基于张量积公式的精确传染病模型
Exact epidemic models from a tensor product formulation
摘要：
本文提出了一个获得网络上随机系统精确转移率矩阵的一般框架，并将其应用于许多著名的流行病学房室模型。人口的状态被描述为$N$个体概率向量空间的张量积空间中的向量，其维度等于流行病学模型$n_c$的隔间数。通过对$n_c$维矩阵的张量积取适当的线性组合，得到了$n_c^N$维马氏链的转移率矩阵.由此产生的过渡率矩阵是一个双局部线性算子的总和，这使洞察系统的微观动力学。更熟悉的和非线性的基于节点的平均场近似恢复限制的精确模型，以不相关（可分离）的状态。我们展示了如何准确的转移率矩阵的易感染（SI）模型可以用来找到解析解SI爆发的树和循环图有限$N$。
英文摘要：
A general framework for obtaining exact transition rate matrices for stochastic systems on networks is presented and applied to many well-known compartmental models of epidemiology. The state of the population is described as a vector in the tensor product space of $N$ individual probability vector spaces, whose dimension equals the number of compartments of the epidemiological model $n_c$. The transition rate matrix for the $n_c^N$-dimensional Markov chain is obtained by taking suitable linear combinations of tensor products of $n_c$-dimensional matrices. The resulting transition rate matrix is a sum over bilocal linear operators, which gives insight in the microscopic dynamics of the system. The more familiar and non-linear node-based mean-field approximations are recovered by restricting the exact models to uncorrelated (separable) states. We show how the exact transition rate matrix for the susceptible-infected (SI) model can be used to find analytic solutions for SI outbreaks on trees and the cycle graph for finite $N$.
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