Why does the epidemic start growing when (N-1)$β/γ$ is greater than 1?

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 $\gamma$.

Reasoning, step by step

1. Start from the infected-compartment equation: $\frac{dI}{dt}=\beta SI-\gamma I$.
2. Substitute the initial condition for a single first infective: $S=N-1$ and $I=1$.
3. Evaluate the growth rate at the initial state: $\left.\frac{dI}{dt}\right|_{t=0}=\beta(N-1)-\gamma$.
4. Require the initial infected population to increase: $\beta(N-1)-\gamma>0$.
5. Rearrange to isolate the dimensionless threshold ratio: $\frac{(N-1)\beta}{\gamma}>1$.

Example

The board states 'Epidemic occurs if: dI/dt>0' and writes '$R_0=(N-1)\beta/\gamma>1$'. The lecturer says that when the factor is greater than one, the epidemic starts growing because more infected people are being produced than removed.

Common misconceptions

• Thinking the threshold is $\frac{(N-1)\beta}{\gamma}>0$ instead of $>1$.
• Assuming the condition applies to the entire duration of the epidemic rather than just the initial onset.

Watch the explanation

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