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The Scientific World Journal
Volume 2013, Article ID 470646, 10 pages
Research Article

Positive Periodic Solutions of an Epidemic Model with Seasonality

1Complex Sciences Center, Shanxi University, Taiyuan 030006, China
2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
3Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
4Department of Applied Mathematics, Xidian University, Xi’an, Shaanxi 710071, China
5Institute of Information Economy, Hangzhou Normal University, Hangzhou 310036, China
6Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
7Department of Physics, University of Fribourg, Chemin du Musée 3, 1700 Fribourg, Switzerland

Received 17 August 2013; Accepted 12 September 2013

Academic Editors: P. Leach, O. D. Makinde, and Y. Xia

Copyright © 2013 Gui-Quan Sun 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.


An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction number is obtained. Moreover, only the basic reproduction number cannot ensure the existence of the positive equilibrium, which needs additional condition . For the nonautonomous case, by introducing the basic reproduction number defined by the spectral radius, we study the uniform persistence and extinction of the disease. The results show that for the periodic system the basic reproduction number is more accurate than the average reproduction number.