An SIRS model with a nonlinear incidence rate |
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| Affiliation: | 1. Department of Mathematics, Southwest China Normal University, Chongqing 400715, PR China;2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NF, Canada A1C 5S7;3. Department of Mathematics, Xiangfan University, Xiangfan 441053, PR China;1. Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom;2. Centro de Astronomia e Astrofísica da Universidade de Lisboa, Campo Grande, Edificío C8, 1749-016 Lisboa, Portugal;3. Department of Computing and Information Management, Hong Kong Institute of Vocational Education, Chai Wan, Hong Kong, PR China;1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, Jiangsu, PR China;2. School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, PR China;3. College of Science, China University of Petroleum (East China), Qingdao 266580, Shandong, PR China |
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| Abstract: | The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease. |
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