# Basic reproduction number

In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted ${\displaystyle R_{0}}$ (pronounced R nought or R zero),[1] of an infection is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection.[2] 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".[3] The basic reproduction number is not necessarily the same as the effective reproduction number ${\displaystyle R}$ (usually written ${\displaystyle R_{t}}$ [t for time], sometimes ${\displaystyle R_{e}}$),[4] which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. ${\displaystyle R_{0}}$ is a dimensionless number (persons infected per person infecting) and not a time rate, which would have units of time−1,[5] or units of time like doubling time.[6]

Graph of herd immunity threshold vs basic reproduction number with selected diseases
An explanation of the ${\displaystyle R}$ number in simple terms from the Welsh Government.

${\displaystyle R_{0}}$ 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. ${\displaystyle R_{0}}$ 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.[7] ${\displaystyle R_{0}}$ does not by itself give an estimate of how fast an infection spreads in the population.

The most important uses of ${\displaystyle R_{0}}$ 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 ${\displaystyle R_{0}>1}$ the infection will be able to start spreading in a population, but not if ${\displaystyle R_{0}<1}$. Generally, the larger the value of ${\displaystyle R_{0}}$, 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 ${\displaystyle 1-1/R_{0}}$.[8] Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is ${\displaystyle 1/R_{0}}$.

The basic reproduction number is affected by several factors, including the duration of infectivity of affected people, the infectiousness 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,[9] but its first modern application in epidemiology was by George Macdonald in 1952,[10] who constructed population models of the spread of malaria. In his work he called the quantity basic reproduction rate and denoted it by ${\displaystyle Z_{0}}$ . "Rate" in this context means per person, which makes ${\displaystyle Z_{0}}$  dimensionless as required. Because this can be misleading to anyone who understands "rate" only in the sense per unit of time, "number" or "ratio" is now preferred.[citation needed]

## Definitions in specific cases

### Contact rate and infectious period

${\displaystyle R_{0}}$  is the average number of people infected from one other person. For example, Ebola has an ${\displaystyle R_{0}}$  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 ${\displaystyle \beta }$  infection-producing contacts per unit time, with a mean infectious period of ${\displaystyle \tau }$ . Then the basic reproduction number is:

${\displaystyle R_{0}=\beta \,\tau }$

This simple formula suggests different ways of reducing ${\displaystyle R_{0}}$  and ultimately infection propagation. It is possible to decrease the number of infection-producing contacts per unit time ${\displaystyle \beta }$  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). Hence, it can also be written as[11]
${\displaystyle R_{0}={\overline {c}}\,T\,\tau ,}$

where ${\displaystyle {\overline {c}}}$  is the rate of contact between susceptible and infected individuals and ${\displaystyle T}$  is the transmissibility, i.e, the probability of infection given a contact. It is also possible to decrease the infectious period ${\displaystyle \tau }$  by finding and then isolating, treating or eliminating (as is often the case with animals) infectious individuals as soon as possible.[citation needed]

### With varying latent periods

Latent period is the transition time between contagion event and disease manifestation. In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction numbers for each transition time into the disease. An example of this is tuberculosis (TB). Blower and coauthors calculated from a simple model of TB the following reproduction number:[12]

${\displaystyle R_{0}=R_{0}^{\text{FAST}}+R_{0}^{\text{SLOW}}}$

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.[13]

### Heterogeneous populations

In populations that are not homogeneous, the definition of ${\displaystyle R_{0}}$  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 ${\displaystyle R_{0}}$  must account for this difference. An appropriate definition for ${\displaystyle R_{0}}$  in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual".[14]

The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts ${\displaystyle \beta }$  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 ${\displaystyle {\dfrac {1}{\gamma }}}$ , then the basic reproduction number is just ${\displaystyle R_{0}={\dfrac {\beta }{\gamma }}}$ . 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. For example, Blower et al.[12] model two forms of tuberculosis infection: in the fast case, the symptoms show up immediately after exposure; in the slow case, the symptoms develop years after the initial exposure (endogenous reactivation). The overall reproduction number is the sum of the two forms of contraction: ${\displaystyle R_{0}=R_{0}^{FAST}+R_{0}^{SLOW}}$ .

## Estimation methods

The basic reproduction number can be estimated through examining detailed transmission chains or through genomic sequencing. However, it is most frequently calculated using epidemiological models.[15] During an epidemic, typically the number of diagnosed infections ${\displaystyle N(t)}$  over time ${\displaystyle t}$  is known. In the early stages of an epidemic, growth is exponential, with a logarithmic growth rate

${\displaystyle K:={\frac {d\ln(N)}{dt}}.}$

For exponential growth, ${\displaystyle N}$  can be interpreted as the cumulative number of diagnoses (including individuals who have recovered) or the present number of infection cases; the logarithmic growth rate is the same for either definition. In order to estimate ${\displaystyle R_{0}}$ , assumptions are necessary about the time delay between infection and diagnosis and the time between infection and starting to be infectious.

In exponential growth, ${\displaystyle K}$  is related to the doubling time ${\displaystyle T_{d}}$  as

${\displaystyle K={\frac {\ln(2)}{T_{d}}}.}$

### Simple model

If an individual, after getting infected, infects exactly ${\displaystyle R_{0}}$  new individuals only after exactly a time ${\displaystyle \tau }$  (the serial interval) has passed, then the number of infectious individuals over time grows as

${\displaystyle n_{E}(t)=n_{E}(0)\,R_{0}^{t/\tau }=n_{E}(0)\,e^{Kt}}$

or
${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+\ln(R_{0})t/\tau .}$

The underlying matching differential equation is
${\displaystyle {\frac {dn_{E}(t)}{dt}}=n_{E}(t){\frac {\ln(R_{0})}{\tau }}.}$

or
${\displaystyle {\frac {d\ln(n_{E}(t))}{dt}}={\frac {\ln(R_{0})}{\tau }}.}$

In this case, ${\displaystyle R_{0}=e^{K\tau }}$  or ${\displaystyle K={\frac {\ln R_{0}}{\tau }}}$ .

For example, with ${\displaystyle \tau =5~\mathrm {d} }$  and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$ , we would find ${\displaystyle R_{0}=2.5}$ .

If ${\displaystyle R_{0}}$  is time dependent

${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+{\frac {1}{\tau }}\int \limits _{0}^{t}\ln(R_{0}(t))dt}$

showing that it may be important to keep ${\displaystyle \ln(R_{0})}$  below 0, time-averaged, to avoid exponential growth.

### Latent infectious period, isolation after diagnosis

In this model, an individual infection has the following stages:

1. Exposed: an individual is infected, but has no symptoms and does not yet infect others. The average duration of the exposed state is ${\displaystyle \tau _{E}}$ .
2. Latent infectious: an individual is infected, has no symptoms, but does infect others. The average duration of the latent infectious state is ${\displaystyle \tau _{I}}$ . The individual infects ${\displaystyle R_{0}}$  other individuals during this period.
3. Isolation after diagnosis: measures are taken to prevent further infections, for example by isolating the infected person.

This is a SEIR model and ${\displaystyle R_{0}}$  may be written in the following form[16]

${\displaystyle R_{0}=1+K(\tau _{E}+\tau _{I})+K^{2}\tau _{E}\tau _{I}.}$

This estimation method has been applied to COVID-19 and SARS. It follows from the differential equation for the number of exposed individuals ${\displaystyle n_{E}}$  and the number of latent infectious individuals ${\displaystyle n_{I}}$ ,
${\displaystyle {\frac {d}{dt}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}={\begin{pmatrix}-1/\tau _{E}&R_{0}/\tau _{I}\\1/\tau _{E}&-1/\tau _{I}\end{pmatrix}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}.}$

The largest eigenvalue of the matrix is the logarithmic growth rate ${\displaystyle K}$ , which can be solved for ${\displaystyle R_{0}}$ .

In the special case ${\displaystyle \tau _{I}=0}$ , this model results in ${\displaystyle R_{0}=1+K\tau _{E}}$ , which is different from the simple model above (${\displaystyle R_{0}=\exp(K\tau _{E})}$ ). For example, with the same values ${\displaystyle \tau =5~\mathrm {d} }$  and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$ , we would find ${\displaystyle R_{0}=1.9}$ , rather than the true value of ${\displaystyle 2.5}$ . The difference is due to a subtle difference in the underlying growth model; the matrix equation above assumes that newly infected patients are currently already contributing to infections, while in fact infections only occur due to the number infected at ${\displaystyle \tau _{E}}$  ago. A more correct treatment would require the use of delay differential equations.[17]

## Effective reproduction number

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 ${\displaystyle R_{e}}$  is used, usually written as ${\displaystyle R_{t}}$ , or 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 ${\displaystyle R_{0}}$  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 ${\displaystyle R_{e}}$  drops below 1, "herd immunity" has been achieved and the number of cases occurring in the population will gradually decrease to zero.[18][19][20]

## Limitations of R0

Use of ${\displaystyle R_{0}}$  in the popular press has led to misunderstandings and distortions of its meaning. ${\displaystyle R_{0}}$  can be calculated from many different mathematical models. Each of these can give a different estimate of ${\displaystyle R_{0}}$ , which needs to be interpreted in the context of that model. Therefore, the contagiousness of different infectious agents cannot be compared without recalculating ${\displaystyle R_{0}}$  with invariant assumptions. ${\displaystyle R_{0}}$  values for past outbreaks might not be valid for current outbreaks of the same disease. Generally speaking, ${\displaystyle R_{0}}$  can be used as a threshold, even if calculated with different methods: if ${\displaystyle R_{0}<1}$ , the outbreak will die out, and if ${\displaystyle R_{0}>1}$ , the outbreak will expand. In some cases, for some models, values of ${\displaystyle R_{0}<1}$  can still lead to self-perpetuating outbreaks. This is particularly problematic if there are intermediate vectors between hosts, such as malaria.[21] Therefore, comparisons between values from the "Values of ${\displaystyle R_{0}}$  of well-known infectious diseases" table should be conducted with caution.

Although ${\displaystyle R_{0}}$  cannot be modified through vaccination or other changes in population susceptibility, it can vary based on a number of biological, sociobehavioral, and environmental factors.[7] It can also be modified by physical distancing and other public policy or social interventions,[22][7] although some historical definitions exclude any deliberate intervention in reducing disease transmission, including nonpharmacological interventions.[3] And indeed, whether nonpharmacological interventions are included in ${\displaystyle R_{0}}$  often depends on the paper, disease, and what if any intervention is being studied.[7] This creates some confusion, because ${\displaystyle R_{0}}$  is not a constant; whereas most mathematical parameters with "nought" subscripts are constants.

${\displaystyle R}$  depends on many factors, many of which need to be estimated. Each of these factors adds to uncertainty in estimates of ${\displaystyle R}$ . Many of these factors are not important for informing public policy. Therefore, public policy may be better served by metrics similar to ${\displaystyle R}$ , but which are more straightforward to estimate, such as doubling time or half-life (${\displaystyle t_{1/2}}$ ).[23][24]

Methods used to calculate ${\displaystyle R_{0}}$  include the survival function, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method,[25] calculations from the intrinsic growth rate,[26] existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection[27] and the final size equation. Few of these methods agree with one another, even when starting with the same system of differential equations.[21] Even fewer actually calculate the average number of secondary infections. Since ${\displaystyle R_{0}}$  is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.[28]

## Sample values for various infectious diseases

Values of R0 and herd immunity thresholds (HITs) of well-known infectious diseases prior to intervention
Disease Transmission R0 HIT[a]
Measles Aerosol 12–18[29][30] 92–94%
Chickenpox (varicella) Aerosol 10–12[31] 90–92%
Mumps Respiratory droplets 10–12[32] 90–92%
Rubella Respiratory droplets 6–7[b] 83–86%
COVID-19 (Delta variant) Respiratory droplets and aerosol 5–8[37] 80–88%
Polio Fecal–oral route 5–7[b] 80–86%
Pertussis Respiratory droplets 5.5[38] 82%
Smallpox Respiratory droplets 3.5–6.0[39] 71–83%
COVID-19 (Alpha variant) Respiratory droplets and aerosol 4–5[37] 75–80%
HIV/AIDS Body fluids 2–5[40] 50–80%
COVID-19 (ancestral strain) Respiratory droplets and aerosol[41] 2.9 ()[42] 65% ()
SARS Respiratory droplets 2–4[43] 50–75%
Diphtheria Saliva 2.6 ()[44] 62% ()
Common cold Respiratory droplets 2–3[45] 50–67%
Ebola (2014 outbreak) Body fluids 1.8 ()[46] 44% ()
Influenza (2009 pandemic strain) Respiratory droplets 1.6 ()[2] 37% ()
Influenza (seasonal strains) Respiratory droplets 1.3 ()[47] 23% ()
Andes hantavirus Respiratory droplets and body fluids 1.2 ()[48] 16% ()[c]
Nipah virus Body fluids 0.5[49] 0%[c]
MERS Respiratory droplets 0.5 ()[50] 0%[c]

## In popular culture

In the 2011 film Contagion, a fictional medical disaster thriller, a blogger's calculations for ${\displaystyle R_{0}}$  are presented to reflect the progression of a fatal viral infection from case studies to a pandemic. The methods depicted were faulty.[22]

## Notes

1. ^ Calculated using p = 1 - 1 / R0.
2. ^ a b From a module of a training course[33] with data modified from other sources.[34][35][36]
3. ^ a b c When R0 < 1.0, the disease naturally disappears.
• 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, ${\displaystyle R(0)}$  and ${\displaystyle R_{0}}$  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.
• Held L, Hens N, O'Neill PD, Wallinga J (November 7, 2019). Handbook of Infectious Disease Data Analysis. CRC Press. p. 347. ISBN 978-1-351-83932-7. According to Guangdong Provincial Center for Disease Control and Prevention, "The effective reproductive number (R or Re is more commonly used to describe transmissibility, which is defined as the average number of secondary cases generated by per [sic] infectious case." For example, by one preliminary estimate during the ongoing pandemic, the effective reproductive number for SARS-CoV-2 was found to be 2.9,[citation needed] whereas for SARS it was 1.77.

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