Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany.
Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany.
Chaos. 2024 Sep 1;34(9). doi: 10.1063/5.0221150.
In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of a constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., >0.95). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and, in some cases, via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.
本工作研究了恒定和时变接种率对易感-暴露-感染-恢复-易感（SEIRS）季节性模型的影响。通过计算 Lyapunov 指数，我们表明，在给定恒定接种率和另一个模型参数的组合下，会出现典型的复杂结构，如虾状结构。在某些特定情况下，恒定接种率不会作为混沌抑制器，并且在高接种率（例如，>0.95）下可能存在混沌带。此外，我们获得了一个控制参数和恒定接种率之间的线性和非线性关系，以建立无病解决方案。我们还验证了无论动力学是混沌还是周期性的，总感染人数都不会改变。通过包含具有定义幅度和频率的周期性函数来引入时变疫苗。对于这种情况，我们研究了不同幅度和频率对混沌吸引子的影响，产生了接触的低、中、高季节性程度。根据时变疫苗函数的参数，可以控制混沌结构并使其变为周期性结构。对于给定的参数集，这些结构主要通过危机进入，在某些情况下通过倍周期进入。之后，我们研究了时变疫苗在混沌和周期性吸引子共存时如何在双稳态动力学中发挥作用。我们发现，这种疫苗通过破坏几乎所有周期性基域来发挥控制作用。我们通过以下事实解释了这一点：在双稳态状态下，混沌吸引子比周期性吸引子表现出更有利于流行病的特征。
