Basic reproduction number

Graph of herd immunity threshold vs basic reproduction number with selected diseases

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]

${\displaystyle R_{0}}$ 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.

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.

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]

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

An explanation of the ${\displaystyle R}$ number in simple terms from the Welsh Government.

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 ${\displaystyle R_{e}}$ or ${\displaystyle R}$ is used. ${\displaystyle R_{t}}$ 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 ${\displaystyle R_{0}}$ 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 in a basic SIR simulation, "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

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]

See also

Notes

References

Further reading