论文标题
根据最大的地方性平衡接种疫苗可实现群的免疫力
Vaccinating according to the maximal endemic equilibrium achieves herd immunity
论文作者
论文摘要
我们考虑了Lajmanovich和Yorke（1976）在有限维度中引入的一般异质种群的简单流行病学SIS模型，以及我们在以前的作品中引入的无限维度概括。在此模型中，基本复制数$ r_0 $由积分运算符的频谱半径给出。如果$ r_0> 1 $，则存在最大的地方性平衡。在这个非常普遍的异质SIS模型中，我们证明，根据这种最大的地方性平衡的特征进行疫苗接种，可确保群的免疫力。此外，这种疫苗接种策略至关重要：由此产生的有效繁殖数完全等于一个。作为应用程序，我们估计，如果$ r_0 = 2 $在一个年龄结构化的社区中，将其混合率安装在社交活动中，那么采用此策略将需要疫苗剂量少约29％，而疫苗剂量比均匀接种$ 1-1-1/r_0 $的人口的策略要少29％。 从动力学系统的角度来看，我们证明了平衡$ g $的非最大程度性等同于其在原始动力学中的线性不稳定性，以及在修改的动力学中无疾病状态的线性不稳定性，我们根据$ g $进行疫苗接种。
We consider the simple epidemiological SIS model for a general heterogeneous population introduced by Lajmanovich and Yorke (1976) in finite dimension, and its infinite dimensional generalization we introduced in previous works. In this model the basic reproducing number $R_0$ is given by the spectral radius of an integral operator. If $R_0>1$, then there exists a maximal endemic equilibrium. In this very general heterogeneous SIS model, we prove that vaccinating according to the profile of this maximal endemic equilibrium ensures herd immunity. Moreover, this vaccination strategy is critical: the resulting effective reproduction number is exactly equal to one. As an application, we estimate that if $R_0 = 2$ in an age-structured community with mixing rates fitted to social activity, applying this strategy would require approximately 29% less vaccine doses than the strategy which consists in vaccinating uniformly a proportion $1 - 1/R_0$ of the population. From a dynamical systems point of view, we prove that the non-maximality of an equilibrium $g$ is equivalent to its linear instability in the original dynamics, and to the linear instability of the disease-free state in the modified dynamics where we vaccinate according to $g$.
