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

Exact solutions and bounds for network SIR and SEIR models using a rooted-tree approximation