Theoretical Physics Group, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK.
J Theor Biol. 2010 Nov 7;267(1):85-94. doi: 10.1016/j.jtbi.2010.08.014. Epub 2010 Aug 17.
We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.
我们使用分析技术研究了具有时变强制的随机易感染恢复（SIR）模型，这些技术使我们能够分离随机性和外部强制的相互作用。该模型被构造成连续时间马尔可夫过程，通过使用逆系统大小的展开，将其分解为确定性动力学和随机校正。强制在确定性动力学中引起了一个极限环，并且给出了关于这个时变解的波动的完整分析。当极限环是年度的时，以及当它是两年一次时，经过倍周期分岔后，我们应用了这种分析。我们方法的综合性使我们能够给出一个统一的动力学图景，该图景统一了过去的工作，但也为预测百日咳和麻疹流行中观察到的振荡周期提供了系统的方法。
