Complex dynamics of epidemic models on adaptive networks |
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Authors: | Xiaoguang Zhang Chunhua Shan Zhen Jin Huaiping Zhu |
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Institution: | 1. Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China;2. Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Taiyuan 030006, Shanxi, China;3. Department of Mathematics and Statistics, The University of Toledo, Toledo, OH 43606, USA;4. LAMPS and Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada |
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Abstract: | There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks. |
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Keywords: | Epidemic model Adaptive network Transient oscillations Bogdanov–Takens bifurcation Homoclinic orbit Multiple limit cycles |
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