A codimension-5 Hopf bifurcation yielding 5 limit cycles in a SIRS epidemic model with nonlinear incidence rate

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 $\frac{kI^p}{1+αI^q}$, where $p>0$ and $q\geq0$ are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifur