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 $N$.

中文翻译：

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张量积公式的精确流行病模型

提出了为网络上的随机系统获得准确的转换率矩阵的通用框架，并将其应用于许多众所周知的流行病学分区模型。人口的状态被描述为$ N $个个体概率向量空间的张量积空间中的向量，其维数等于流行病模型$ n_c $的区室数。通过采用$ n_c $维矩阵的张量积的合适线性组合，可以获得$ n_c ^ N $维Markov链的跃迁速率矩阵。生成的跃迁速率矩阵是双局部线性算子的总和，它使您可以深入了解系统的微观动力学。通过将精确模型限制为不相关（可分离）状态，可以恢复更为熟悉的基于节点的非线性平均场近似。