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

Abstract

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.

The state-dependent impulsive control for a general predator–prey model

“…It is known from the properties of U and V, for system (4), that the existence and uniqueness of the solutions hold true, and the solutions are continuously dependent on and differentiable with respect to the initial value. It is also easy to verify that the solution of (4) with a positive initial value must be positive.…”

“…The impulsive differential equations have even been intensively studied in recent two decades, especially for state-dependent impulsive equations, which means the impulsive control takes place when a certain state is met [4,5,8,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]26]. However, because of the discontinuity of the system, the issue becomes more complicated but not less important than a continuous one.…”

Bifurcation Analysis of a Tumour-Immune Model with Nonlinear Killing Rate as State-Dependent Feedback Control

“…Yang et al developed a novel impulsive tumour-immune system with drug responses, they studied the effects of drug dosages on the evolution of tumours and proposed a more effective method to maintain a high tumour cell depletion rate [Yang et al, 2020]. Actually, the methods of state-dependent feedback control have been used in a variety of fields, including integrated pest management [Tang & Cheke, 2008, infectious disease control [Cheng et al, 2019;Zhang et al, 2020], the neurological system [Touboul & Brette, 2009], and etc.…”