions in Eq. 3.7 and Sec. 4.1, and the r=0 analysis in the Appendix (A1–A6) ; the final-size relation and pcrit are from Eqs. 3.3 and 3.5 .