  We consider the Erd\"{o}s--R\'{e}nyi random graph $G_{n,p}$ and we analyze
the simple irreversible epidemic process on the graph, known in the literature
as bootstrap percolation. We give a quantitative version of some results by
Janson et al. (2012), providing a fine asymptotic analysis of the final size
$A_n^*$ of active nodes, under a suitable super-critical regime. More
specifically, we establish large deviation principles for the sequence of
random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate
functions and allowing the scaling function $f$ to vary in the widest possible
range.
