An epidemic follows the Kermack-McKendrick model with $\frac{dS}{dt} = -\beta SI$ and $\frac{dI}{dt} = \beta SI - \gamma I$. If the population is $N=1000$ and initial susceptibles are $S_0 = 999$, what is the threshold value of $\beta$ required for an outbreak to occur if $\gamma = 0.1$?