Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results
We establish two structural properties of the Jacobian of a positive ODE on any invariant boundary face, i.e. on the face defined by a siphon of an associated Chemical Reaction Network. First, the Jacobian is block lower-triangular at every point of such a face, with a transversal diagonal block, which governs the species that vanish on the face, and a tangential one. Second, the transversal block is always a Metzler matrix. These two facts yield an elegant proof, in the spirit of Chemical Reaction Network theory, and a generalization of the most cited result in Mathematical Epidemiology (ME), the Next Generation Matrix (NGM) theorem, and they clarify its hypotheses. The Metzler property is exactly what makes the threshold work, it implies that regular splitting of the transversal block always exist, and, via Perron--Frobenius theory, that a boundary fixed point can never lose its stability through a Hopf bifurcation in the directions transversal to its face. Among other results, we review the ``symbolic-numeric" approach of Vassena and Stadler, which tackles bifurcation problems by viewing the characteristic polynomial of the Jacobian at fixed points as a formal polynomial in the "symbolic reactivities", and identifies its coefficients as ``Child Selection minors of the stoichiometric matrix". We also review two applications of this approach, to an SIRWS model and to a Capasso-Ruan-Wang family of SIRS models, using the Mathematica package Epid-CRN, which implements tools from both Chemical Reaction Network and Mathematical Epidemiology.