A codimension-5 Hopf bifurcation yielding 5 limit cycles in a SIRS epidemic model with nonlinear incidence rate
Abstract
In a recent paper published in the Journal of Differential Equations (384, 2024), Cui and Zhao investigated an SIRS epidemic model with the nonlinear incidence rate kIp/(1+αIq), where p>0 and q≥0 are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifurcation in this model is at most two, while the codimension of the Hopf bifurcation was left as an open problem. In this work, we investigate this problem while keeping p and q free throughout the analysis. A strategic nondimensionalization and parametrization remove the exponential dependence of the positive equilibrium and convert the generalized-Hopf conditions into algebraic focus equations coupled to explicit semialgebraic admissibility regions. Using algebraic focus reduction, strict semialgebraic admissibility, and validated interval computation, we rigorously prove the existence of admissible nondegenerate generalized Hopf points generating three, four, and five small-amplitude limit cycles. In particular, we rigorously certify a strict-interior codimension-five weak focus and a full-rank local unfolding, thereby proving that five small-amplitude limit cycles can bifurcate from a single positive equilibrium. We further derive exact algebraic obstructions to codimension six and find no admissible candidate in extensive continuation and projected-component searches. These results provide analytical and numerical evidence for the conjecture that five is maximal, but no codimension-six nonexistence theorem is claimed. The same parametrization also yields a simple proof that the Bogdanov-Takens codimension is at most two.
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BibTeX
@misc{yu2026codimension5,
title = {{A codimension-5 Hopf bifurcation yielding 5 limit cycles in a SIRS epidemic model with nonlinear incidence rate}},
author = {Pei Yu and Wanyue Tang and Yanni Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38874},
eprint = {2609.38874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38874},
}
RIS
TY - UNPB
TI - A codimension-5 Hopf bifurcation yielding 5 limit cycles in a SIRS epidemic model with nonlinear incidence rate
AU - Pei Yu
AU - Wanyue Tang
AU - Yanni Zeng
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38874
UR - https://arxiv.org/abs/2609.38874
AB - In a recent paper published in the Journal of Differential Equations (384, 2024), Cui and Zhao investigated an SIRS epidemic model with the nonlinear incidence rate kI^p/(1+αI^q), where p>0 and q≥0 are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifurcation in this model is at most two, while the codimension of the Hopf bifurcation was left as an open problem. In this work, we investigate this problem while keeping p and q free throughout the analysis. A strategic nondimensionalization and parametrization remove the exponential dependence of the positive equilibrium and convert the generalized-Hopf conditions into algebraic focus equations coupled to explicit semialgebraic admissibility regions. Using algebraic focus reduction, strict semialgebraic admissibility, and validated interval computation, we rigorously prove the existence of admissible nondegenerate generalized Hopf points generating three, four, and five small-amplitude limit cycles. In particular, we rigorously certify a strict-interior codimension-five weak focus and a full-rank local unfolding, thereby proving that five small-amplitude limit cycles can bifurcate from a single positive equilibrium. We further derive exact algebraic obstructions to codimension six and find no admissible candidate in extensive continuation and projected-component searches. These results provide analytical and numerical evidence for the conjecture that five is maximal, but no codimension-six nonexistence theorem is claimed. The same parametrization also yields a simple proof that the Bogdanov-Takens codimension is at most two.
ER -