In epidemiology, the basic reproduction number (sometimes called basic reproductive ratio, or incorrectly basic reproductive rate, and denoted R0, r nought) of an infection can be thought of as the number of cases one case generates on average over the course of its infectious period, in an otherwise uninfected population.
This metric is useful because it helps determine whether or not an infectious disease can spread through a population. The roots of the basic reproduction concept can be traced through the work of Alfred Lotka, Ronald Ross, and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria.
the infection will die out in the long run. But if
the infection will be able to spread in a population.
Generally, the larger the value of R0, the harder it is to control the epidemic. For simple models and a 100%-effective vaccine, the proportion of the population that needs to be vaccinated to prevent sustained spread of the infection is given by 1 − 1/R0. The basic reproduction number is affected by several factors including the duration of infectivity of affected patients, the infectiousness of the organism, and the number of susceptible people in the population that the affected patients are in contact with.
In populations that are not homogeneous, the definition of R0 is more subtle. The definition must account for the fact that a typical infected individual may not be an average individual. As an extreme example, consider a population in which a small portion of the individuals mix fully with one another while the remaining individuals are all isolated. 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. This is because the typical infected individual is in the fully mixed portion and thus is able to successfully cause infections. In general, if the individuals who become infected early in an epidemic may be more (or less) likely to transmit than a randomly chosen individual late in the epidemic, then our computation of R0 must account for this tendency. An appropriate definition for R0 in this case is "the expected number of secondary cases produced by a typical infected individual early in an epidemic".
Say that an infectious individual makes β contacts per unit time producing new infections with a mean infectious period of 1/γ. Therefore, the basic reproduction number is
In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction number for each transition time into the disease. An example of this is tuberculosis. Blower et al. calculated from a simple model of TB the following reproduction number:
In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW tuberculosis.
R0 is also used as a measure of individual reproductive success in population ecology, evolutionary invasion analysis and life history theory. It represents the average number of offspring produced over the lifetime of an individual (under ideal conditions).
For simple population models, R0 can be calculated, provided an explicit decay rate (or "death rate") is given. In this case, the reciprocal of the decay rate (usually 1/d) gives the average lifetime of an individual. When multiplied by the average number of offspring per individual per timestep (the "birth rate" b), this gives R0 = b / d. For more complicated models that have variable growth rates (e.g. because of self-limitation or dependence on food densities), the maximum growth rate should be used.
When calculated from mathematical models, particularly ordinary differential equations, what is often claimed to be R0 is, in fact, simply a threshold, not the average number of secondary infections. There are many methods used to derive such a threshold from a mathematical model, but few of them always give the true value of R0. This is particularly problematic if there are intermediate vectors between hosts, such as malaria.
What these thresholds will do is determine whether a disease will die out (if R0 < 1) or whether it may become epidemic (if R0 > 1), but they generally can not compare different diseases. Therefore, the values from the table above should be used with caution, especially if the values were calculated from mathematical models.
Methods include the survival function, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method, calculations from the intrinsic growth rate, existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection  and the final size equation. Few of these methods agree with one another, even when starting with the same system of differential equations. Even fewer actually calculate the average number of secondary infections. Since R0 is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.
In the 2011 film Contagion, a fictional medical disaster thriller, R0 calculations are presented to reflect the progression of a fatal viral infection from case studies to a pandemic.