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In the Reed-Frost model, an example of an SIR epidemic model, one can examinea statistic that counts the number of concurrently infected individuals. Thisstatistic 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 weaklyto the integral of a Brownian motion with parabolic drift. This same statisticexhibits a deterministic scaling limit when $p = (1+\lambda \varepsilon_n)/n$whenever $\varepsilon_n\to 0$ and $n^{1/3}\varepsilon_n\to\infty$.

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