and $U(Y_*) \simeq \log{\epsilon}$, we estimate the slope of the straight line connecting two points $\left(1,U(1)\right)$ and $\left(Y_*,U(Y_*)\right)$ in the $(Y,U)$ plane as $(U(Y_*)-U(1))/(Y_*-1)\simeq 
\sqrt{{\epsilon}}(\log {\epsilon})$, which approaches zero in the limit ${\epsilon}\rightarrow 0$. Thus, the transition from $Y=1$ to $Y=Y_*$ may be assumed to be free Brownian motion with the diffusion constant $D=(\lambda+1)/8$. The transition rate from $Y=1$ to $Y_*$ is then estimated as $T=2D/Y_*^2  = {\epsilon}+ O({\epsilon}^2)$. We thus obtain $$\begin{aligned}
q(\lambda) &=& {\epsilon}+O({\epsilon}^2).
\label{qep}\end{aligned}$$ In Fig. \[fig-sy\_lgv-sp-Nxxx-fit\], we compare the theoretical result with those obtained in numerical simulations of and . We measured the probability that $\rho > 0.003$, which is denoted as $p(\rho > 0.003)$. Recall that $\lim_{N \to \infty} p(\rho >0.003) =q(\lambda)$ when $\rho_*(\lambda) > 0.003$. Since the experimental result suggests $p(\rho >0.003)={\epsilon}+O({\epsilon}^2)$ in the limit $N \to \infty$, we claim that the theoretical result (\[qep\]) is in good agreement with the experimental result.

Concluding remarks
==================

In this paper, we have achieved a novel understanding of the intrinsic unpredictability of epidemic outbreaks by analyzing the Langevin equation , which effectively describes this singular phenomenon. Further, trajectories in the outbreak phase are divided into two groups: trajectories in one group are absorbed into zero, and the others diverge in . The division corresponds to the non-trivial limiting density given in . On the basis of this description, we calculated the probability of an epidemic outbreak near the transition point. Before ending the paper, we make a few remarks. First, the probability $q(\lambda)$ was studied in the mathematical literature (see [@yan2008distribution] and [@britton2010stochastic] as reviews.) To the best of our knowledge, the method proposed in this paper has never been used in previous studies. It m