Exact equations for SIR epidemics on tree graphs
  We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this ``deterministic'' representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically. 
Kieran J. Sharkey, Istvan Z. Kiss, Robert R. Wilkinson, Peter L. Simon
q-bio.PE math.PR