Basic reproduction number

(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".

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.

does not by itself give an estimate of how fast an infection spreads in the population.

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.

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

[9] Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is

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

[10] The roots of the basic reproduction concept can be traced through the work of Ronald Ross, Alfred Lotka and others,[11] but its first modern application in epidemiology was by George Macdonald in 1952,[12] who constructed population models of the spread of malaria.

In his work he called the quantity basic reproduction rate and denoted it by

In these models, population members are assigned to 'compartments' with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered).

A disease may be able to spread in the fully mixed portion even though a randomly selected individual would lead to fewer than one secondary case.

[14] The basic reproduction number can be computed as a ratio of known rates over time: if a contagious individual contacts

In reality, varying proportions of the population are immune to any given disease at any given time.

drops below, herd immunity has been achieved and the number of cases occurring in the population will gradually decrease to zero.

in the popular press has led to misunderstandings and distortions of its meaning.

[10] Therefore, the contagiousness of different infectious agents cannot be compared without recalculating

This is particularly problematic if there are intermediate vectors between hosts (as is the case for zoonoses), such as malaria.

of well-known contagious diseases" table should be conducted with caution.

cannot be modified through vaccination or other changes in population susceptibility, it can vary based on a number of biological, sociobehavioral, and environmental factors.

include the survival function, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method,[22] calculations from the intrinsic growth rate,[23] existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection[24] and the final size equation.

[25] Few of these methods agree with one another, even when starting with the same system of differential equations.

[18] Even fewer actually calculate the average number of secondary infections.

is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.

mentioned in the previous section, estimates have been made for a number of genera, and are shown in this table.

for species, strains, and variants are typically less accurate than for genera, and so are provided in separate tables below for diseases of particular interest (influenza and COVID-19).

In the 2011 film Contagion, a fictional medical disaster thriller, a blogger's calculations for

are presented to reflect the progression of a fatal viral infection from isolated cases to a pandemic.