  In this paper, we develop a node-based approximate model for Markovian
contagion dynamics on networks. We prove that our approximate model is exact
for SIR (susceptible-infectious-recovered) and SEIR
(susceptible-exposed-infectious-recovered) dynamics on tree graphs with a
single source of infection and that the model otherwise gives upper bounds on
the probabilities of each node being susceptible. Our analysis of SEIR
contagion dynamics is generalised to SEIR models with arbitrarily many distinct
classes of exposed state. In the case of trees with a single source of
infection, our approach yields a system of partially-decoupled linear
differential equations that exactly describes the evolution of node-state
probabilities. We use this to state explicit closed-form solutions for SIR
dynamics on a chain.
