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Define mathematical epidemiology models within the CRN framework

Develop a formal, rigorous definition of mathematical epidemiology ordinary differential equation models as a subclass of chemical reaction network systems composed only of (i) monomolecular transfer reactions, (ii) bimolecular autocatalytic infections of the form S+I→2I (as in SIR-type models), and (iii) bimolecular autocatalytic infections of the form S+I→I+E with equations s′=−β_e s i, e′=β_e s i (as in SEIR-type models), and require that such models possess at least one boundary equilibrium (disease-free) and one interior equilibrium (endemic).

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Background

The paper argues that mathematical epidemiology and chemical reaction network theory can benefit from a unified language and proposes, for the first time, a precise definition of ME models in CRN terms.

This definition aims to leverage CRN structure (stoichiometry, kinetics, reversibility) while capturing essential ME features such as infection reactions and the coexistence of disease-free and endemic equilibria.

References

Open Problem Mathematical epidemiology ODE models could be defined (hopefully with benefits) as particular CRN models formed with only three types of reactions: Transfers (monomolecular reactions); Bimolecular auto-catalytic reactions of the type S+I→2 I as encountered in SIR, etc; Bimolecular auto-catalytic reactions of the type S+I→I+E s′=−β_e s i … , e′=β_e s i … as encountered in SEIR. In addition, they should have at least one boundary fixed point and one interior fixed point.

— Advancing Mathematical Epidemiology and Chemical Reaction Network Theory via Synergies Between Them (2411.00488 - Avram et al., 1 Nov 2024) in Open Problem (label o:MEm), Subsection “Some Recent Interactions Between CRN and ME Methods”