Backward bifurcation arising from decline of immunity against emerging infectious diseases

The paper sets up an age-structured model (1)–(2), identifies the DFE P0 and R0, and proves a transcritical bifurcation at R0=1 using Lyapunov–Schmidt reduction; it derives the quadratic coefficient a explicitly (its Eq. (3)) and shows that 〈Fφβ̄(P0,β̄∗)x̂, ξ̂〉>0, so the sign of a determines forward vs backward criticality . The candidate solution uses a center-manifold/normal-form approach in the same Banach space X=R^4×L^1(0,∞), shows the simple zero eigenvalue at R0=1 and that b>0, and computes exactly the same a (via slaving of S and r along characteristics), concluding the same criticality rule. The paper’s operator A, its kernel basis x̂ and A* eigenfunction ξ̂, and the positivity of 〈Fφβ̄ x̂, ξ̂〉 match the candidate’s left/right eigenvectors and b>0 claim . Thus, both establish the same local bifurcation and classification, differing mainly in technique (Lyapunov–Schmidt vs. center manifold).