EPIDEMICS ON RANDOM INTERSECTION GRAPHS
EPIDEMICS ON RANDOM INTERSECTION GRAPHS
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DOI：
10.1214/13-aap942
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发表时间：
2014-06-01
影响因子：
1.8
通讯作者：
Trapman, Pieter
中科院分区：
文献类型：
--
作者：
Ball, Frank G.;Sirl, David J.;Trapman, Pieter
在本文中，我们考虑了一个模型的传播的随机SIR（易感->传染->隔离）流行病的网络上的个人描述的随机交叉图。个体属于随机数量的小集团，每个小集团的大小都是随机的，当且仅当两个个体都属于一个小集团时，感染才能在两个个体之间传播。集团的大小和集团的个人属于遵循混合泊松分布的数量。一个无限型的分支过程近似（与类型被给定的个人的传染期的长度）的早期阶段的流行病的发展和充分严格的证明相关的极限定理的人口规模趋于无穷大。这导致了一个阈值参数R-*，因此在一个大的人群中，当且仅当R-*>1时，初始感染者很少的流行病才能引起大的爆发。近似的无限型分支过程的生存概率的函数方程的确定;如果R-*1，它被证明有精确的一个非零解。利用一个单一类型的分支过程，接近一个典型的个人的易感性集的大小，证明了一个法律的大数这样一个大的爆发的大小。
In this paper we consider a model for the spread of a stochastic SIR (Susceptible -> Infectious -> Recovered) epidemic on a network of individuals described by a random intersection graph. Individuals belong to a random number of cliques, each of random size, and infection can be transmitted between two individuals if and only if there is a clique they both belong to. Both the clique sizes and the number of cliques an individual belongs to follow mixed Poisson distributions. An infinite-type branching process approximation (with type being given by the length of an individual's infectious period) for the early stages of an epidemic is developed and made fully rigorous by proving an associated limit theorem as the population size tends to infinity. This leads to a threshold parameter R-*, so that in a large population an epidemic with few initial infectives can give rise to a large outbreak if and only if R-*>1. A functional equation for the survival probability of the approximating infinite-type branching process is determined; if R-*1, it is shown to have precisely one nonzero solution. A law of large numbers for the size of such a large outbreak is proved by exploiting a single-type branching process that approximates the size of the susceptibility set of a typical individual.