Huang Yu-Jhe, Huang Hsuan Te, Juang Jonq, Wu Cheng-Han
Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 300 Taiwan, ROC.
J Nonlinear Sci. 2022;32(1):15. doi: 10.1007/s00332-021-09776-4. Epub 2021 Dec 28.
In this paper, we propose and analyze a nonsmoothly two-dimensional map arising in a seasonal influenza model. Such map consists of both linear and nonlinear dynamics depending on where the map acts on its domain. The map exhibits a complicated and unpredictable dynamics such as fixed points, period points, chaotic attractors, or multistability depending on the ranges of a certain parameters. Surprisingly, bistable states include not only the coexistence of a stable fixed point and stable period three points but also that of stable period three points and a chaotic attractor. Among other things, we are able to prove rigorously the coexistence of the stable equilibrium and stable period three points for a certain range of the parameters. Our results also indicate that heterogeneity of the population drives the complication and unpredictability of the dynamics. Specifically, the most complex dynamics occur when the underlying basic reproduction number with respect to our model is an intermediate value and the large portion of the population in the same compartment changes in states the following season.
在本文中，我们提出并分析了一个出现在季节性流感模型中的非光滑二维映射。这样的映射由线性和非线性动力学组成，这取决于映射在其定义域上的作用位置。该映射表现出复杂且不可预测的动力学，例如不动点、周期点、混沌吸引子或多重稳定性，这取决于某些参数的取值范围。令人惊讶的是，双稳状态不仅包括稳定不动点和稳定周期三点的共存，还包括稳定周期三点和混沌吸引子的共存。除此之外，我们能够严格证明在一定参数范围内稳定平衡点和稳定周期三点的共存。我们的结果还表明，种群的异质性驱动了动力学的复杂性和不可预测性。具体而言，当相对于我们模型的潜在基本再生数为中间值且同一隔室中的大部分种群在下一季状态发生变化时，会出现最复杂的动力学。
