Basic reproduction number

In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted ${\displaystyle R_{0}}$ (pronounced R nought or R zero),^[1] of an infection is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection.^[2] The definition assumes that no other individuals are infected or immunized (naturally or through vaccination). Some definitions, such as that of the Australian Department of Health, add the absence of "any deliberate intervention in disease transmission".^[3] The basic reproduction number is not necessarily the same as the effective reproduction number ${\displaystyle R}$ (usually written ${\displaystyle R_{t}}$ [t for time], sometimes ${\displaystyle R_{e}}$),^[4] which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. ${\displaystyle R_{0}}$ is a dimensionless number (persons infected per person infecting) and not a time rate, which would have units of time⁻¹,^[5] or units of time like doubling time.^[6]

${\displaystyle R_{0}}$ is not a biological constant for a pathogen as it is also affected by other factors such as environmental conditions and the behaviour of the infected population. ${\displaystyle R_{0}}$ values are usually estimated from mathematical models, and the estimated values are dependent on the model used and values of other parameters. Thus values given in the literature only make sense in the given context and it is not recommended to compare values based on different models.^[7] ${\displaystyle R_{0}}$ does not by itself give an estimate of how fast an infection spreads in the population.

The most important uses of ${\displaystyle R_{0}}$ are determining if an emerging infectious disease can spread in a population and determining what proportion of the population should be immunized through vaccination to eradicate a disease. In commonly used infection models, when ${\displaystyle R_{0}>1}$ the infection will be able to start spreading in a population, but not if ${\displaystyle R_{0}<1}$. Generally, the larger the value of ${\displaystyle R_{0}}$, the harder it is to control the epidemic. For simple models, the proportion of the population that needs to be effectively immunized (meaning not susceptible to infection) to prevent sustained spread of the infection has to be larger than ${\displaystyle 1-1/R_{0}}$.^[8] This is the so-called Herd immunity threshold or herd immunity level. Here, herd immunity means that the disease cannot spread in the population because each infected person, on average, can only transmit the infection to less than one other contact.^[9] Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is ${\displaystyle 1/R_{0}}$. However, this threshold is based on simple models that assume a fully mixed population with no structured relations between the individuals. For example, if there is some correlation between people's immunization (e.g., vaccination) status, then the formula ${\displaystyle 1-1/R_{0}}$ may underestimate the herd immunity threshold.^[9]

The basic reproduction number is affected by several factors, including the duration of infectivity of affected people, the contagiousness of the microorganism, and the number of susceptible people in the population that the infected people contact.^[10]