Codimension-two bifurcations analysis and tracking control on a discrete epidemic modelby Na Yi, Qingling Zhang, Peng Liu, Yanping LIN

J Syst Sci Complex


Computer Science (miscellaneous) / Information Systems


J Syst Sci Complex (2011) 24: 1033–1056




Na YI · Qingling ZHANG · Peng LIU · Yanping LIN

DOI: 10.1007/s11424-011-9041-0

Received: 26 February 2009 / Revised: 26 October 2009 c©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2011

Abstract In this paper, the dynamic behaviors of a discrete epidemic model with a nonlinear incidence rate obtained by Euler method are discussed, which can exhibit the periodic motions and chaotic behaviors under the suitable system parameter conditions. Codimension-two bifurcations of the discrete epidemic model, associated with 1:1 strong resonance, 1:2 strong resonance, 1:3 strong resonance and 1:4 strong resonance, are analyzed by using the bifurcation theorem and the normal form method of maps. Moreover, in order to eliminate the chaotic behavior of the discrete epidemic model, a tracking controller is designed such that the disease disappears gradually. Finally, numerical simulations are obtained by the phase portraits, the maximum Lyapunov exponents diagrams for two different varying parameters in 3-dimension space, the bifurcation diagrams, the computations of Lyapunov exponents and the dynamic response. They not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.

Key words Chaos, codimension-two bifurcations, discrete epidemic model, tracking control. 1 Introduction

Epidemic models have been a popular topic of biology research for many years. It is wellknown that many continuous epidemic models have played a great deal of role on investigating the transmitting law of epidemic diseases and predicting the development trend of their spread (see [1–3]). Since the epidemic statistics are compiled from the given time intervals instead of the


Institute of Systems Science, Northeastern University, Shenyang 110004, China; School of Sciences, Liaoning

Shihua University, Fushun 113001, China. Email:

Qingling ZHANG (Corresponding author)

Institute of Systems Science, Northeastern University, Shenyang 110004, China.


Peng LIU

School of Management, Shenyang University of Technology, Shenyang 110870, China.


Yanping LIN

Department of Mathematics, The Hong Kong Polytechnic University, Hong Kong, China.

Email: ∗This research is supported by the National Natural Science Foundation of China under Grant Nos. 60974004 and 71001074, and the Science Research Foundation of Department of Education of Liaoning Province of China under Grant No. W2010302. This paper was recommended for publication by Editor Jinhu LU¨. 1034 NA YI, et al. continuous time for the epidemic, in particular, for the fast-spreading epidemics, discrete models are more realistic than continuous ones. Some discrete models have been used to formulate some epidemic models in recent years (see [4–6]). Innocenzo, et al.[7] studied the dynamical evolutions and codimention-one transcritical bifurcation of a discrete oscillating SIRS epidemic model. Gao, et al.[8] studied flip bifurcation and chaos of a discrete epidemic model with density-dependent birth pulses and seasonal prevention. Ramani, et al.[9] investigated a discrete epidemic model with oscillating. Mendez and Fort[10] considered the dynamical evolution of a deterministic discrete epidemic model. For other literature of biology systems, the reader is referred to [11–13] and the references therein. The literature mentioned above investigated the stabilities and the analysis of codimension-one bifurcations of discrete epidemic models.

However, they did not consider the analysis of codimention-two bifurcations and control for discrete epidemic models.

Researchers often study bifurcation phenomena as one systemic parameter varies. Many practical models contain several systemic parameters. Complicated bifurcations such as codimension-two bifurcations likely occur when more than one systemic parameter is varied at the same time. Codimension-two bifurcations, also known as double crises, occur when different codimension-one bifurcations intersect in the two-dimensional plane of the systemic parameters. Many literature considered the codimention-two bifurcations for different fields. Luo, et al.[14] considered 1:2 strong resonance bifurcation of a vibroimpact system. Kaslik and Balint[15] studied codimension-two bifurcations of a delayed discrete-time Hopfield neural network system, including fold-Neimark-Sacker bifurcation, double Neimark-Sacker bifurcation and 1:1 strong resonance bifurcation. Algaba, et al.[16] investigated Bogdanov-Takens bifurcation of a simple electronic circuit. Peng, et al.[17] studied some local bifurcations and global bifurcations of a discrete economic model, including codimension-one bifurcations, codimension-two bifurcations, and hetero-clinic, homo-clinic bifurcations, etc. Ruan and Wang[18] discussed Bogdanov-Takens bifurcation of a continuous epidemic model with a nonlinear incidence rate. In our problem, the codimension-two bifurcations of a discrete epidemic model are studied.

Alexander and Moghadas[2] analyzed a continuous SIV model with a generalized nonlinear incidence rate and showed the existence of bistability and periodicity using the Poincare index theory. Based on the continuous epidemic model of [2], we consider a discrete epidemic model with a nonlinear incidence rate obtained by forward Euler scheme. The dynamic behaviors of the discrete epidemic model can exhibit some rich phenomena, such as codimension-two bifurcations and chaotic behavior. In this paper, our interest focuses on the codimention-two bifurcations analysis of the discrete epidemic model. The two-parameter bifurcations of the fixed points in the discrete epidemic model, associated with 1:1 strong resonance, 1:2 strong resonance, 1:3 strong resonance and 1:4 strong resonance, are analyzed. In order to eliminate the occurrence of chaos for the discrete epidemic model, a tracking controller is designed.