A cocktail of chemical reaction networks and mathematical epidemiology tools for positive ODE stability problems

TL;DR

This paper integrates Chemical Reaction Networks theory and Mathematical Epidemiology to address stability of positive ODEs, introducing a CRN-based generalization of the NGM theorem and reviewing symbolic-numeric bifurcation analysis methods.

Contribution

It presents a novel CRN-flavored generalization of the NGM theorem and reviews a symbolic-numeric approach for bifurcation analysis using Epid-CRN tools.

Findings

Introduces a CRN-based generalization of the NGM theorem.

Reviews the symbolic-numeric bifurcation approach with Mathematica tools.

Demonstrates applications of the approach in stability analysis.

Abstract

We continue recent attempts to put together concepts and results of Chemical Reaction Networks theory (CRNT) and Mathematical Epidemiology (ME), for solving problems of stability of positive ODEs. We provide first an elegant CRN-flavored generalization of the most cited result in ME, the Next Generation Matrix (NGM) theorem. We review next 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 using the Mathematica package Epid-CRN tools from both CRNT and ME.