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In this paper, we study degenerate Hopf bifurcations in a class ofparametrized retarded functional differential equations. Specifically, we areinterested in the case where the eigenvalue crossing condition of the classicalHopf bifurcation theorem is violated. Our approach is based on center manifoldreduction and Poincare-Birkhoff normal forms, and a singularity theoreticalclassification of this degenerate Hopf bifurcation. Our results are thenapplied to a recently developed SIS model incorporating a delayed behavioralresponse. We show that the phenomenon of endemic bubbles, which ischaracterized by a branch of periodic solutions which bifurcates from theendemic equilibrium at some value of the basic reproduction number R0, and thenreconnects to the endemic equilibrium at a larger value of R0, originates in acodimension-two organizing center where the eigenvalue crossing condition forthe Hopf bifurcation theorem is violated.

A degenerate Hopf bifurcation in retarded functional differential equations, and applications to endemic bubbles