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

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.