A minimal model for adaptive SIS epidemics

The paper’s general aNIMFA analysis derives (i) the DFE y0=0 with z0=fcr(0)/(ω fbr(0)+fcr(0)) and R0=τ z0 via the next-generation method (its Eq. (6)), and shows the DFE Jacobian is lower triangular with eigenvalues −1+τ z0 and −ζ fbr(0)−ξ fcr(0), so (in the generic case fcr(0)>0) the DFE is locally stable iff R0<1; see Sections 3.1–3.4 and Eqs. (6)–(7) . It also proves existence of at least one endemic equilibrium for some (τ,ω)-region (Theorem 2, using Eq. (4) with zE from Eq. (5)) , and excludes periodic orbits by Bendixson–Dulac with φ=1/(yz), obtaining F(y,z)=−τ−ξ fcr(y)/(y z^2)<0 on (0,1)^2 . The model solution reproduces the same DFE/R0 computation and the same Dulac function and divergence, and gives a constructive parameterization that yields an EE for any admissible (fbr,fcr), matching the paper’s existence claim but via a different proof. Forward invariance of [0,1]^2 matches Lemma 1 of the paper . Minor caveat, noted in both: the next-generation R0 requires fcr(0)>0; special corner cases are handled separately in the paper’s linearization (Section 3.4) .