Finding bifurcations in mathematical epidemiology via reaction network methods

The paper proves that any bifurcation at an endemic equilibrium requires Gii = f′i − h′ − μi > 0 (Theorem 4.2, split into zero-eigenvalue and Hopf cases), and notes that at least one of f(s,·) or h(·) must be nonlinear in i. The candidate solution derives the same necessary condition via an explicit Routh–Hurwitz analysis of the Jacobian characteristic polynomial (showing a1>0, a2>0, and a1 a2 − a3>0 when e = Gii ≤ 0). The Hopf exclusion argument differs from the paper’s proof technique but reaches the same conclusion. Minor caveat: the candidate overstates a degeneracy (claiming a3=0 with e=0 additionally requires μs=0), which is unnecessary, but this does not affect the main result. Overall, both are correct on the central claim. See Theorem 4.2 and Lemmas 4.6 and 4.8 in the paper for the official statements and proofs .