Feng Zhilan, Hill Andrew N, Smith Philip J, Glasser John W
Department of Mathematics, Purdue University, West Lafayette, IN, USA.
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention, CDC, Atlanta, GA, USA.
J Theor Biol. 2015 Dec 7;386:177-87. doi: 10.1016/j.jtbi.2015.09.006. Epub 2015 Sep 14.
The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.
许多疫苗接种计划的目标是实现群体免疫，即高于这一水平时，感染者（如来自疫区的旅行者）引入的病原体将不会引发疫情。通过一个简单的元种群模型，我们证明，如果亚群体在影响其基本繁殖数的特征上存在差异，或者其成员优先混合，那么加权平均亚群体免疫不能像公共卫生从业者通常所做的那样，与按比例混合的同质群体免疫阈值进行比较。然后我们回顾了人均接触率异质性对基本元种群繁殖数的影响。例如，就人口密度影响接触而言，城市和农村亚群体的接触率可能不同。亚群体在影响其基本繁殖数的特征上的其他差异也会有类似作用。与最近的结果一致，我们表明亚群体之间的异质优先混合比同质优先混合更能增加基本元种群繁殖数。接下来，我们完善了关于亚群体免疫异质性和优先混合对有效元种群繁殖数影响的早期结果。最后，我们提出将这个繁殖数相对于亚群体免疫的偏导数向量作为一种全新的工具，用于指导疫苗接种工作。
