Finding bifurcations in mathematical epidemiology via reaction network methods

TL;DR

This paper applies Chemical Reaction Network Theory methods to mathematical epidemiology models, revealing bifurcation behaviors and oscillations, and providing conditions for their occurrence in various epidemic models.

Contribution

It adapts CRNT methods to ME models, proving the existence of bifurcations and oscillations, and offers parametrizations and conditions for these phenomena.

Findings

All ME models can admit Hopf bifurcations under certain conditions.

Periodic oscillations in closed systems imply oscillations when demography is added.

Necessary and sufficient conditions for bifurcations in specific epidemic models.

Abstract

Mathematical Epidemiology (ME) shares with Chemical Reaction Network Theory (CRNT) the basic mathematical structure of its dynamical systems. Despite this central similarity, methods from CRNT have been seldom applied to solving problems in ME. We explore here the applicability of CRNT methods to find bifurcations at endemic equilibria of ME models. We adapt three CRNT methods to the features of ME. First, we prove that essentially all ME models admit Hopf bifurcations for certain monotone choices of the interaction functions. Second, we offer a parametrization of equilibria Jacobians of ME systems where few interactions are not in mass action form. Third, for a quite general class of models, we show that periodic oscillations in closed systems imply periodic oscillations when demography is added. Finally, we apply such results to two families of networks: a general SIR model with a nonlinear force of infection and treatment rate and a recent SIRnS model with a gradual increase in infectiousness. We give both necessary conditions and sufficient conditions for the occurrence of bifurcations at endemic equilibria of both families.