= Solution
The infectious count in the <SIR model> obeys $\dot I=I(\beta S-\gamma)$. At the initial state, $\dot I(0)=\beta N-\gamma$, so initial epidemic growth requires \b[the <epidemic threshold for the closed SIR model>]
$$
\boxed{N>\frac\gamma\beta,\qquad\mathcal R_0=\frac{\beta N}{\gamma}>1.}
$$
Here $\mathcal R_0$ is the <basic reproduction number> in this <mass-action infection> convention. Necessity also follows globally: $S(t)$ is nonincreasing, so if $\beta N\le\gamma$, then $\beta S(t)-\gamma\le0$ for all time and $I(t)$ never increases. When $\beta=0$, growth is impossible.
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