Threshold dynamics and bifurcation of a state-dependent feedback nonlinear control SIR model

[+] Author and Article Information
Tianyu Cheng

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, P.R. China

Sanyi Tang

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, P.R. China
sytang@snnu.edu.cn; sanyitang219@hotmail.com

Robert A. Cheke

Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK

1Corresponding author.

ASME doi:10.1115/1.4043001 History: Received December 10, 2018; Revised February 25, 2019


A classic 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 in 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 $R_c$ for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number $R_0$ of the SIR model is larger than one. To show that the threshold dynamics are determined by the $R_c$, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincar\'{e} 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.

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