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Abstract and Applied Analysis
Volume 2014, Article ID 678723, 10 pages
http://dx.doi.org/10.1155/2014/678723
Research Article

Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate

1School of Computer Science and Software Engineering, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Basic Science, Military Transportation University, Tianjin 300161, China
3School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 28 December 2013; Revised 7 May 2014; Accepted 8 May 2014; Published 16 June 2014

Academic Editor: Kaifa Wang

Copyright © 2014 Yongzhen Pei 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

A delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on the disease spread. From theorems, it is obtained that the basic reproduction number governs the dynamic behavior of the system. The existence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproduction number. By use of new computational techniques for delay differential equations, we prove that the system is permanent. Our results indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion, and to illustrate possible behavioral scenarios of the model.