Advances in Difference Equations
Volume 2010 (2010), Article ID 281612, 42 pages
Research Article

On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination Controls

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 Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL 32901, USA
3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4Department of Telecommunications and Systems Engineering, Autonomous University of Barcelona, Bellaterra, 08193 Barcelona, Spain
5Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain

Received 17 August 2010; Revised 9 November 2010; Accepted 2 December 2010

Academic Editor: A. Zafer

Copyright © 2010 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.


This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.