What is reproduction number?

The reproduction number, written R, is the average number of people that one infected person goes on to infect. It is the single most useful number for telling whether an outbreak will grow or shrink. If R is above 1 the number of cases rises, and if it is below 1 the outbreak dies out.

How it works

The basic number R0 applies to a population where nobody is immune and no measures are in place. In practice the effective number, often written Rt, changes over time as people recover and become immune, and as behavior or policies alter how often people meet. A simple approximation is R = contacts per period x chance of transmission per contact x length of the infectious period.

This is a feedback loop. Each new case adds to the pool of infectious people, which drives further cases, so growth is exponential when R is above 1. As more people become immune, the share of contacts that can be infected falls, which pushes R down and eventually ends the growth.

Example

Say each infected person meets 10 people a day, each contact has a 3% chance of passing on the infection, and a person is infectious for 4 days. Then R = 10 x 0.03 x 4 = 1.2. Starting from 100 cases, one generation later there are about 120 new cases, then 144, then 173. If distancing halves the contacts to 5 a day, R = 5 x 0.03 x 4 = 0.6, and each generation is only 60% as large as the one before: 100, 60, 36. These numbers are illustrative.

How to model it in Machinations

Use a Pool for susceptible people, one for infectious and one for recovered, linked by a Converter that moves people from susceptible to infectious. A Register computes the infection rate from contacts, chance and the share still susceptible, and a state connection feeds it back. Make contacts a Lever, then Play and chart the three Pools.