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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 974868, 13 pages
http://dx.doi.org/10.1155/2015/974868
Research Article

Bifurcation Analysis and Chaos Control in a Discrete Epidemic System

1Department of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China
2Department of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 14 June 2015; Accepted 9 August 2015

Academic Editor: Garyfalos Papashinopoulos

Copyright © 2015 Wei Tan 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

The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior , the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the system , . Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.