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

TL;DR

This paper investigates a special type of Hopf bifurcation in retarded functional differential equations, revealing how endemic bubbles in epidemiological models originate from a degenerate bifurcation point where classical conditions fail.

Contribution

It introduces a classification of degenerate Hopf bifurcations in retarded equations and applies this to explain endemic bubbles in a delayed SIS epidemic model.

Findings

Endemic bubbles arise from a codimension-two bifurcation point.

Classical eigenvalue crossing conditions are violated in this degenerate case.

The approach combines center manifold reduction and normal form analysis.

Abstract

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.