Epidemics on critical random graphs with heavy-tailed degree distribution

We study the susceptible–infected–recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. 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 contain non-negative jumps corresponding to some vertices of large degree. These weak convergence techniques allow us to describe the height profile of the $\alpha$-stable continuum random graph (Goldschmidt et al., 2022; Conchon-Kerjan and Goldschmidt, 2023), extending results known in the Brownian case (Miermont and Sen, 2022). We also prove model-independent results that can be used on other critical random graph models.