Finding bifurcations in mathematical epidemiology via reaction network methods

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. In particular, we adapt three CRNT methods to the features of ME. Firstly, we prove that essentially all ME models admit Hopf bifurcations for certain monotone choices of the interaction functions. Secondly, we offer a parametrization of equilibria Jacobians of ME systems where few interactions are not in mass action form. Thirdly, 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 treatment rate and a recent SIRnS model with a gradual increase in infectiousness. We give necessary and sufficient conditions for the occurrence of bifurcations at endemic equilibria of both families.