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The epidemic starts growing when $(N-1)\beta/\gamma > 1$ because this condition ensures that the initial rate of new infections exceeds the rate of removals. With one initial infective ($I=1$) and $S=N-1$, the infected equation $\frac{dI}{dt}=\beta SI-\gamma I$ becomes positive only if the infection-production term $\beta(N-1)$ is greater than the removal term $\gamma$.

Conditions: Applies at the start of the outbreak.; Uses the single-infective initialization $I(0)=1$.; Assumes the SIR dynamics shown on the board.; $\gamma>0$ is assumed implicitly by division by $\gamma$.

The epidemic starts growing when $(N-1)\beta/\gamma > 1$ because this condition ensures that the initial rate of new infections exceeds the rate of removals. With one initial infective ($I=1$) and $S=N-1$, the infected equation $\frac{dI}{dt}=\beta SI-\gamma I$ becomes positive only if the infection-production term $\beta(N-1)$ is greater than the removal term $\gamma$.

Conditions: Applies at the start of the outbreak.; Uses the single-infective initialization $I(0)=1$.; Assumes the SIR dynamics shown on the board.; $\gamma>0$ is assumed implicitly by division by $\gamma$.