Dice Question Streamline Icon: https://streamlinehq.com

Hopf and Bogdanov–Takens bifurcation in SIR2S, and double contact of manifolds

Establish the existence of Hopf and Bogdanov–Takens bifurcations in the SIR2S model (with y2 = 0), and prove that the Hopf manifold H4 = 0 has a double contact with the Bogdanov–Takens manifold c01 = 0 = c02 in that model.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors’ computations for SIR2S indicate double-zero eigenvalues and vanishing Hurwitz determinants along certain parameter manifolds, suggesting degenerate Hopf and BT bifurcations.

They formalize the task of proving these bifurcations and their geometric relationship between stability manifolds.

References

Open Problem 8. 1. For any y1 > 0, p > 0, i > 0 the model SIR2, with 12 = 0, admits endemic equilibria with Jacobian having a double zero, where A, B, y are rational functions of the other parameters and i, provided certain rational equalities and inequalities, necessary for the positivity of 3, y and provided only in SIR2S.nb, are satisfied. 2. On the double zero manifold of part 1., the Hurwitz determinant H4 is doubly identically 0, suggesting a degenerate Hopf bifurcation occurs. Establish the existence of Hopf and Bogdanov-Takens bifurcation for SIR2S, and prove that the Hopf manifold H4 = 0 has a double contact with the BT manifold co1 = 0 = c02, for SIR2S, with 2 = 0.

— Stability in Reaction Network Models via an Extension of the Next Generation Matrix Method (2411.11867 - Avram et al., 3 Nov 2024) in Section 5.2