Editing Basic reproduction number (section)

{{short description|Metric in epidemiology}}
{{About|the rate of spread of an epidemic|the average number of offspring born to a female|Net reproduction rate}}
{{redirect|R number||R-value (disambiguation){{!}}R-value}}
{{Use mdy dates|date=February 2020}}

[[File:R Naught Ebola and Flu Diagram.svg|thumb|<math>R_0</math> is the average number of people infected from one other person. For example, Ebola has an <math>R_0</math> of two, so on average, a person who has Ebola will pass it on to two other people.]]

In [[epidemiology]], the '''basic reproduction number''', or '''basic reproductive number''' (sometimes called '''basic reproduction ratio''' or '''basic reproductive rate'''), denoted <math>R_0</math> (pronounced ''R nought'' or ''R zero''),<ref>{{Cite book|title=Vaccinology : an essential guide| vauthors = Milligan GN, Barrett AD |publisher=Wiley Blackwell|year=2015|isbn=978-1-118-63652-7|location=Chichester, West Sussex| pages=310 |oclc=881386962}}</ref> of an [[infection]] is the [[Expected value|expected number]] of cases directly generated by one case in a population where all individuals are [[Susceptible individual|susceptible]] to infection.<ref name="Fraser"/> The definition assumes that no other individuals are infected or [[Immunization|immunized]] (naturally or through [[vaccination]]). Some definitions, such as that of the [[Department of Health (Australia)|Australian Department of Health]], add the absence of "any deliberate intervention in disease transmission".<ref name=":0">{{Cite book|title=Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease|vauthors=Becker NG, Glass K, Barnes B, Caley P, Philp D, McCaw JM, McVernon J, Wood J|display-authors=6|url=https://www1.health.gov.au/internet/publications/publishing.nsf/Content/mathematical-models|chapter-url=https://www1.health.gov.au/internet/publications/publishing.nsf/Content/mathematical-models~mathematical-models-models.htm~mathematical-models-2.2.htm|publisher=National Centre for Epidemiology and Population Health|date=April 2006|isbn=1-74186-357-0|chapter=The reproduction number|access-date=2020-02-01|archive-date=February 1, 2020|archive-url=https://web.archive.org/web/20200201033944/https://www1.health.gov.au/internet/publications/publishing.nsf/Content/mathematical-models|url-status=dead}}</ref> The basic reproduction number is not necessarily the same as the [[effective reproduction number]] <math>R</math> (usually written <math>R_t</math> [''t'' for "time"], sometimes <math>R_e</math>),<ref>{{cite journal | vauthors = Adam D | title = A guide to R - the pandemic's misunderstood metric | journal = Nature | volume = 583 | issue = 7816 | pages = 346–348 | date = July 2020 | pmid = 32620883 | doi = 10.1038/d41586-020-02009-w | bibcode = 2020Natur.583..346A | doi-access = free }}</ref> which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. <math>R_0</math> is a [[dimensionless number]] (persons infected per person infecting) and not a time rate, which would have units of time<sup>−1</sup>,<ref>{{Cite web|url=https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf|title=Notes On R0| vauthors = Jones J |website=Stanford University}}</ref> or units of time like [[doubling time]].<ref>{{Cite web|url=https://www.forbes.com/sites/startswithabang/2020/03/17/why-exponential-growth-is-so-scary-for-the-covid-19-coronavirus/|title=Why 'Exponential Growth' Is So Scary For The COVID-19 Coronavirus| vauthors = Siegel E | website=Forbes |language=en |access-date=2020-03-19}}</ref>

<math>R_0</math> 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. <math>R_0</math> 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 not recommended to compare values based on different models.<ref name = "Delamater">{{cite journal | vauthors = Delamater PL, Street EJ, Leslie TF, Yang YT, Jacobsen KH | title = Complexity of the Basic Reproduction Number (R<sub>0</sub>) | journal = Emerging Infectious Diseases | volume = 25 | issue = 1 | pages = 1–4 | date = January 2019 | pmid = 30560777 | pmc = 6302597 | doi = 10.3201/eid2501.171901 }}</ref> <math>R_0</math> does not by itself give an estimate of how fast an infection spreads in the population.

The most important uses of <math>R_0</math> 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 [[Mathematical modelling of infectious disease|infection models]], when <math>R_0 > 1</math> the infection will be able to start spreading in a population, but not if <math>R_0 < 1</math>. Generally, the larger the value of <math>R_0</math>, 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 <math>1 - 1 / R_0</math>.<ref>{{cite journal |last1=Fine |first1=P. |last2=Eames |first2=K. |last3=Heymann |first3=D. L. |title='Herd Immunity': A Rough Guide |journal=Clinical Infectious Diseases |date=1 April 2011 |volume=52 |issue=7 |pages=911–916 |doi=10.1093/cid/cir007 |pmid=21427399 |doi-access=free }}</ref> This is the so-called [[herd immunity]] threshold or herd immunity level. Here, herd immunity means that the disease cannot spread in the population because each infected person, on average, can only transmit the infection to less than one other contact.<ref name=":1">{{Cite journal |last1=Hiraoka |first1=Takayuki |last2=K. Rizi |first2=Abbas |last3=Kivelä |first3=Mikko |last4=Saramäki |first4=Jari |date=2022-05-12 |title=Herd immunity and epidemic size in networks with vaccination homophily |url=https://link.aps.org/doi/10.1103/PhysRevE.105.L052301 |journal=Physical Review E |volume=105 |issue=5 |pages=L052301 |doi=10.1103/PhysRevE.105.L052301|pmid=35706197 |arxiv=2112.07538 |bibcode=2022PhRvE.105e2301H |s2cid=245130970 }}</ref> Conversely, the proportion of the population that remains susceptible to infection in the [[endemic (epidemiology)|endemic equilibrium]] is <math>1 / R_0</math>. However, this threshold is based on simple models that assume a fully mixed population with no [[Compartmental models in epidemiology|structured relations]] between the individuals. For example, if there is some correlation between people's immunization (e.g., vaccination) status, then the formula <math>1 - 1 / R_0</math> may underestimate the herd immunity threshold.<ref name=":1" />

{{herd_immunity_threshold_vs_r0.svg}}

The basic reproduction number is affected by several factors, including the duration of [[infectivity]] of affected people, the contagiousness of the [[microorganism]], and the number of susceptible people in the population that the infected people contact.<ref name="Vegvari"/>