Basic reproduction number

In epidemiology, the basic reproduction number (sometimes called basic reproductive rate or basic reproductive ratio) of an infection is the mean number of secondary cases a typical single infected case will cause in a population with no immunity to the disease in the absence of interventions to control the infection. It is often denoted "R"₀. This metric is useful because it helps determine whether or not an infectious disease will 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 applicationin epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria.

When

:"R"₀ < 1

the infection will die out in the long run (provided infection rates are constant). But if

:"R"₀ > 1

the infection will be able to spread in a population. Large values of "R"₀ may indicate the possibility of a major epidemic.

Generally, the larger the value of "R"₀, the harder it is to control the epidemic. In particular, the proportion of the population that needs to be vaccinated to provide herd immunity and 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

"R"₀ is also used as a measure of individual reproductive success in population ecology [cite book | author = de Boer, Rob J | title = Theoretical Biology | url = http://theory.bio.uu.nl/rdb/books/tb.pdf | accessdate = 2007-11-13] , 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, "R"₀ 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 $R\_0\; =\; b\; /\; d$. 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 "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.

What these thresholds will do is determine whether a disease will die out (if "R"₀<1) or become endemic (if "R"₀>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 [cite book |author=Diekmann O and Heesterbeek JAP|title=Mathematical epidemiology of infectious diseases: model building, analysis and interpretation|publisher=New York: Wiley|year=2000] , calculations from the intrinsic growth rate [cite journal |author=Chowell G, Hengartnerb NW, Castillo-Chaveza C, Fenimorea PW and Hyman JM|title=The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda|journal=Journal of Theoretical Biology |volume=229 |issue=1 |pages=119–126 |year=2004 |doi=10.1016/j.jtbi.2004.03.006] , existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection [cite journal |author=Ajelli M, Iannelli M, Manfredi P and Ciofi degli Atti, ML|title=Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas|journal=Vaccine|volume=26
issue=13 |pages=1697–1707 |year=2008 |doi=10.1016/j.vaccine.2007.12.058] 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 "R"₀ is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness. [cite journal |author=Heffernan JM, Smith RJ, Wahl LM |title=Perspectives on the Basic Reproductive Ratio |url= http://www.mathstat.uottawa.ca/~rsmith/R0Review.pdf |journal=Journal of the Royal Society Interface |volume=2 |issue=4 |pages=281–93 |year=2005 |pmid=16849186 |doi=10.1098/rsif.2005.0042]

References