On the Similarity between Epidemiologic Strains, Minimal Self-Replicable Siphons, and autocatalytic cores in (Chemical) Reaction Networks: Towards a Unifying Framework

Abstract

We aim to study boundary stability and persistence of positive odes in mathematical epidemiology models by importing structural tools from chemical reaction networks. This is largely a review work, which attempts to bring closer together the fields of mathematical epidemiology (ME), and chemical reaction networks (CRNs), based on several observations. We started by observing the conceptual correspondence between epidemiologic strains and both critical minimal siphons and minimal autocatalytic sets (cores) in an underlying CRN, and confirmed this in all the models we studied. We leverage this to provide a definition of the disease free equilibrium (DFE) face/infected set as the union of either all minimal siphons, or of all cores (they coincide always in our examples). Next, we provide a proposed definition of ME models, as models which have a unique boundary fixed point on the DFE face, and for which the Jacobian of the infected subnetwork admits a regular splitting, which allows defining the famous next generating matrix (NGM). We then define the interaction graph on minimal siphons (IGMS), whose vertices are minimal siphons, and whose edges indicate the existence of reactions producing species in one siphon from species in another. When this graph is acyclic, we say the model exhibits a Acyclic Minimal Siphon Decomposition (AMSD). For AMSD models whose minimal siphons partition the infection species, we show that the NGM is block triangular after permutation, which implies the classical max structure of the reproduction number R0 for multi-strain models. We implement algorithms to compute IGMS and detect AMSD in the Epid-CRN Mathematica package (https://github.com/florinav/EpidCRNmodels) (which contains already modules to identify minimal siphons, criticality, drainability, self-replicability, etc).