Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

This paper investigates an SIR model with nonlinear state-dependent feedback control. The saturation incidence rate is introduced into the classical SIR model, and the specific expression of the equilibrium point of the model without pulse control is derived, and the stability of the equilibrium point is analyzed. When the number of susceptible individuals reaches a threshold Sh, comprehensive prevention and control strategies such as treatment, isolation, and vaccination will be implemented. The existence and global stability of the disease-free periodic solution (DFPS) of the model with state-dependent feedback control are discussed, and the properties of the Poincar & eacute; map are analyzed. Based on the bifurcation theory related to a one-parameter family of maps associated with the Poincar & eacute; map, we define the control reproduction number R, based on different parameters and observe the effect of these parameters on bifurcation. Numerical simulations show that transcritical and pitchfork bifurcations, as well as backward bifurcations, may occur under certain conditions. Additionally, a sensitivity analysis of R, is conducted, and the optimal control strategy for the SIR model without pulse control is discussed. These results can provide some suggestions for virus control and vaccine production, among other issues.