Why does R-naught greater than one lead to exponential growth?

Conditions

• Assumes the same average transmission per case across generations.
• The worked example uses $R_0=2.5$.

Reasoning, step by step

1. Start with one initial case.
2. Multiply the number of cases in each successive generation by $R_0$.
3. Observe that if $R_0 > 1$, the number of cases increases from one generation to the next.
4. Conclude that this repeated multiplication produces exponential growth.

Example

The video shows the sequence $1 \xrightarrow{\times 2.5} 2.5 \xrightarrow{\times 2.5} 6.25 \xrightarrow{\times 2.5} 15.6$, illustrating how case counts explode when $R_0=2.5$.

Common misconceptions

• Thinking that $R_0 > 1$ means the outbreak grows linearly rather than exponentially.
• Assuming that every single infected person infects exactly $R_0$ people, rather than it being an average.

Watch the explanation

Explore next

• Why is R-naught equal to one the dividing line for an outbreak?
• How do you compute the herd immunity threshold from R-naught?