标题： 大数定律在Erdős-Rényi图上具有随机顶点权重的SIR模型中的应用 显示英文标题

标题： Law of large numbers for the SIR model with random vertex weights on Erdős-Rényi graph

摘要： 本文我们关注的是在 Erdős-Rényi 图$G(n,p)$上具有随机顶点权重的 SIR 模型。Erdős-Rényi 图$G(n,p)$是从具有$n$个顶点的完全图$C_n$中通过独立地以概率$(1-p)$删除每条边生成的。我们在每个顶点上分配了正随机变量$\rho$的独立同分布拷贝作为顶点权重。对于 SIR 模型，每个顶点处于三种状态之一：`易感'、`感染'和`移除'。一个感染的顶点以与这两个顶点权重乘积成比例的速率感染给定的易感邻居。一个感染的顶点以恒定速率变为移除状态。一个移除的顶点将不再被感染。 我们假设在$t=0$处没有移除的顶点，感染顶点的数量服从伯努利分布$B(n,\theta)$。我们的主要结果是该模型的大数定律。 我们给出两个确定性函数$H_S(\psi_t), H_V(\psi_t)$对于$t\geq 0$并证明对于任何$t\geq 0$，$H_S(\psi_t)$是易感顶点的极限比例，而$H_V(\psi_t)$是感染顶点在时刻$t$感染给一个给定的易感邻居的平均能力的极限，当$n$趋向于无穷大时。 显示英文摘要

摘要： In this paper we are concerned with the SIR model with random vertex weights on Erd\H{o}s-R\'{e}nyi graph $G(n,p)$. The Erd\H{o}s-R\'{e}nyi graph $G(n,p)$ is generated from the complete graph $C_n$ with $n$ vertices through independently deleting each edge with probability $(1-p)$. We assign i. i. d. copies of a positive r. v. $\rho$ on each vertex as the vertex weights. For the SIR model, each vertex is in one of the three states `susceptible', `infective' and `removed'. An infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices. An infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at $t=0$ there is no removed vertex and the number of infective vertices follows a Bernoulli distribution $B(n,\theta)$. Our main result is a law of large numbers of the model. We give two deterministic functions $H_S(\psi_t), H_V(\psi_t)$ for $t\geq 0$ and show that for any $t\geq 0$, $H_S(\psi_t)$ is the limit proportion of susceptible vertices and $H_V(\psi_t)$ is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment $t$ as $n$ grows to infinity.