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Volume 10, Number 5, 2020, Pages 2278-2298                                                                DOI:10.11948/20200301
Multiple recurrent outbreak cycles in an autonomous epidemiological model due to multiple limit cycle bifurcation
Pei Yu,Maoan Han,Wenjing Zhang
Keywords:Epidemiology, infectious diseases, recurrent outbreaks, multiple attractors, multiple limit cycles.
      Multiple recurrent outbreak cycles have been commonly observed in infectious diseases such as measles and chicken pox. This complex outbreak dynamics in epidemiologicals is rarely captured by deterministic models. In this paper, we investigate a simple 2-dimensional SI epidemiological model and propose that the coexistence of multiple attractors attributes to the complex outbreak patterns. We first determine the conditions on parameters for the existence of an isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically prove that the maximum number of the coexisting limit cycles is three, and provide a corresponding set of parameters for the existence of the three limit cycles. Simulation results demonstrate the case with the maximum coexisting attractors, which contains one stable disease free equilibrium and two stable endemic periodic solutions separated by one unstable periodic solution. Therefore, different disease outcomes can be predicted by a single nonlinear deterministic model based on different initial data.
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