  We study the susceptible-infected-recovered (SIR) epidemic on a random graph
chosen uniformly over all graphs with certain critical, heavy-tailed degree
distributions. For this model, each vertex infects all its susceptible
neighbors and recovers the day after it was infected. When a single individual
is initially infected, the total proportion of individuals who are eventually
infected approaches zero as the size of the graph grows towards infinity. Using
different scaling, we prove process level scaling limits for the number of
individuals infected on day $h$ on the largest connected components of the
graph. The scaling limits are contain non-negative jumps corresponding to some
vertices of large degree, that is these vertices are super-spreaders. Using
weak convergence techniques, we can describe the height profile of the
$\alpha$-stable continuum random graph (Goldschmidt et. al. 2018,
Conchon-Kerjan - Goldschmidt 2020), extending results known in the Brownian
case (Miermont - Sen 2019). We also prove abstract results that can be used on
other critical random graph models.
