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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 $\gamma$.

The basic reproductive number $R_0$ is the expected number of secondary infections produced by one infected individual in the initial population setting used by the model. In this lecture, it is written as $(N-1)\beta/\gamma$ or $\hat{\beta}/\gamma$.

Conditions: Defined for the SIR model under the initial condition $I(0)=1$.; $\hat{\beta}$ is the lumped exposure-rate parameter introduced in this segment.; The quantity is the same $R_0$ derived from the initial-growth condition.; The model context is the SIR framework displayed on the board.

The basic reproductive number $R_0$ is the expected number of secondary infections produced by one infected individual in the initial population setting used by the model. In this lecture, it is written as $(N-1)\beta/\gamma$ or $\hat{\beta}/\gamma$.

Conditions: Defined for the SIR model under the initial condition $I(0)=1$.; $\hat{\beta}$ is the lumped exposure-rate parameter introduced in this segment.; The quantity is the same $R_0$ derived from the initial-growth condition.; The model context is the SIR framework displayed on the board.