The divergence criterion has been shown to be helpful in distinguishing between sub- and supercritical Hopf bifurcations, but its applicability is limited to systems whose divergence is sign definite. A step-by-step computational procedure which allows one to extend the applicability of the divergence criterion is derived by altering the system to an equivalent one with sign definite divergence. The procedure is based on multiplying the original vector field by a positive quadratic function in a neighborhood of the bifurcating rest point. This procedure is then applied to several examples of planar systems that exhibit the Hopf bifurcation. Specifically, it is demonstrated that only supercritical bifurcations occur in a system modeling specific immune responses with handling time. It is also shown that the FitzHugh-Nagumo equations and the chemostat equations with substrate inhibition and linear yield coefficient may exhibit both sub- and supercritical Hopf bifurcations. In both cases, simple analytic criteria for determining the criticality of the bifurcation are presented.

RESUMEN

We investigate an epidemic model of two diseases. The primary disease is assumed to be a slowly progressing disease, and the density of individuals infected with it is structured by age since infection. Hosts that are already infected with the primary disease can become coinfected with a secondary disease. We show that in addition to the disease-free equilibrium, there exists a unique dominance equilibrium corresponding to each disease. Without coinfection there are no coexistence equilibria; however, with coinfection the number of coexistence equilibria may vary. For some parameter values, there exist two coexistence equilibria. We also observe competitor-mediated oscillatory coexistence. Furthermore, weakly subthreshold (which occur when exactly one of the reproduction numbers is below one) and strongly subthreshold (which occur when both reproduction numbers are below one) coexistence equilibria may exist. Some of those are a result of a two-parameter backward bifurcation. Bistability occurs in several regions of the parameter space. Despite the presence of coinfection, coexistence of the two diseases appears possible only for relatively small values of the reproduction numbers---for large values of the reproduction numbers the typical outcome of competition is the dominance of one of the diseases, including bistable dominance where the competition outcome is initial condition dependent.