Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6.
Department of Ecology and Evolutionary Biology, Department of Mathematics, and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA.
J R Soc Interface. 2019 Jul 26;16(156):20190151. doi: 10.1098/rsif.2019.0151. Epub 2019 Jul 31.
Mathematical models of childhood diseases date back to the early twentieth century. In several cases, models that make the simplifying assumption of homogeneous time-dependent transmission rates give good agreement with data in the absence of secular trends in population demography or transmission. The prime example is afforded by the dynamics of measles in industrialized countries in the pre-vaccine era. Accurate description of the transient dynamics following the introduction of routine vaccination has proved more challenging, however. This is true even in the case of measles which has a well-understood natural history and an effective vaccine that confers long-lasting protection against infection. Here, to shed light on the causes of this problem, we demonstrate that, while the dynamics of homogeneous and age-structured models can be qualitatively similar in the absence of vaccination, they diverge subsequent to vaccine roll-out. In particular, we show that immunization induces changes in transmission rates, which in turn reshapes the age distribution of infection prevalence, which effectively modulates the amplitude of seasonality in such systems. To examine this phenomenon empirically, we fit transmission models to measles notification data from London that span the introduction of the vaccine. We find that a simple age-structured model provides a much better fit to the data than does a homogeneous model, especially in the transition period from the pre-vaccine to the vaccine era. Thus, we propose that age structure and heterogeneities in contact rates are critical features needed to accurately capture transient dynamics in the presence of secular trends.
儿童疾病的数学模型可以追溯到 20 世纪初。在某些情况下，假设时间相关的传播率是均匀的模型在没有人口统计学或传播的长期趋势的情况下，与数据吻合得很好。在疫苗接种前的工业化国家麻疹的动态就是一个很好的例子。然而，准确描述常规疫苗接种后出现的暂态动力学更具挑战性。即使是麻疹，其具有明确的自然史和有效的疫苗，可以提供对感染的长期保护，情况也是如此。在这里，为了阐明这个问题的原因，我们证明，虽然均匀和年龄结构模型的动力学在没有接种疫苗的情况下可能在定性上相似，但在疫苗推出后它们会出现分歧。具体来说，我们表明，免疫接种会改变传播率，这反过来又会重塑感染流行的年龄分布，从而有效地调节此类系统季节性的幅度。为了从经验上检验这一现象，我们根据跨越疫苗接种引入期的伦敦麻疹报告数据拟合了传播模型。我们发现，一个简单的年龄结构模型比均匀模型更能很好地拟合数据，尤其是在从疫苗前到疫苗时代的过渡时期。因此，我们提出年龄结构和接触率的异质性是准确捕捉存在长期趋势的暂态动力学所需的关键特征。
