Threshold Dynamics and Bifurcation of a State-Dependent Feedback Nonlinear Control Susceptible–Infected–Recovered Model1

A classic susceptible–infected–recovered (SIR) model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease-free periodic solution (DFPS) are addressed, and the threshold condition is provided, which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincaré map of the proposed model, and the main results indicate that under certain conditions, a stable or unstable interior periodic solution could be generated through transcritical, pitchfork, and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve, which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency, and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.