Grenfell B T, Kleczkowski A, Gilligan C A, Bolker B M
Zoology Department, Cambridge University, UK.
Stat Methods Med Res. 1995 Jun;4(2):160-83. doi: 10.1177/096228029500400205.
There is currently considerable interest in the role of nonlinear phenomena in the population dynamics of infectious diseases. Childhood diseases such as measles are particularly well documented dynamically, and have recently been the subject of analyses (of both models and notification data) to establish whether the pattern of epidemics is chaotic. Though the spatial dynamics of measles have also been extensively studied, spatial and nonlinear dynamics have only recently been brought together. The present review concentrates mainly on describing this synthesis. We begin with a general review of the nonlinear dynamics of measles models, in a spatially homogeneous environment. Simple compartmental models (specifically the SEIR model) can behave chaotically, under the influence of strong seasonal 'forcing' of infection rate associated with patterns of schooling. However, adding observed heterogeneities such as age structure can simplify the deterministic dynamics back to limit cycles. By contrast all current strongly seasonally forced stochastic models show large amplitude irregular fluctuations, with many more 'fadeouts' of infection that is observed in real communities of similar size. This indicates that (social and/or geographical) spatial heterogeneity is needed in the models. We review the exploration of this problem with nonlinear spatiotemporal models. The few studies to date indicate that spatial heterogeneity can help to increase the realism of models. However, a review of nonlinear analyses of spatially subdivided measles data show that more refinements of the models (particularly in representing the impact of human demographic changes on infection dynamics) are required. We conclude with a discussion of the implication of these results for the dynamics of infectious diseases in general and, in particular, the possibilities of cross fertilization between human disease epidemiology and the study of plant and animal diseases.
目前，非线性现象在传染病种群动态中的作用引起了广泛关注。麻疹等儿童疾病在动态方面有特别详尽的记录，最近已成为（模型和通报数据）分析的对象，以确定流行病模式是否具有混沌性。尽管麻疹的空间动态也已得到广泛研究，但空间动态和非线性动态直到最近才被结合起来。本综述主要集中于描述这种综合情况。我们首先对空间均匀环境中麻疹模型的非线性动态进行一般性综述。简单的 compartments 模型（特别是 SEIR 模型）在与学校模式相关的感染率的强烈季节性“驱动”影响下可能表现出混沌性。然而，加入诸如年龄结构等观察到的异质性可以将确定性动态简化回极限环。相比之下，所有当前受强烈季节性驱动的随机模型都显示出大幅不规则波动，感染的“消退”比在类似规模的真实社区中观察到的要多得多。这表明模型中需要（社会和/或地理）空间异质性。我们回顾了用非线性时空模型对这个问题的探索。迄今为止的少数研究表明，空间异质性有助于提高模型的现实性。然而，对空间细分的麻疹数据的非线性分析综述表明，需要对模型进行更多改进（特别是在表示人口结构变化对感染动态的影响方面）。我们最后讨论这些结果对一般传染病动态的意义，特别是人类疾病流行病学与动植物疾病研究之间交叉融合的可能性。
