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

Aili Wang; Yanni Xiao; Robert A. Cheke

doi:10.3934/dcdsb.2014.19.2915

Article Contents

2014, Volume 19, Issue 9: 2915-2940. Doi: 10.3934/dcdsb.2014.19.2915

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

1.
Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049
2.
Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent ME44TB

Received: November 30, 2013

Revised: April 30, 2014

Published: October 2014

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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.

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Mathematics Subject Classification: Primary: 92D30, 92B05; Secondary: 34C05.

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