  In this paper, we study degenerate Hopf bifurcations in a class of
parametrized retarded functional differential equations. Specifically, we are
interested in the case where the eigenvalue crossing condition of the classical
Hopf bifurcation theorem is violated. Our approach is based on center manifold
reduction and Poincare-Birkhoff normal forms, and a singularity theoretical
classification of this degenerate Hopf bifurcation. Our results are then
applied to a recently developed SIS model incorporating a delayed behavioral
response. We show that the phenomenon of endemic bubbles, which is
characterized by a branch of periodic solutions which bifurcates from the
endemic equilibrium at some value of the basic reproduction number R0, and then
reconnects to the endemic equilibrium at a larger value of R0, originates in a
codimension-two organizing center where the eigenvalue crossing condition for
the Hopf bifurcation theorem is violated.
