Basic reproduction number
(1918 pandemic strain)
In epidemiology, the basic reproduction number (sometimes called basic reproductive rate, basic reproductive ratio and denoted R0, r nought) of an infection is the number of cases one case generates on average over the course of its infectious period.
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.
- R0 < 1
the infection will die out in the long run. But if
- R0 > 1
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, 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 reproductive rate 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.
Other uses 
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.
Limitations of R0 
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 endemic (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 popular culture 
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.
Further reading 
See also 
- Epi Info software program
- Epidemic model
- Epidemiological methods
- Epidemiological Transition
- Unless noted R0 values are from: History and Epidemiology of Global Smallpox Eradication From the training course titled "Smallpox: Disease, Prevention, and Intervention". The CDC and the World Health Organization. Slide 16-17.
- Wallinga J, Teunis P (2004). "Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures". Am. J. Epidemiol. 160 (6): 509–16. doi:10.1093/aje/kwh255. PMID 15353409.
- Mills CE, Robins JM, Lipsitch M (2004). "Transmissibility of 1918 pandemic influenza". Nature 432 (7019): 904–6. doi:10.1038/nature03063. PMID 15602562.
- "Pandemic Potential of a Strain of Influenza A (H1N1): Early Findings". Science 324 (5934): 1557–1561. 19 June 2009. doi:10.1126/science.1176062.Free text
- de Boer, Rob J. Theoretical Biology. Retrieved 2007-11-13.
- Diekmann O and Heesterbeek JAP (2000). Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. New York: Wiley.
- Chowell G, Hengartnerb NW, Castillo-Chaveza C, Fenimorea PW and Hyman JM (2004). "The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda". Journal of Theoretical Biology 229 (1): 119–126. doi:10.1016/j.jtbi.2004.03.006. PMID 15178190.
- Ajelli M, Iannelli M, Manfredi P and 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. PMID 18314231.
- Heffernan JM, Smith RJ, Wahl LM (2005). "Perspectives on the Basic Reproductive Ratio". Journal of the Royal Society Interface 2 (4): 281–93. doi:10.1098/rsif.2005.0042. PMC 1578275. PMID 16849186.
- Jones, James Holland. "Notes on R0". Retrieved 12 March 2013.