Modelling the Effect of Vaccination and Human Behaviour on the Spread of Epidemic Diseases on Temporal Networks

The paper proves the epidemic threshold c̄t = 2(1−η)(1−σ)λ(1−γt v)[1−pq(1−γq v)] − β and local asymptotic stability of the disease-free equilibrium by linearizing the (yi,yq) planar reduction (eigenvalues −β and c̄t−ct, cf. eqs. (7)–(9) and Theorem 1), then establishes global dynamics and the explicit endemic equilibrium via a Bendixson–Dulac argument on the (ys,yi) planar system (Theorem 2 and eq. (12)) . The candidate solution uses the same planar reduction (taking yq = 1 − ys − yi), identical abbreviations α = 2(1−η)(1−σ)λ(1−γt v) and κ = 1 − pq(1−γq v), the same threshold c̄t = ακ − β, the same Dulac function 1/(ys yi), and derives the same endemic equilibrium. It adds routine details (explicit Jacobian at the DFE, LaSalle at ct = c̄t). Hence, both are correct and substantially the same proof.