Abstract / Summary
We study the bifurcation structure of a class of SIRS epidemic models with a nonlinear incidence rate of the form k I p S 1 + ω I q , where the exponents p and q are treated as general parameters. By reducing the model to a planar S - I system, we analyze the existence and degeneracy of positive equilibria and classify their local dynamics. We show that the system can undergo saddle-node bifurcation, Hopf bifurcation, and codimension-two Bogdanov–Takens bifurcation under suitable parameter conditions. A novel feature revealed in this work is the occurrence of isolas of equilibria in the parameter space. In particular, we identify and characterize an isola containing a Hopf bifurcation point, referred to as a Hopf–isola, which, to the best of our knowledge, has not been reported previously for SIRS models. Analytical conditions for the existence of equilibrium isolas and isola centers are derived in representative parameter regimes, while the dependence of the isola structure on the general exponents p and q is further explored numerically. We also investigate how the positive-equilibrium branches are organized with respect to the composite transmission parameter K and show that variation in the infection-force coefficient can induce a transition from a horizontal mushroom bifurcation structure to a Hopf–isola without involving a transcritical bifurcation. These findings demonstrate that nonlinear incidence with general exponents can generate rich and previously unexplored bifurcation phenomena in SIRS-type systems.