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Relay transitions and invasion thresholds in multi-strain rumor models: a chemical reaction network approach

Florin Avram, Andrei-Dan Halanay

TL;DR

The boundary dynamics via the lattice of invariant faces generated by minimal siphons are organised, and it is established that stability transitions take the form of \emph{relays}: for each distance-one cover in the siphon lattice, a single invasion inequality simultaneously governs the loss of transversal stability of the resident equilibrium.

Abstract

The historical quest for unifying the concepts and methods of Chemical Reaction Networks theory (CRNT), Mahematical Epidemiology (ME) and ecology has received increased attention in the last years and has led in particular to the development of the symbolic package EpidCRN, for automatic analysis of positive ODEs, which implements tools from all these disciplines like siphons, reproduction functions and invasion numbers, Child-Selection expansions, etc. We illustrate below the convenience of using this package on some recent online social network (OSN) rumor spreading models, with emphasis on showing how CRNT throws a new light on their analysis. Specifically, we organise the boundary dynamics via the lattice of invariant faces generated by minimal siphons, and establish that stability transitions take the form of \emph{relays}: for each distance-one cover in the siphon lattice, a single invasion inequality simultaneously governs the loss of transversal stability of the resident equilibrium and the existence of a successor equilibrium on the adjacent face. For the base OSN model ($ω=0$) all boundary and interior equilibria admit explicit rational formulas, and the relay table is fully verified using invasion numbers computed symbolically by EpidCRN. For the variant with waning spreading impulse ($ω>0$), the relay structure is analysed via transversal Jacobian blocks; three equilibria involve irrational coordinates and their stability is predicted by the relay framework subject to direct Routh--Hurwitz verification. The relay mechanism is then situated in its normal-form context (siphon-induced transcritical bifurcations), distinguished from classical transcritical bifurcations along four structural axes, and compared with Hofbauer invasion graphs.

Relay transitions and invasion thresholds in multi-strain rumor models: a chemical reaction network approach

TL;DR

The boundary dynamics via the lattice of invariant faces generated by minimal siphons are organised, and it is established that stability transitions take the form of \emph{relays}: for each distance-one cover in the siphon lattice, a single invasion inequality simultaneously governs the loss of transversal stability of the resident equilibrium.

Abstract

The historical quest for unifying the concepts and methods of Chemical Reaction Networks theory (CRNT), Mahematical Epidemiology (ME) and ecology has received increased attention in the last years and has led in particular to the development of the symbolic package EpidCRN, for automatic analysis of positive ODEs, which implements tools from all these disciplines like siphons, reproduction functions and invasion numbers, Child-Selection expansions, etc. We illustrate below the convenience of using this package on some recent online social network (OSN) rumor spreading models, with emphasis on showing how CRNT throws a new light on their analysis. Specifically, we organise the boundary dynamics via the lattice of invariant faces generated by minimal siphons, and establish that stability transitions take the form of \emph{relays}: for each distance-one cover in the siphon lattice, a single invasion inequality simultaneously governs the loss of transversal stability of the resident equilibrium and the existence of a successor equilibrium on the adjacent face. For the base OSN model () all boundary and interior equilibria admit explicit rational formulas, and the relay table is fully verified using invasion numbers computed symbolically by EpidCRN. For the variant with waning spreading impulse (), the relay structure is analysed via transversal Jacobian blocks; three equilibria involve irrational coordinates and their stability is predicted by the relay framework subject to direct Routh--Hurwitz verification. The relay mechanism is then situated in its normal-form context (siphon-induced transcritical bifurcations), distinguished from classical transcritical bifurcations along four structural axes, and compared with Hofbauer invasion graphs.
Paper Structure (47 sections, 12 theorems, 76 equations, 3 figures, 2 tables)

This paper contains 47 sections, 12 theorems, 76 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Reproduction functions as organizing objects.
  4. Relation to invasion graphs in ecology.
  5. Structure of the paper
  6. A model of propagation of rumors in online social networks (OSN) of Fakih, Halanay and Avram HA, with 8 species 13 parameter s, and 6 equilibria: OSN.wl
  7. The script ODE2RNp[RHS,var,prF] gets a "perturbed" mass-action decomposition (RN,rts) of the ODE
  8. The script minSiph[var,RN] gets the minimal siphons
  9. The symbolic Jacobian and its block structure
  10. The scripts NGM[RHS, var,sip] and NGMRN[RN, rts, var] find (F,V,K), reproduction functions and resident equilibria for ME-type siphons (admitting regular splitting s)
  11. The case $\omega=0$
  12. Capturing global boundary behavior via the boundary transcritical relay graph when $\omega=0$
  13. Explanation of Table \ref{['tab:OSN_equilibria_om0']}
  14. Verification of tangential Hurwitzness (H2) entry by entry.
  15. Algorithmic detection of relay transitions along the siphon lattice
  16. ...and 32 more sections

Key Result

Theorem 1

Let $\dot x = f(x)$ be a smooth positive ODE on $\mathbb{R}^n_{\ge 0}$ with minimal siphons $\sigma_1,\ldots,\sigma_k$. A distance-one cover in the siphon lattice is a pair $(\Sigma',\Sigma)$ with $\Sigma = \Sigma'\cup\sigma$ for some minimal siphon $\sigma\not\subseteq\Sigma'$ and no siphon strictl Let $M_\sigma(E^*)$ be the transversal Jacobian block (Definition d:transblock) and $R_\sigma(E^*)$

Figures (3)

  • Figure 1: A flow-diagram showing the compartmental modeling of two rumors, 1 and 2, spreading among the users of a social media platform. Each of the 13 rates of the model is associated to an edge, but the three "decay edges " for $x_1,U,W$ are omitted in the diagram. 8 of the edges correspond to mass action reactions (the five lower "conversion transitions" with linear rates, and the quadratic "epidemic SEIR" transitions), which allow reconstructing parts of the ODE. The two fractional, non mass-action rates are actually flows coming from outside, catalyzed by $U$ (which was indicated by the dashed arrows). Note that if $\omega=0$, then $R$ does not affect the rest of the species, and may be dropped.
  • Figure 2: Relay graph for the nine equilibria ($\omega=0$; see Definition \ref{['d:relay-type']}). Existence conditions appear above each node and extra stability conditions below. Solid arrows: full relays (single threshold simultaneously creates successor and destabilises predecessor): DFE$\to$gOSN, $E_1\to$EE, $E_2\to$EE, $\mathrm{EE}_g\to$EE. Dashed arrows: multiple relays (predecessor has multiple instability directions): gOSN$\to E_{jg}$, gOSN$\to$RFE, $E_{jg}\to\mathrm{EE}_g$, RFE$\to E_j$. Dotted arrows: cross-branch multiple relay s driven by W-invasion ($R_0>1+\frac{\beta}{\beta_w}$): $E_{1g}\to E_1$, $E_{2g}\to E_2$, $\mathrm{EE}_g\to\mathrm{EE}$.
  • Figure 3: Relay order between the six equilibria ($\omega>0$). Existence conditions appear above each node and extra stability conditions below. Solid arrow (OSND$\to$gOSN): full relay. Dashed arrows: multiple relay s (existence at head $\Rightarrow$ instability at tail). Dotted arrows: predicted multiple relay s (pending direct verification, marked $\dagger$). The gOSN/RFE split (dashed separator) is governed by $R_0^W=1$. Unlike the $\omega=0$ case (Figure \ref{['fig:relay0']}), there are no gOSN-branch endemic equilibria with $W=0$: the $R\to W$ coupling forces $W>0$ at $E_1$, $E_2$, EE. Relays from RFE are analogous but not yet fully analysed.

Theorems & Definitions (62)

  • Definition 1: Positive / non-negative ODEs
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2: DFE siphon, lattice of minimal siphons, inhabited siphons
  • Remark 4
  • Definition 3: Lotka-Volterra siphons
  • Remark 5
  • Definition 4: ME-type siphon
  • Remark 6
  • ...and 52 more