basic reproduction number

{{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}}

In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted $R_0$ (pronounced R nought or R zero),{{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}} of an infection is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. The definition assumes that no other individuals are infected or immunized (naturally or through vaccination). Some definitions, such as that of the Australian Department of Health, add the absence of "any deliberate intervention in disease transmission".{{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}} The basic reproduction number is not necessarily the same as the effective reproduction number $R$ (usually written $R_t$ [t for "time"], sometimes $R_e$ ),{{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 }} which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. $R_0$ is a dimensionless number (persons infected per person infecting) and not a time rate, which would have units of time⁻¹,{{Cite web|url=https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf|title=Notes On R0| vauthors = Jones J |website=Stanford University}} or units of time like doubling time.{{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}}

$R_0$ 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. $R_0$ 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 not recommended to compare values based on different models.{{cite journal | vauthors = Delamater PL, Street EJ, Leslie TF, Yang YT, Jacobsen KH | title = Complexity of the Basic Reproduction Number (R₀) | journal = Emerging Infectious Diseases | volume = 25 | issue = 1 | pages = 1–4 | date = January 2019 | pmid = 30560777 | pmc = 6302597 | doi = 10.3201/eid2501.171901 }} $R_0$ 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 $1 - 1 / R_0$ .{{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 }} 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.{{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 }} Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is $1 / R_0$ . However, this threshold is based on simple models that assume a fully mixed population with no structured relations between the individuals. For example, if there is some correlation between people's immunization (e.g., vaccination) status, then the formula $1 - 1 / R_0$ may underestimate the herd immunity threshold.

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.

History

The roots of the basic reproduction concept can be traced through the work of Ronald Ross, Alfred Lotka and others,{{cite journal | vauthors = Smith DL, Battle KE, Hay SI, Barker CM, Scott TW, McKenzie FE | title = Ross, macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens | journal = PLOS Pathogens | volume = 8 | issue = 4 | pages = e1002588 | date = 2012-04-05 | pmid = 22496640 | pmc = 3320609 | doi = 10.1371/journal.ppat.1002588 | doi-access = free }} but its first modern application in epidemiology was by George Macdonald in 1952,{{cite journal | vauthors = Macdonald G | title = The analysis of equilibrium in malaria | journal = Tropical Diseases Bulletin | volume = 49 | issue = 9 | pages = 813–29 | date = September 1952 | pmid = 12995455 }} who constructed population models of the spread of malaria. In his work he called the quantity basic reproduction rate and denoted it by $Z_0$ .

Overview of <math>R_0</math> estimation methods

= Compartmental models =

Compartmental models are a general modeling technique often applied to the mathematical modeling of infectious diseases. In these models, population members are assigned to 'compartments' with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered). These models can be used to estimate $R_0$

= Epidemic models on networks =

Epidemics can be modeled as diseases spreading over networks of contact and disease transmission between people.{{Cite book |url=http://networksciencebook.com/ |title=Network Science by Albert-László Barabási}} Nodes in these networks represent individuals and links (edges) between nodes represent the contact or disease transmission between them. If such a network is a locally tree-like network, then the basic reproduction can be written in terms of the average excess degree of the transmission network such that:

$R_0 = \frac{\beta}{\beta+\gamma} \frac{{\langle k^2 \rangle} -{\langle k \rangle}}{{\langle k \rangle}},$

where

${\beta}$

is the per-edge transmission rate,

${\gamma}$

is the recovery rate,

{\langle k \rangle}

is the mean-degree (average degree) of the network and

{\langle k^2 \rangle}

is the second moment of the transmission network degree distribution.

= 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 infected early in an epidemic are on average either more likely or less likely to transmit the infection than individuals infected late in the epidemic, then the computation of $R_0$ must account for this difference. An appropriate definition for $R_0$ in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual".{{cite journal | vauthors = Diekmann O, Heesterbeek JA, Metz JA | title = On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations | journal = Journal of Mathematical Biology | volume = 28 | issue = 4 | pages = 365–82 | date = 1990 | pmid = 2117040 | doi = 10.1007/BF00178324 | hdl-access = free | s2cid = 22275430 | hdl = 1874/8051 }}

The basic reproduction number can be computed as a ratio of known rates over time: if a contagious individual contacts $\beta$ other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of $\dfrac{1}{\gamma}$ , then the basic reproduction number is just $R_0 = \dfrac{\beta}{\gamma}$ . Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease.

Effective reproduction number

File:R number – the rate of infection; a government video.webm.]]

In reality, varying proportions of the population are immune to any given disease at any given time. To account for this, the effective reproduction number $R_e$ or $R$ is used. $R_t$ is the average number of new infections caused by a single infected individual at time t in the partially susceptible population. It can be found by multiplying $R_0$ by the fraction S of the population that is susceptible. When the fraction of the population that is immune increases (i. e. the susceptible population S decreases) so much that $R_e$ drops below 1, herd immunity has been achieved and the number of cases occurring in the population will gradually decrease to zero.{{cite journal | vauthors = Garnett GP | title = Role of herd immunity in determining the effect of vaccines against sexually transmitted disease | journal = The Journal of Infectious Diseases | volume = 191 | issue = Suppl 1 | pages = S97-106 | date = February 2005 | pmid = 15627236 | doi = 10.1086/425271 | doi-access = free }}{{cite journal | vauthors = Rodpothong P, Auewarakul P | title = Viral evolution and transmission effectiveness | journal = World Journal of Virology | volume = 1 | issue = 5 | pages = 131–4 | date = October 2012 | pmid = 24175217 | pmc = 3782273 | doi = 10.5501/wjv.v1.i5.131 | doi-access = free }}{{cite book| vauthors = Dabbaghian V, Mago VK |date=2013|title=Theories and Simulations of Complex Social Systems|url=https://books.google.com/books?id=AdLBBAAAQBAJ&pg=PA134|publisher=Springer|pages=134–35|isbn=978-3642391491|access-date=29 March 2015}}

Limitations of <math>R_0</math>

Use of $R_0$ in the popular press has led to misunderstandings and distortions of its meaning. $R_0$ can be calculated from many different mathematical models. Each of these can give a different estimate of $R_0$ , which needs to be interpreted in the context of that model.{{cite journal | vauthors = Vegvari C | title = Commentary on the use of the reproduction number R during the COVID-19 pandemic | journal = Stat Methods Med Res | date = 2022 | volume = 31 | issue = 9 | pages = 1675–1685 | doi = 10.1177/09622802211037079 | pmid = 34569883| pmc = 9277711 }} Therefore, the contagiousness of different infectious agents cannot be compared without recalculating $R_0$ with invariant assumptions. $R_0$ values for past outbreaks might not be valid for current outbreaks of the same disease. Generally speaking, $R_0$ can be used as a threshold, even if calculated with different methods: if $R_0 < 1$ , the outbreak will die out, and if $R_0 > 1$ , the outbreak will expand. In some cases, for some models, values of $R_0 < 1$ can still lead to self-perpetuating outbreaks. This is particularly problematic if there are intermediate vectors between hosts (as is the case for zoonoses), such as malaria.{{cite journal | vauthors = Li J, Blakeley D, Smith RJ | title = The failure of R0 | journal = Computational and Mathematical Methods in Medicine | volume = 2011 | issue = 527610 | pages = 527610 | year = 2011 | pmid = 21860658 | pmc = 3157160 | doi = 10.1155/2011/527610 | doi-access = free }} Therefore, comparisons between values from the "Values of $R_0$ of well-known contagious diseases" table should be conducted with caution.

Although $R_0$ cannot be modified through vaccination or other changes in population susceptibility, it can vary based on a number of biological, sociobehavioral, and environmental factors. It can also be modified by physical distancing and other public policy or social interventions, although some historical definitions exclude any deliberate intervention in reducing disease transmission, including nonpharmacological interventions. And indeed, whether nonpharmacological interventions are included in $R_0$ often depends on the paper, disease, and what if any intervention is being studied. This creates some confusion, because $R_0$ is not a constant; whereas most mathematical parameters with "nought" subscripts are constants.

$R$ depends on many factors, many of which need to be estimated. Each of these factors adds to uncertainty in estimates of $R$ . Many of these factors are not important for informing public policy. Therefore, public policy may be better served by metrics similar to $R$ , but which are more straightforward to estimate, such as doubling time or half-life ( $t_{1/2}$ ).{{cite thesis | vauthors = Balkew TM |date=December 2010 |title=The SIR Model When S(t) is a Multi-Exponential Function |publisher=East Tennessee State University |url=https://dc.etsu.edu/etd/1747 }}{{cite book | veditors = Ireland MW |date=1928 |title=The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases |location=Washington: U.S. |publisher=U.S. Government Printing Office |pages=116–7}}

Methods used to calculate $R_0$ include the survival function, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method,{{cite book |vauthors=Diekmann O, Heesterbeek JA |chapter=The Basic Reproduction Ratio |pages=73–98 |title=Mathematical Epidemiology of Infectious Diseases : Model Building, Analysis and Interpretation| publisher=New York: Wiley|year=2000 |isbn=0-471-49241-8 |chapter-url=https://books.google.com/books?id=5VjSaAf35pMC&pg=PA73 }} calculations from the intrinsic growth rate,{{cite journal | vauthors = Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, 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 = 119–26 | date = July 2004 | pmid = 15178190 | doi = 10.1016/j.jtbi.2004.03.006 | arxiv = q-bio/0503006 | bibcode = 2004JThBi.229..119C | s2cid = 7298792 }} existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection{{cite journal | vauthors = Ajelli M, Iannelli M, Manfredi P, 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 = 1697–707 | date = March 2008 | pmid = 18314231 | doi = 10.1016/j.vaccine.2007.12.058 | name-list-style = amp }} and the final size equation.{{Citation |last=von Csefalvay |first=Chris |title=2 - Simple compartmental models: The bedrock of mathematical epidemiology |date=2023-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780323953894000116 |work=Computational Modeling of Infectious Disease |pages=19–91 |editor-last=von Csefalvay |editor-first=Chris |publisher=Academic Press |language=en |doi=10.1016/b978-0-32-395389-4.00011-6 |isbn=978-0-323-95389-4 |access-date=2023-03-02}} 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 | vauthors = Heffernan JM, Smith RJ, Wahl LM | title = Perspectives on the basic reproductive ratio | journal = Journal of the Royal Society, Interface | volume = 2 | issue = 4 | pages = 281–93 | date = September 2005 | pmid = 16849186 | pmc = 1578275 | doi = 10.1098/rsif.2005.0042 }}

Sample values for various contagious diseases

Despite the difficulties in estimating $R_0$ mentioned in the previous section, estimates have been made for a number of genera, and are shown in this table. Each genus may be composed of many species, strains, or variants. Estimations of $R_0$ for species, strains, and variants are typically less accurate than for genera, and so are provided in separate tables below for diseases of particular interest (influenza and COVID-19).

class="wikitable sortable" : 1em; font-size:85%;" \|+Values of R₀ and herd immunity thresholds (HITs) of contagious diseases prior to intervention ! Disease !! Transmission !! R₀ !! HIT{{efn\|name=calc\|Calculated using {{math\|p {{=}} 1 − {{sfrac\|1\|R₀}}}}.}}
Measles	Aerosol	12–18{{cite journal \| vauthors = Guerra FM, Bolotin S, Lim G, Heffernan J, Deeks SL, Li Y, Crowcroft NS \| title = The basic reproduction number (R₀) of measles: a systematic review \| journal = The Lancet. Infectious Diseases \| volume = 17 \| issue = 12 \| pages = e420–e428 \| date = December 2017 \| pmid = 28757186 \| doi = 10.1016/S1473-3099(17)30307-9 }}	{{#expr:100-100/12 round 0}}–{{#expr:100-100/18 round 0}}%
Chickenpox (varicella)	Aerosol	10–12{{cite book \|author1=Ireland's Health Services \| title = Health Care Worker Information \| url = https://www.hse.ie/eng/health/immunisation/hcpinfo/guidelines/chapter23.pdf \| access-date = 2020-03-27}}	{{#expr:100-100/10 round 0}}–{{#expr:100-100/12 round 0}}%
Mumps	Respiratory droplets	10–12[https://www1.health.gov.au/internet/main/publishing.nsf/Content/cda-phlncd-mumps.htm Australian government Department of Health] Mumps Laboratory Case Definition (LCD)	{{#expr:100-100/10 round 0}}–{{#expr:100-100/12 round 0}}%
COVID-19 (see values for specific strains below)	Respiratory droplets and aerosol	2.9-9.5{{cite journal \|last1=Liu \|first1=Y \|title=The effective reproductive number of the Omicron variant of SARS-CoV-2 is several times relative to Delta \|journal=Journal of Travel Medicine \|date=9 March 2022 \|doi=10.1093/jtm/taac037 \|issn=1708-8305 \|at=Table 1 \|volume=29\|issue=3 \|pmid=35262737 \|pmc=8992231 }}	{{#expr:100-100/2.85 round 0}}–{{#expr:100-100/9.5 round 0}}%
Rubella	Respiratory droplets	6–7{{efn\|name=cdc-who-2001\|From a module of a training course{{cite AV media \| author1 = Centers for Disease Control and Prevention \| author2 = World Health Organization \|date=2001 \|title=Smallpox: disease, prevention, and intervention (training course) \|chapter=History and epidemiology of global smallpox eradication \|medium=Presentation \|url=https://stacks.cdc.gov/view/cdc/27929 \|url-status=live \|at=Slide 17 \|access-date=2021-06-17 \|publisher=Centers for Disease Control and Prevention \|id=cdc:27929 \|publication-place=Atlanta \|publication-date=2014-08-25 \|archive-url=https://web.archive.org/web/20170317185052/https://emergency.cdc.gov/agent/smallpox/training/overview/pdf/eradicationhistory.pdf \|archive-date=2017-03-17}} with data modified from other sources.{{cite journal \|last1=Fine \|first1=Paul E. M. \|title=Herd Immunity: History, Theory, Practice \|journal=Epidemiologic Reviews \|date=1993 \|volume=15 \|issue=2 \|pages=265–302 \|doi=10.1093/oxfordjournals.epirev.a036121 \|pmid=8174658 }}{{cite journal \|last1=Luman \|first1=ET \|last2=Barker \|first2=LE \|last3=Simpson \|first3=DM \|last4=Rodewald \|first4=LE \|last5=Szilagyi \|first5=PG \|last6=Zhao \|first6=Z \|title=National, state, and urban-area vaccination-coverage levels among children aged 19–35 months, United States, 1999 \|journal=American Journal of Preventive Medicine \|date=May 2001 \|volume=20 \|issue=4 \|pages=88–153 \|doi=10.1016/s0749-3797(01)00274-4 \|pmid=12174806 }}{{cite journal \|last1=Jiles \|first1=RB \|last2=Fuchs \|first2=C \|last3=Klevens \|first3=RM \|title=Vaccination coverage among children enrolled in Head Start programs or day care facilities or entering school \|journal=Morbidity and Mortality Weekly Report \|date=22 September 2000 \|volume=49 \|issue=9 \|pages=27–38 \|pmid=11016876 \|url=https://www.cdc.gov/mmwr/preview/mmwrhtml/ss4909a2.htm }}}}	{{#expr:100-100/6 round 0}}–{{#expr:100-100/7 round 0}}%
Polio	Fecal–oral route	5–7{{efn\|name=cdc-who-2001}}	{{#expr:100-100/5 round 0}}–{{#expr:100-100/7 round 0}}%
Pertussis	Respiratory droplets	5.5{{cite journal \| vauthors = Kretzschmar M, Teunis PF, Pebody RG \| title = Incidence and reproduction numbers of pertussis: estimates from serological and social contact data in five European countries \| journal = PLOS Medicine \| volume = 7 \| issue = 6 \| pages = e1000291 \| date = June 2010 \| pmid = 20585374 \| pmc = 2889930 \| doi = 10.1371/journal.pmed.1000291 \| doi-access = free }}	{{#expr:100-100/5.5 round 0}}%
Smallpox	Respiratory droplets	3.5–6.0{{cite journal \| vauthors = Gani R, Leach S \| title = Transmission potential of smallpox in contemporary populations \| journal = Nature \| volume = 414 \| issue = 6865 \| pages = 748–51 \| date = December 2001 \| pmid = 11742399 \| doi = 10.1038/414748a \| bibcode = 2001Natur.414..748G \| url = https://www.nature.com/articles/414748a \| url-access=subscription \| access-date = 18 March 2020 \| s2cid = 52799168 }}	{{#expr:100-100/3.5 round 0}}–{{#expr:100-100/6 round 0}}%
HIV/AIDS	Body fluids	2–5{{cite web \|url=https://netec.org/2020/01/30/playing-the-numbers-game-r0/ \|title=Playing the Numbers Game: R0 \|date=January 30, 2020 \|publisher=National Emerging Special Pathogen Training and Education Center \|quote=[...] while infections that require sexual contact like HIV have a lower R₀ (2-5). \|access-date=27 December 2020 \|archive-url=https://web.archive.org/web/20200512013302/https://netec.org/2020/01/30/playing-the-numbers-game-r0/ \|archive-date=12 May 2020}}	{{#expr:100-100/2 round 0}}–{{#expr:100-100/5 round 0}}%
SARS	Respiratory droplets	2–4{{cite tech report \|title=Consensus document on the epidemiology of severe acute respiratory syndrome (SARS) \|id=WHO/CDS/CSR/GAR/2003.11 \|institution=World Health Organization \|department=Department of Communicable Disease Surveillance and Response \|hdl=10665/70863 \|hdl-access=free \|page=26 \|quote=A number of researchers have estimated the basic reproduction number by fitting models to the initial growth of epidemics in a number of countries. Their observations indicate that the SARS-CoV is less transmissible than initially thought with estimates of Ro in the range of 2-4.}}	{{#expr:100-100/2 round 0}}–{{#expr:100-100/4 round 0}}%
Diphtheria	Saliva	data-sort-value="2.6" \| {{Estimate\|2.6\|1.7\|4.3\|type=cri\|mini=yes}}{{cite journal \| vauthors = Truelove SA, Keegan LT, Moss WJ, Chaisson LH, Macher E, Azman AS, Lessler J \| title = Clinical and Epidemiological Aspects of Diphtheria: A Systematic Review and Pooled Analysis \| journal = Clinical Infectious Diseases \| volume = 71 \| issue = 1 \| pages = 89–97 \| date = June 2020 \| pmid = 31425581 \| pmc = 7312233 \| doi = 10.1093/cid/ciz808 }}	data-sort-value="62" \| {{Estimate\|{{#expr:100-100/2.6 round 0}}\|{{#expr:100-100/1.7 round 0}}\|{{#expr:100-100/4.3 round 0}}\|unit=%\|mini=yes}}
Common cold (e.g., rhinovirus)	Respiratory droplets	2–3{{cite web \| vauthors = Freeman C \|title=Magic formula that will determine whether Ebola is beaten \|url=https://www.telegraph.co.uk/news/worldnews/ebola/11213280/Magic-formula-that-will-determine-whether-Ebola-is-beaten.html \|archive-url=https://ghostarchive.org/archive/20220112/https://www.telegraph.co.uk/news/worldnews/ebola/11213280/Magic-formula-that-will-determine-whether-Ebola-is-beaten.html \|archive-date=January 12, 2022 \|url-status=live \|url-access=subscription \|website=The Telegraph \|date=November 6, 2014 \|publisher=Telegraph.Co.Uk \|access-date=30 March 2020}}{{cbignore}}{{medcn\|date=December 2021}}	{{#expr:100-100/2 round 0}}–{{#expr:100-100/3 round 0}}%
Mpox	Physical contact, body fluids, respiratory droplets, sexual (MSM)	data-sort-value="2.1" \| {{Estimate\|{{Round\|2.13\|1}}\|{{Round\|1.1\|1}}\|{{Round\|2.67\|1}}\|mini=yes}}{{Cite journal \|vauthors=Grant R, Nguyen LL, Breban R \|date=2020-09-01 \|title=Modelling human-to-human transmission of monkeypox \|journal=Bulletin of the World Health Organization \|volume=98 \|issue=9 \|pages=638–640 \|doi=10.2471/BLT.19.242347 \|issn=0042-9686 \|pmc=7463189 \|pmid=33012864 \|url=http://www.who.int/bulletin/volumes/98/9/19-242347.pdf \|archive-url=https://web.archive.org/web/20201211172704/http://www.who.int/bulletin/volumes/98/9/19-242347.pdf \|archive-date=11 December 2020 \|url-status=dead}}{{Cite journal \|vauthors=Al-Raeei M \|date=February 2023 \|title=The study of human monkeypox disease in 2022 using the epidemic models: herd immunity and the basic reproduction number case \|journal=Annals of Medicine & Surgery \|volume=85 \|issue=2 \|pages=316–321 \|doi=10.1097/MS9.0000000000000229 \|issn=2049-0801 \|pmc=9949786 \|pmid=36845803 }}	data-sort-value="53" \| {{Estimate\|{{#expr:100-100/2.13 round 0}}\|{{#expr:100-100/1.28 round 0}}\|{{#expr:100-100/2.67 round 0}}\|unit=%\|mini=yes}}
Ebola (2014 outbreak)	Body fluids	data-sort-value="1.78" \| {{Estimate\|{{Round\|1.78\|1}}\|{{Round\|1.44\|1}}\|{{Round\|1.80\|1}}\|mini=yes}}{{cite journal \| vauthors = Wong ZS, Bui CM, Chughtai AA, Macintyre CR \| title = A systematic review of early modelling studies of Ebola virus disease in West Africa \| journal = Epidemiology and Infection \| volume = 145 \| issue = 6 \| pages = 1069–1094 \| date = April 2017 \| pmid = 28166851 \| doi = 10.1017/S0950268817000164 \| pmc = 9507849 \| quote = The median of the R₀ mean estimate for the ongoing epidemic (overall) is 1.78 (interquartile range: 1.44, 1.80) \| doi-access = free }}	data-sort-value="44" \| {{Estimate\|{{#expr:100-100/1.78 round 0}}\|{{#expr:100-100/1.44 round 0}}\|{{#expr:100-100/1.80 round 0}}\|unit=%\|mini=yes}}
Influenza (seasonal strains)	Respiratory droplets	data-sort-value="1.13" \| {{Estimate\|1.3\|1.2\|1.4\|mini=yes}}{{cite journal \| vauthors = Chowell G, Miller MA, Viboud C \| title = Seasonal influenza in the United States, France, and Australia: transmission and prospects for control \| journal = Epidemiology and Infection \| volume = 136 \| issue = 6 \| pages = 852–64 \| date = June 2008 \| pmid = 17634159 \| doi = 10.1017/S0950268807009144 \| url= \| publisher = Cambridge University Press \| pmc = 2680121 \| quote = The reproduction number across influenza seasons and countries lied in the range 0.9–2.0 with an overall mean of 1.3, and 95% confidence interval (CI) 1.2–1.4. }}	data-sort-value="23" \| {{Estimate\|{{#expr:100-100/1.3 round 0}}\|{{#expr:100-100/1.2 round 0}}\|{{#expr:100-100/1.4 round 0}}\|unit=%\|mini=yes}}
Andes hantavirus	Respiratory droplets and body fluids	data-sort-value="1.19" \| {{Estimate\|{{Round\|1.19\|1}}\|{{Round\|0.82\|1}}\|{{Round\|1.56\|1}}\|type=cri\|mini=yes}}{{cite journal \|last1=Martínez \|first1=Valeria P. \|last2=Di Paola \|first2=Nicholas \|last3=Alonso \|first3=Daniel O. \|last4=Pérez-Sautu \|first4=Unai \|last5=Bellomo \|first5=Carla M. \|last6=Iglesias \|first6=Ayelén A. \|last7=Coelho \|first7=Rocio M. \|last8=López \|first8=Beatriz \|last9=Periolo \|first9=Natalia \|last10=Larson \|first10=Peter A. \|last11=Nagle \|first11=Elyse R. \|last12=Chitty \|first12=Joseph A. \|last13=Pratt \|first13=Catherine B. \|last14=Díaz \|first14=Jorge \|last15=Cisterna \|first15=Daniel \|last16=Campos \|first16=Josefina \|last17=Sharma \|first17=Heema \|last18=Dighero-Kemp \|first18=Bonnie \|last19=Biondo \|first19=Emiliano \|last20=Lewis \|first20=Lorena \|last21=Anselmo \|first21=Constanza \|last22=Olivera \|first22=Camila P. \|last23=Pontoriero \|first23=Fernanda \|last24=Lavarra \|first24=Enzo \|last25=Kuhn \|first25=Jens H. \|last26=Strella \|first26=Teresa \|last27=Edelstein \|first27=Alexis \|last28=Burgos \|first28=Miriam I. \|last29=Kaler \|first29=Mario \|last30=Rubinstein \|first30=Adolfo \|last31=Kugelman \|first31=Jeffrey R. \|last32=Sanchez-Lockhart \|first32=Mariano \|last33=Perandones \|first33=Claudia \|last34=Palacios \|first34=Gustavo \|display-authors=6 \|title='Super-Spreaders' and Person-to-Person Transmission of Andes Virus in Argentina \|journal=New England Journal of Medicine \|date=3 December 2020 \|volume=383 \|issue=23 \|pages=2230–2241 \|doi=10.1056/NEJMoa2009040 \|pmid=33264545 \|s2cid=227259435 \|doi-access=free }}	data-sort-value="16" \| {{Estimate\|{{#expr:100-100/1.19 round 0}}\|0\|{{#expr:100-100/1.56 round 0}}\|unit=%\|mini=yes}}{{efn\|name=lowr}}
Nipah virus	Body fluids	{{Round\|0.48\|1}}{{cite journal \| vauthors = Luby SP \| title = The pandemic potential of Nipah virus \| journal = Antiviral Research \| volume = 100 \| issue = 1 \| pages = 38–43 \| date = October 2013 \| pmid = 23911335 \| doi = 10.1016/j.antiviral.2013.07.011 }}	0%{{efn\|name=lowr\|When R₀ < 1.0, the disease naturally disappears.}}
MERS	Respiratory droplets	data-sort-value="0.47" \| {{Estimate\|{{Round\|0.47\|1}}\|{{Round\|0.29\|1}}\|{{Round\|0.80\|1}}\|mini=yes}}{{cite journal \| vauthors = Kucharski AJ, Althaus CL \| title = The role of superspreading in Middle East respiratory syndrome coronavirus (MERS-CoV) transmission \| journal = Euro Surveillance \| volume = 20 \| issue = 25 \| pages = 14–8 \| date = June 2015 \| pmid = 26132768 \| doi = 10.2807/1560-7917.ES2015.20.25.21167 \| doi-access = free }}	0%{{efn\|name=lowr}}

Estimates for strains of influenza.

class="wikitable sortable" : 1em; font-size:85%;" \|+Values of R₀ and herd immunity thresholds (HITs) for specific influenza strains ! Disease !! Transmission !! R₀ !! HIT{{efn\|name=calc}}
Influenza (1918 pandemic strain) \|Respiratory droplets \|2{{Cite web\|date=2021-12-21\|title=Omicron transmission: how contagious diseases spread\|url=https://www.nebraskamed.com/COVID/how-quickly-do-diseases-spread\|access-date=2022-01-25\|website=Nebraska Medicine\|language=en}} \|50%
Influenza (2009 pandemic strain)	Respiratory droplets	data-sort-value="1.58" \| {{Estimate\|{{Round\|1.58\|1}}\|{{Round\|1.34\|1}}\|{{Round\|2.04\|1}}\|mini=yes}}{{cite journal \| vauthors = Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, Griffin J, Baggaley RF, Jenkins HE, Lyons EJ, Jombart T, Hinsley WR, Grassly NC, Balloux F, Ghani AC, Ferguson NM, Rambaut A, Pybus OG, Lopez-Gatell H, Alpuche-Aranda CM, Chapela IB, Zavala EP, Guevara DM, Checchi F, Garcia E, Hugonnet S, Roth C \| display-authors = 6 \| title = Pandemic potential of a strain of influenza A (H1N1): early findings \| journal = Science \| volume = 324 \| issue = 5934 \| pages = 1557–61 \| date = June 2009 \| pmid = 19433588 \| pmc = 3735127 \| doi = 10.1126/science.1176062 \| bibcode = 2009Sci...324.1557F }}	data-sort-value="37" \| {{Estimate\|{{#expr:100-100/1.58 round 0}}\|{{#expr:100-100/1.34 round 0}}\|{{#expr:100-100/2.04 round 0}}\|unit=%\|mini=yes}}
Influenza (seasonal strains)	Respiratory droplets	data-sort-value="1.13" \| {{Estimate\|1.3\|1.2\|1.4\|mini=yes}}	data-sort-value="23" \| {{Estimate\|{{#expr:100-100/1.3 round 0}}\|{{#expr:100-100/1.2 round 0}}\|{{#expr:100-100/1.4 round 0}}\|unit=%\|mini=yes}}

Estimates for variants of SARS-CoV-2.

class="wikitable sortable" : 1em; font-size:85%;" \|+Values of R₀ and herd immunity thresholds (HITs) for variants of SARS-CoV-2 ! Disease !! Transmission !! R₀ !! HIT{{efn\|name=calc}}
COVID-19 (Omicron variant)	Respiratory droplets and aerosol	9.5	{{#expr:100-100/9.5 round 0}}%
COVID-19 (Delta variant)	Respiratory droplets and aerosol	{{Round\|5.08\|1}}{{Cite journal \|last1=Liu \|first1=Ying \|last2=Rocklöv \|first2=Joacim \|date=2021-10-01 \|title=The reproductive number of the Delta variant of SARS-CoV-2 is far higher compared to the ancestral SARS-CoV-2 virus \|url=https://academic.oup.com/jtm/article/28/7/taab124/6346388 \|journal=Journal of Travel Medicine \|volume=28 \|issue=7 \|doi=10.1093/jtm/taab124 \|pmid=34369565 \|pmc=8436367 \|issn=1708-8305}}	{{#expr:100-100/5.08 round 0}}%
COVID-19 (Alpha variant)	Respiratory droplets and aerosol	data-sort-value="4.5" \| 4–5{{cite news \|last1=Gallagher \|first1=James \|title=Covid: Is there a limit to how much worse variants can get? \|url=https://www.bbc.com/news/health-57431420 \|access-date=21 July 2021 \|work=BBC News \|date=12 June 2021}}{{medcn\|date=December 2021}}	{{#expr:100-100/4 round 0}}–{{#expr:100-100/5 round 0}}%
COVID-19 (ancestral strain) \| Respiratory droplets and aerosol{{cite journal \|last1=Prather \|first1=Kimberly A. \|last2=Marr \|first2=Linsey C. \|last3=Schooley \|first3=Robert T. \|last4=McDiarmid \|first4=Melissa A. \|last5=Wilson \|first5=Mary E. \|last6=Milton \|first6=Donald K. \|title=Airborne transmission of SARS-CoV-2 \|journal=Science \|date=16 October 2020 \|volume=370 \|issue=6514 \|pages=303.2–304 \|doi=10.1126/science.abf0521 \|pmid=33020250 \|bibcode=2020Sci...370..303P \|s2cid=222145689 \|doi-access= }}	data-sort-value="2.87" \| {{Estimate\|{{Round\|2.87\|1}}\|{{Round\|2.39\|1}}\|{{Round\|3.44\|1}}\|mini=yes}}{{cite journal \|last1=Billah \|first1=Arif \|last2=Miah \|first2=Mamun \|last3=Khan \|first3=Nuruzzaman \|title=Reproductive number of coronavirus: A systematic review and meta-analysis based on global level evidence \|journal=PLOS ONE \|date=11 November 2020 \|volume=15 \|issue=11 \|pages=e0242128 \|doi=10.1371/journal.pone.0242128 \|pmid=33175914 \|pmc=7657547 \|bibcode=2020PLoSO..1542128B \|doi-access=free }}	data-sort-value="65" \| {{Estimate\|{{#expr:100-100/2.87 round 0}}\|{{#expr:100-100/2.39 round 0}}\|{{#expr:100-100/3.44 round 0}}\|unit=%\|mini=yes}}

In popular culture

In the 2011 film Contagion, a fictional medical disaster thriller, an epidemiologist explains the concept of $R_0$ .{{citation |title= The Misunderstood Number That Predicts Epidemics | vauthors = Byrne M |date= October 6, 2014 |work=vice.com |url= https://www.vice.com/en_us/article/pgazpv/meet-r-nought-the-magic-number-that-spreads-infectious-diseases |access-date= 2020-03-23}}

class="wikitable sortable" : 1em; font-size:85%;" \|+Values of R₀ and herd immunity thresholds (HITs) for specific influenza strains ! Disease !! Transmission !! R₀ !! HIT{{efn\|name=calc}}
Influenza (1918 pandemic strain) \|Respiratory droplets \|2{{Cite web\|date=2021-12-21\|title=Omicron transmission: how contagious diseases spread\|url=https://www.nebraskamed.com/COVID/how-quickly-do-diseases-spread\|access-date=2022-01-25\|website=Nebraska Medicine\|language=en}} \|50%
Influenza (2009 pandemic strain)	Respiratory droplets	data-sort-value="1.58" \| {{Estimate\|{{Round\|1.58\|1}}\|{{Round\|1.34\|1}}\|{{Round\|2.04\|1}}\|mini=yes}}{{cite journal \| vauthors = Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, Griffin J, Baggaley RF, Jenkins HE, Lyons EJ, Jombart T, Hinsley WR, Grassly NC, Balloux F, Ghani AC, Ferguson NM, Rambaut A, Pybus OG, Lopez-Gatell H, Alpuche-Aranda CM, Chapela IB, Zavala EP, Guevara DM, Checchi F, Garcia E, Hugonnet S, Roth C \| display-authors = 6 \| title = Pandemic potential of a strain of influenza A (H1N1): early findings \| journal = Science \| volume = 324 \| issue = 5934 \| pages = 1557–61 \| date = June 2009 \| pmid = 19433588 \| pmc = 3735127 \| doi = 10.1126/science.1176062 \| bibcode = 2009Sci...324.1557F }}	data-sort-value="37" \| {{Estimate\|{{#expr:100-100/1.58 round 0}}\|{{#expr:100-100/1.34 round 0}}\|{{#expr:100-100/2.04 round 0}}\|unit=%\|mini=yes}}
Influenza (seasonal strains)	Respiratory droplets	data-sort-value="1.13" \| {{Estimate\|1.3\|1.2\|1.4\|mini=yes}}	data-sort-value="23" \| {{Estimate\|{{#expr:100-100/1.3 round 0}}\|{{#expr:100-100/1.2 round 0}}\|{{#expr:100-100/1.4 round 0}}\|unit=%\|mini=yes}}