Exact epidemic models from a tensor product formulation

TL;DR

This paper introduces a tensor product framework to derive exact transition rate matrices for stochastic epidemic models on networks, enabling precise analysis beyond mean-field approximations.

Contribution

It presents a novel tensor-based method to obtain exact Markov chain transition matrices for network epidemics, capturing microscopic dynamics.

Findings

Exact transition matrices for epidemic models derived using tensor products.

Analytic solutions for SI outbreaks on trees and cycle graphs.

Insight into microscopic dynamics through bilocal linear operators.

Abstract

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 $N$.