Advances in Difference Equations
Volume 2011 (2011), Article ID 748608, 32 pages
doi:10.1155/2011/748608
Research Article

On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination

M. De la Sen,1,2 Ravi P. Agarwal,3,4 A. Ibeas,5 and S. Alonso-Quesada1,2

1Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain
2Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain
3Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
4Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
5Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Received 17 January 2011; Accepted 23 February 2011

Academic Editor: Claudio Cuevas

Copyright © 2011 M. De la Sen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.