Analysis of a SIR model with pulse vaccination and temporary immunity: Stability, bifurcation and a cylindrical attractor. (December 2019)

• Record Type: Journal Article
• Authors: Church, Kevin E.M.; Liu, Xinzhi
• Is Part Of: Nonlinear analysis. Volume 50(2019)
• Page Start: 240
• Page End: 266
• Publication Date: 2019-12
• Subjects: SIR model -- Pulse vaccination -- Temporary immunity -- Delay -- Bifurcation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2019.04.015
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical level. Using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution as the vaccination coverage is decreased and a Hopf point is detected. This leads to a bifurcation to an attracting, invariant cylinder. As the vaccination coverage is further decreased, the geometry of the cylinder contracts along its length until it finally collapses to a periodic orbit when the vaccination coverage goes to zero. In the intermediate regime, phase locking on the cylinder is observed.