In epidemiology, the basic reproduction number (sometimes called basic reproductive ratio, or incorrectly basic reproductive rate, and denoted R_{0}, pronounced R nought or R zero^{[18]}) of an infection can be thought of as the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection.^{[19]} The definition describes the state where no other individuals are infected or immunized (naturally or through vaccination). Some definitions, such as that of the Australian Department of Health, add absence of "any deliberate intervention in disease transmission".^{[20]} The basic reproduction number is not to be confused with the effective reproduction numberR which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. By definition R_{0} cannot be modified through vaccination campaigns. Also it is important to note that R_{0} is a dimensionless number and not a rate, which would have units of time^{[21]} like doubling time^{[22]}.
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. Furthermore 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 recommended not to use obsolete values or compare values based on different models.^{[23]}R_{0} does not by itself give an estimate of how fast an infection spreads in the population.
The most important uses of 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 R_{0} > 1 the infection will be able to start spreading in a population, but not if R_{0} < 1. Generally, the larger the value of 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 1 − 1/R_{0}.^{[24]} Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is 1/R_{0}.
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.
The roots of the basic reproduction concept can be traced through the work of Ronald Ross, Alfred Lotka and others,^{[25]} but its first modern application in epidemiology was by George MacDonald in 1952,^{[26]} who constructed population models of the spread of malaria. In his work he called the quantity basic reproduction rate and denoted it by Z_{0}. Calling the quantity a "rate" can be misleading, insofar as it can be interpreted as number per unit of time. The expressions number or ratio are now preferred.
Definitions in specific cases
Reproductive number as it relates to contact rate and infectious period
R_{0} is the average number of people infected from one other person, for example, Ebola has an R_{0} of two, so on average, a person who has Ebola will pass it on to two other people.
Suppose that infectious individuals make an average of $\beta$ infection-producing contacts per unit time, with a mean infectious period of $\tau$. Then the basic reproduction number is:
$R_{0}=\beta \,\tau$
This simple formula suggests different ways of reducing R_{0} and ultimately infection propagation. It is possible to decrease the number of infection-producing contacts per unit time $\beta$ by reducing the number of contacts per unit time (for example staying at home if the infection requires contact with others to propagate) or the proportion of contacts that produces infection (for example wearing some sort of protective equipment). It is also possible to decrease the infectious period $\tau$ by finding and then isolating, treating or eliminating (as is often the case with animals) infectious individuals as soon as possible.
With varying latent periods
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 and coauthors calculated from a simple model of TB the following reproduction number:^{[27]}
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.
Heterogeneous populations
In populations that are not homogeneous, the definition of R_{0} 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 R_{0} must account for this tendency. An appropriate definition for R_{0} in this case is "the expected number of secondary cases produced by a typical infected individual early in an epidemic".^{[28]}
During an epidemic, typically the number of diagnosed infections $N(t)$ over time $t$ is known. In the early stages of an epidemic, growth is exponential, with a logarithmic growth rate^{[citation needed]}
$K={\frac {d\ln(N)}{dt}}.$
For exponential growth, $N$ can be interpreted as the cumulative number of diagnoses (including individuals who have recovered) or the present number of diagnosed patients; the logarithmic growth rate is the same for either definition. In order to estimate $R_{0}$, assumptions are necessary about the time delay between infection and diagnosis and the time between infection and starting to be infectious.
Latent infectious period, isolation after diagnosis
In this model, an individual infection has the following stages:
Exposed: an individual is infected, but has no symptoms and does not yet infect others. The duration of the exposed state is $\tau _{E}$.
Latent infectious: an individual is infected, has no symptoms, but does infect others. The duration of the latent infectious state is $\tau _{I}$. The individual infects $R_{0}$ other individuals during this period.
isolation after diagnosis: measures are taken to prevent further infections, for example by isolating the patient.
This is a SEIR model and R_{0} may be written in the following form^{[29]}
This estimation method has been applied to COVID-19 and SARS.^{[30]} It follows from the differential equation for the number of exposed individuals $n_{E}$ and the number of latent infectious individuals $n_{I}$,
The largest eigenvalue of the matrix is the logarithmic growth rate $K$, which can be solved for $R_{0}$.
Limitations of R_{0}
When calculated from mathematical models, particularly ordinary differential equations, what is often claimed to be R_{0} 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 R_{0}. This is particularly problematic if there are intermediate vectors between hosts, such as malaria.^{[31]}
What these thresholds will do is determine whether a disease will die out (if R_{0} < 1) or whether it may become epidemic (if R_{0} > 1), but they generally cannot 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^{[further explanation needed]} include the survival function, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method,^{[32]} calculations from the intrinsic growth rate,^{[33]} existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection^{[34]} and the final size equation. Few of these methods agree with one another, even when starting with the same system of differential equations.^{[citation needed]} Even fewer actually calculate the average number of secondary infections. Since R_{0} is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.^{[35]}
In popular culture
In the 2011 film Contagion, a fictional medical disaster thriller, a blogger's calculations for R_{0} are presented to reflect the progression of a fatal viral infection from case studies to a pandemic. The methods depicted were faulty.^{[36]}
Compartmental models in epidemiology describe disease dynamics over time in a population of susceptible (S), infectious (I), and recovered (R) people using the SIR model. Note that in the SIR model, R(0) and R_{0} are different quantities - the former describes the number of recovered at t = 0 whereas the latter describes the ratio between the frequency of contacts to the frequency of recovery.
According to Guangdong Provincial Center for Disease Control and Prevention, "The effective reproductive number (R or R_{e}^{[37]}) is more commonly used to describe transmissibility, which is defined as the average number of secondary cases generated by per [sic] infectious case. In the absence of control measures, R = R_{0}χ, where χ is the proportion of the susceptible population." For example, by one preliminary estimate during the ongoing pandemic, the effective reproductive number for 2019-nCoV was found as 2.9, whereas for SARS it was 1.77.^{[38]}
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^Kucharski, Adam and Althaus, Christian L. (2015). "The role of superspreading in Middle East respiratory syndrome coronavirus (MERS-CoV) transmission". Eurosurveillance. 20 (26): 14–8. doi:10.2807/1560-7917.ES2015.20.25.21167. PMID26132768.CS1 maint: multiple names: authors list (link)
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^Macdonald, G. (September 1952). "The analysis of equilibrium in malaria". Tropical Diseases Bulletin. 49 (9): 813–829. ISSN0041-3240. PMID12995455.
^Blower, S. M.; Mclean, A. R.; Porco, T. C.; Small, P. M.; Hopewell, P. C.; Sanchez, M. A. (1995). "The intrinsic transmission dynamics of tuberculosis epidemics". Nature Medicine. 1: 815–821. doi:10.1038/nm0895-815.
^O Diekmann; J.A.P. Heesterbeek; J.A.J. Metz (1990). "On the definition and the computation of the basic reproduction ratio R_{0} in models for infectious diseases in heterogeneous populations". Journal of Mathematical Biology. 28 (4): 356–382. doi:10.1007/BF00178324. hdl:1874/8051. PMID2117040.
^Diekmann O, Heesterbeek JA (2000). "The Basic Reproduction Ratio". Mathematical Epidemiology of Infectious Diseases : Model Building, Analysis and Interpretation. New York: Wiley. pp. 73–98. ISBN0-471-49241-8.
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^Ajelli M; Iannelli M; Manfredi P & Ciofi degli Atti, ML (2008). "Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas". Vaccine. 26 (13): 1697–1707. doi:10.1016/j.vaccine.2007.12.058. PMID18314231.