Hopf bifurcation and stability for an SIRS epidemic model incorporating immune waning, logistic growth and saturated treatment

We propose a delayed SIRS (Susceptible–Infected–Recovered–Susceptible) epidemic model incorporating the temporary immunity delay, logical growth rate of the total population as recruitment rate of susceptible individuals, general nonlinear incidence function, and Holling-II type saturated treatment. The model admits both disease-free and endemic equilibria, and backward bifurcation is shown to occur under certain parameter conditions. Using geometric approaches based on the second compound matrix, we establish the global stability of the endemic equilibrium in the delay-free case. Taking the immune waning delay as the bifurcation parameter, we analyze the existence, direction, and stability of both local and global Hopf bifurcation via normal form theory, the center manifold theorem, and a global Hopf bifurcation theorem. Numerical simulations reveal that both the carrying capacity and treatment resource parameters can induce backward bifurcation, whereas the saturation parameter of treatment may give rise to an endemic bubble. Furthermore, periodic solutions emerging from Hopf bifurcation and stability switches under different incidence functions are demonstrated. In addition, chaotic dynamics are also observed.