  We study propagation of avalanches in a certain excitable network. The model
is a particular case of the one introduced in [23], and is mathematically
equivalent to an endemic variation of the Reed-Frost epidemic model introduced
in [27]. Two types of heuristic approximation are frequently used for models of
this type in applications, a branching process for avalanches of a small size
at the beginning of the process and a deterministic dynamical system once the
avalanche spreads to a significant fraction of a large network. In this paper
we prove several results concerning the exact relation between the avalanche
model and these limits, including rates of convergence and rigorous bounds for
common characteristics of the model.
