Chen Shiyang, Epureanu Bogdan
Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109, USA.
J Theor Biol. 2017 Feb 21;415:137-144. doi: 10.1016/j.jtbi.2016.12.013. Epub 2016 Dec 20.
The interaction between nonlinearity and seasonal forcing in childhood infectious diseases often leads to multiyear cycles with large amplitude. Regular biennial cycles in particular were observed in measles reports throughout the world. The objective of this paper is to understand the mechanism of such biennial cycles, especially the conditions under which the large amplitude biennial oscillation might appear. It is proposed that such biennial cycles are caused by parametric resonance, which might occur when varying the parameter at a frequency close to twice the natural frequency of the system. The analysis is carried out by solving an SIR model semi-analytically using method of multiple scales (MMS). This analysis shows how parametric resonance occurs due to the interaction between nonlinearity and periodic forcing. Using the MMS solution, the boundary between the resonance region and the non-resonance region in the parameter space is obtained. The effects of different parameters on the triggering of parametric resonance are studied, such as transmission rate, recovery rate, birth rate and amplitude of seasonality. The effects of stochasticity on the onset of parametric resonance are also studied.
儿童传染病中非线性与季节性强迫之间的相互作用常常导致大振幅的多年周期。特别是在世界各地的麻疹报告中观察到了有规律的两年周期。本文的目的是了解这种两年周期的机制，尤其是大振幅两年振荡可能出现的条件。有人提出，这种两年周期是由参数共振引起的，当以接近系统固有频率两倍的频率改变参数时可能会发生这种共振。通过使用多尺度法（MMS）对SIR模型进行半解析求解来进行分析。该分析表明了由于非线性与周期强迫之间的相互作用而产生参数共振的方式。利用MMS解，得到了参数空间中共振区域与非共振区域之间的边界。研究了不同参数对参数共振触发的影响，如传播率、恢复率、出生率和季节性振幅。还研究了随机性对参数共振发生的影响。
