Analytic solution of Markovian epidemics without re-infections on heterogeneous networks

Most epidemic processes on networks can be modelled by a compartmental model, that specifies the spread of a disease in a population. The corresponding compartmental graph describes how the viral state of the nodes (individuals) changes from one compartment to another. If the compartmental graph does not contain directed cycles (e.g. the famous SIR model satisfies this property), then we provide an analytic, closed-form solution of the continuous-time Markovian compartmental model on heterogeneous networks. The eigenvalues of the Markovian process are related to cut sets in the contact graph between nodes with different viral states. We illustrate our finding by analytically solving the continuous-time Markovian SI and SIR processes on heterogeneous networks. We show that analytic extensions to e.g. non-Markovian dynamics, temporal networks, simplicial contagion and more advanced compartmental models are possible. Our exact and explicit formula contains sums over all paths between two states in the SIR Markov graph, which prevents the computation of the exact solution for arbitrary large graphs.