A new relationship between Erd\H{o}s-R\'{e}nyi graphs, epidemic models and Brownian motion with parabolic drift

In the Reed-Frost model, an example of an SIR epidemic model, one can examine a statistic that counts the number of concurrently infected individuals. This statistic can be reformulated as a statistic on the \ER random graph $G(n,p)$. Within the critical window of Aldous and Martin-L\"{o}f, i.e. when $p = p(n) = n^{-1}+\lambda n^{-4/3}$, the cumulative sum of this statistic converges weakly to the integral of a Brownian motion with parabolic drift. This same statistic exhibits a deterministic scaling limit when $p = (1+\lambda \varepsilon_n)/n$ whenever $\varepsilon_n\to 0$ and $n^{1/3}\varepsilon_n\to\infty$.