Basic reproduction number \(R_0\)
The basic reproduction number, \(R_0\), measures the ability of an infection to spread when it is introduced into a population that is effectively fully susceptible.
Why do we need \(R_0\)?
At the beginning of an outbreak we want to know whether a small number of infections will tend to grow or decline. \(R_0\) combines the mechanisms that create new infections and remove infectious individuals into a threshold quantity.
Where does \(R_0=\beta/\gamma\) come from?
For the simple frequency-dependent SIR model,
\[\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I.\]At the very beginning of an outbreak, almost everyone is susceptible, so
\[\frac{S}{N}\approx1.\]Therefore,
\[\frac{dI}{dt}\approx\beta I-\gamma I=(\beta-\gamma)I.\]This already shows that infection grows initially if \(\beta>\gamma\). But the same threshold can be given a more intuitive interpretation.
In a fully susceptible population, one infectious individual generates new infections at approximately rate \(\beta\). If recovery occurs at constant per-capita rate \(\gamma\), the mean infectious period is
\[\frac{1}{\gamma}.\]Hence
\[\underbrace{\beta}_{\substack{\text{infections per}\text{unit time}}}\times\underbrace{\frac{1}{\gamma}}_{\substack{\text{mean time}\text{infectious}}}=\boxed{R_0=\frac{\beta}{\gamma}}.\]So \(R_0\) can be read as infection production rate × average infectious duration in this simple model.
The threshold \(R_0=1\)
infection tends to decline
threshold
infection can initially grow
The mathematical reason is especially clear if we rewrite the early-outbreak approximation using \(R_0=\beta/\gamma\):
\[\frac{dI}{dt}\approx(\beta-\gamma)I=\gamma(R_0-1)I.\]So \(R_0=1\) is the point at which the sign of the initial growth rate changes.
Visualising the early-outbreak threshold
The following schematic graph shows the local early-outbreak behaviour predicted by the approximation \(dI/dt\approx\gamma(R_0-1)I\). All three curves start with the same small infectious population.
This graph should be interpreted as an early-outbreak illustration, not as the complete SIR epidemic trajectory. When \(R_0>1\), infection initially grows, but in a full SIR epidemic it does not increase forever: depletion of susceptible individuals eventually reduces transmission and \(I(t)\) can peak and decline.
Likewise, \(R_0=1\) means zero initial growth in a nearly fully susceptible population. It should not generally be described as an endemic equilibrium. An endemic equilibrium is a separate concept in which infection persists at a positive equilibrium level, and its existence depends on the model.
Why does \(R_0\) not mean everyone gets infected?
\(R_0\) describes transmission under the initial, fully susceptible conditions. As an epidemic develops, susceptibility falls and conditions change. Therefore \(R_0=3\) does not mean every infectious individual throughout the entire epidemic will infect exactly three people, nor does it mean that the whole population must become infected.
Effective reproduction number
Once part of the population is no longer susceptible, the reproduction number changes. In the simple SIR model, a useful time-dependent quantity is
\[\boxed{R_{\mathrm{eff}}(t)=R_0\frac{S(t)}{N}}.\]Even when \(R_0>1\), the epidemic begins to decline when the susceptible fraction becomes small enough that
\[R_{\mathrm{eff}}(t)<1.\]\(R_0\) depends on the model
The formula \(R_0=\beta/\gamma\) belongs to the simple SIR/SIS-type formulation used here. More detailed models may contain exposed stages, several infectious groups, age-dependent contact, vectors, spatial structure or heterogeneous mixing. Their reproduction numbers can have different formulas and may require methods such as the next-generation matrix.
Thus \(R_0\) is not a fixed universal property of a pathogen alone. Its value depends on the pathogen, host population, contact conditions and the mathematical model used to represent them.