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

Florin Avram, Rim Adenane, Andrei-Dan Halanay

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.

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

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.
Paper Structure (22 sections, 8 theorems, 44 equations, 2 figures, 1 table)

This paper contains 22 sections, 8 theorems, 44 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contents
  3. A bird's eye view of positive ODEs
  4. Generalization of the NGM theorem: Boundary-face invariance forces $J_{x y}=0$ for polynomial mass-action fields
  5. Compliments to Routh-Hurwitz analysis: the Child Selections Cauchy Binet expansion VasGBVasStaVasBSVGSV and the Banaji inheritance theorems BanajiBanBorMotifsGutShiuBBH25
  6. From critical fragments to child selections: a historical overview
  7. Summary of the Vassena-Stadler approach to bifurcation analysis via the Cauchy--Binet expansion of symbolic Jacobians
  8. CS-submatrices.
  9. The UPF/NF dichotomy, unstable cores and spectral consequences
  10. Key results from Vas implemented in BiRNe and EpidCRN
  11. Application to immunity boosting in SIRWS (4 species, 10 reactions): SIRWS.wl
  12. Model.
  13. Computational output.
  14. Detailed analysis of pair #4.
  15. Comments on deriving heuristically the Boros--Rost Hopf witness $\{W,S_2,i\}$
  16. ...and 7 more sections

Key Result

Theorem 1

(generalizing DiekVanVan08,JA) Let $(x,y)\in\mathbb{R}^{m}_{\ge 0}\times\mathbb{R}^{n}_{\ge 0}$ and let be a polynomial positive (equivalently, mass-action representable) vector field. Assume that the face $F_x:=\{x=0\}$ is invariant, i.e. (equivalently, $x$ is a siphon). Then:

Figures (2)

  • Figure 1: Contraction and enlargement between the red dashed SIRWS network on left $\mathcal{N}_{\mathrm{SIRWS}}$ and the Boros--Rost witness $\mathcal{N}_{\mathrm{BR}}$ on right. The projection $\pi$ collapses the immune pathway $i\to R\to W$ into a direct interaction $i\to W$, which is represented however as a consumption, since $\nu i W$ consumes W, but dotted, to indicate catalysis by $i$. The enlargement $\mathcal{E}$ (in the sense of Banaji--Boros--Hofbauer) reverses this, inserting $R$ as an intermediate between $i$ and $W$ and recovering the full SIRWS dynamics. The Hopf bifurcation transfers from $\mathcal{N}_{\mathrm{BR}}$ to $\mathcal{N}_{\mathrm{SIRWS}}$ by the inheritance theorems BBH25.
  • Figure 2: Flow diagram of model \ref{['Vys']}. $\Lambda,\mu_s, \mu_i, \mu_r$ are the birth rate of $S$ and the death rates of $S,I,R$, respectively.The linear conversion reactions $S \underset{\gamma_s s}{\longrightarrow} R, R \underset{\gamma_r r}{\longrightarrow} S, I \underset{i_r i}{\longrightarrow} R, I \underset{i_s i}{\longrightarrow} S$ model vaccinations, loss (waning) of immunity, recovery, and very fast loss of immunity. The infection and treatment rates $S+I \underset{\beta s i F(i)}{\longrightarrow} 2I, I \underset{T[i]}{\longrightarrow} R$ are functional.

Theorems & Definitions (36)

  • Definition 1: Positive / non-negative ODE
  • Theorem 1: NGM theorem for forward invariant faces/siphons
  • proof
  • Remark 1
  • Definition 2: ME siphon-faces
  • Remark 2
  • Definition 3: Child selections as matchings in the reactivity matrix
  • Lemma 1: Nonzero Cauchy--Binet monomials are exactly child selections
  • proof
  • Remark 3: Child selections encode the permutation data in Cauchy--Binet
  • ...and 26 more