  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.
