标题： 具有异质性的SIS模型中有效繁殖数的拓扑性质 显示英文标题

标题： Topological Properties of the Effective Reproduction Number in an Heterogeneous SIS Model

摘要： 本研究结果为简单流行病学SIS模型中疫苗的最优分配奠定了基础，在该模型中考虑了一个非常一般的异质人群。 在当前设定下，每个个体具有类型 x，属于一个通用空间，而接种策略是一个函数 $\eta$，其中 $\eta$(x) $\in$ [0, 1] 表示类型为 x 的个体中未接种者的比例。 我们将考虑与接种策略 $\eta$相关的两个损失函数：要么是有效繁殖数，这是流行病学中许多模型中的经典量，在这里由依赖于 $\eta$的紧算子的谱半径给出；要么是在最大地方病状态下的整体感染个体比例。通过在接种策略集 $\Delta$上考虑弱-*拓扑，使其成为一个紧集，我们可以利用算子族的集体紧性概念证明这两个损失函数的连续性。 我们还证明了它们相对于SIS模型参数的稳定性。 最后，我们在“几乎”不可约的情况下考虑它们的单调性和相关性质。 显示英文摘要

摘要： This present results lay the foundations for the study of the optimal allocation of vaccine in the simple epidemiological SIS model where one consider a very general heterogeneous population. In the present setting each individual has a type x belonging to a general space, and a vaccination strategy is a function $\eta$ where $\eta$(x) $\in$ [0, 1] represents the proportion of non-vaccinated among individuals of type x. We shall consider two loss functions associated to a vaccination strategy $\eta$: either the effective reproduction number, a classical quantity appearing in many models in epidemiology, and which is given here by the spectral radius of a compact operator that depends on $\eta$; or the overall proportion of infected individuals after vaccination in the maximal endemic state.By considering the weak-* topology on the set $\Delta$ of vaccination strategies, so that it is a compact set, we can prove that those two loss functions are continuous using the notion of collective compactness for a family of operators. We also prove their stability with respect to the parameters of the SIS model. Eventually, we consider their monotonicity and related properties in particular when the model is ``almost'' irreducible.