Backward bifurcations and multistationarity

The paper’s Theorem 1 establishes a “moving fold” criterion via a center-manifold reduction u̇ = u g(u, α1, α2), applies the implicit function theorem to Φ = (g, g_u), derives U′(0) and A′(0), and uses the local parabola geometry to prove: (i) ce<0 gives a backward bifurcation for α2>0; (ii) ce>0 gives a forward bifurcation for α2>0; (iii) if c<0 and e>0 there are two positive steady states with R0<1; (iv) if c>0 and e>0 there are two positive steady states with R0>1. The candidate solution reproduces precisely this framework, adds a transparent “threshold” coordinate μ capturing sign(R0−1), computes the same U′(0) (with e = ½U′(0) exactly as in the paper), and gives the same stability conclusions from f_u = u g_u. Aside from notation, the arguments are essentially the same and consistent with the paper’s hypotheses (A1)–(A5) and the van den Driessche–Watmough threshold mechanism. No substantive logical gaps or contradictions were found in either; the model’s write-up is slightly more explicit in Taylor expansions, while the paper’s proof is more concise.