Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Global dynamics of a piece-wise epidemic model with switching vaccination strategy

Pages: 2915 - 2940, Volume 19, Issue 9, November 2014      doi:10.3934/dcdsb.2014.19.2915

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Aili Wang - Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China (email)
Yanni Xiao - Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China (email)
Robert A. Cheke - Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent ME44TB, United Kingdom (email)

Abstract: A piece-wise epidemic model of a switching vaccination program, implemented once the number of people exposed to a disease-causing virus reaches a critical level, is proposed. In addition, variation or uncertainties in interventions are examined with a perturbed system version of the model. We also analyzed the global dynamic behaviors of both the original piece-wise system and the perturbed version theoretically, using generalized Jacobian theory, Lyapunov constants for a non-smooth vector field and a generalization of Dulac's criterion. The main results show that, as the critical value varies, there are three possibilities for stabilization of the piece-wise system: (i) at the disease-free equilibrium; (ii) at the endemic states for the two subsystems or (iii) at a generalized equilibrium which is a novel global attractor for non-smooth systems. The perturbed system exhibits new global attractors including a pseudo-focus of parabolic-parabolic (PP) type, a pseudo-equilibrium and a crossing cycle surrounding a sliding mode region. Our findings demonstrate that an infectious disease can be eradicated either by increasing the vaccination rate or by stabilizing the number of infected individuals at a previously given level, conditional upon a suitable critical level and the parameter values.

Keywords:  Piece-wise epidemic model, vaccination, generalized equilibrium, global dynamics, perturbed system, limit cycle.
Mathematics Subject Classification:  Primary: 92D30, 92B05; Secondary: 34C05.

Received: December 2013;      Revised: May 2014;      Available Online: September 2014.