Basic reproduction number

Values of R₀ of well-known infectious diseases

Disease	Transmission	R0
Measles	Airborne	12–18
Diphtheria	Saliva	1.7-4.3
Smallpox	Airborne droplet	3.5-6
Polio	Fecal–oral route	5–7
Rubella	Airborne droplet	5–7
Mumps	Airborne droplet	4–7
Pertussis	Airborne droplet	5.5
HIV/AIDS	Body fluids	2–5
SARS	Airborne droplet	2–5
COVID-19	Airborne droplet	1.4–3.9
Influenza (1918 pandemic strain)	Airborne droplet	2–3
Ebola (2014 Ebola outbreak)	Body fluids	1.5–2.5
MERS	Airborne droplet	0.3–0.8

A video discussing the basic reproduction number (at about 4 min) and case fatality rate in the context of the 2019–20 coronavirus pandemic.

In epidemiology, the basic reproduction number (sometimes called basic reproductive ratio, or incorrectly basic reproductive rate, and denoted R₀, pronounced R nought or R zero) of an infection can be thought of as the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. The definition describes the state where no other individuals are infected or immunized (naturally or through vaccination). Some definitions, such as that of the Australian Department of Health, add absence of "any deliberate intervention in disease transmission". The basic reproduction number is not to be confused with the effective reproduction number R which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. By definition R₀ cannot be modified through vaccination campaigns.

R₀ 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. Furthermore R₀ 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. R₀ does not by itself give an estimate of how fast an infection spreads in the population.

The most important uses of R₀ 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 R₀ > 1 the infection will be able to start spreading in a population, but not if R₀ < 1. Generally, the larger the value of R₀, 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/R₀. Conversely, the proportion of the population that remains susceptible to infection in the endemic equilibrium is 1/R₀.

The basic reproduction number 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.

History

The roots of the basic reproduction concept can be traced through the work of Ronald Ross, Alfred Lotka and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria. In his work he called the quantity basic reproduction rate and denoted it by Z₀. Calling the quantity a "rate" can be misleading, insofar as it can be interpreted as number per unit of time. The expressions number or ratio are now preferred.

Definitions in specific cases

Reproductive number as it relates to contact rate and infectious period

R₀ is the average number of people infected from one other person, for example, Ebola has an R₀ of two, so on average, a person who has Ebola will pass it on to two other people.

Say that an infectious individual makes β contacts per unit time producing new infections with a mean infectious period of γ. Therefore, the basic reproduction number is

$R_{0}=\beta \gamma$

This simple formula suggests different ways of reducing R₀ and ultimately infection propagation. It is possible to decrease the number of infection producing contacts per unit time β 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). It is also possible to decrease the infectious period γ 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

In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction number for each transition time into the disease. An example of this is tuberculosis. Blower et al. calculated from a simple model of TB the following reproduction number:

$R_{0}=R_{0}^{\mathrm {FAST} }+R_{0}^{\mathrm {SLOW} }$

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.

Inhomogeneous populations

In populations that are not homogeneous, the definition of R₀ 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 who become infected early in an epidemic may be more (or less) likely to transmit than a randomly chosen individual late in the epidemic, then our computation of R₀ must account for this tendency. An appropriate definition for R₀ in this case is "the expected number of secondary cases produced by a typical infected individual early in an epidemic".

Estimation methods

See also: Compartmental models in epidemiology

During an epidemic, typically the number of diagnosed infections $N(t)$ over time $t$ is known. In the early stages of an epidemic, growth is exponential, with a logarithmic growth rate

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

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

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 duration of the exposed state is $\tau _{E}$.
2. Latent infectious: an individual is infected, has no symptoms, but does infect others. The duration of the latent infectious state is $\tau _{I}$. The individual infects $R_{0}$ other individuals during this period.
3. isolation after diagnosis: measures are taken to prevent further infections, for example by isolating the patient.

This is a SEIR model and R₀ may be written in the following form

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

This estimation method has been applied to COVID-19 and SARS. It follows from the differential equation for the number of exposed individuals $n_{E}$ and the number of latent infectious individuals $n_{I}$,

${\frac {d}{dt}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}={\begin{pmatrix}-1/\tau _{E}&R_{0}/\tau _{I}\\1/\tau _{E}&-1/\tau _{I}\end{pmatrix}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}.$

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

Other uses

R₀ is also used as a measure of individual reproductive success in population ecology, 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, R₀ 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 R₀ = 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 R₀

When calculated from mathematical models, particularly ordinary differential equations, what is often claimed to be R₀ 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 R₀. 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 R₀ < 1) or whether it may become epidemic (if R₀ > 1), but they generally cannot 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, calculations from the intrinsic growth rate, existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection 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 R₀ is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.

In popular culture

In the 2011 film Contagion, a fictional medical disaster thriller, R₀ calculations are presented to reflect the progression of a fatal viral infection from case studies to a pandemic.

In the TV series Travelers, R₀ calculations are used to inform a group on the progression of a viral infection from various sources during a pandemic.

See also

• E-epidemiology
• Epi Info software program
• Epidemic model
• Epidemiological method
• Epidemiological transition

Notes

• 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, R(0) and R₀ 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.
• According to Guangdong Provincial Center for Disease Control and Prevention, "The effective reproductive number (R) is more commonly used to describe transmissibility, which is defined as the average number of secondary cases generated by per [sic] infectious case. In the absence of control measures, R = R₀χ, where χ is the proportion of the susceptible population." For example, the effective reproductive number for 2019-nCoV was found as 2.9, whereas for SARS it was 1.77.

Further reading

• Heesterbeek, J.A.P. (2002). "A brief history of R₀ and a recipe for its calculation". Acta Biotheoretica. 50 (3): 189–204. doi:10.1023/A:1016599411804. PMID 12211331.
• Heffernan, J.M; Smith, R.J; Wahl, L.M (2005). "Perspectives on the basic reproductive ratio". Journal of the Royal Society Interface. 2 (4): 281–293. doi:10.1098/rsif.2005.0042. PMC 1578275. PMID 16849186.
• Van Den Driessche, P.; Watmough, James (2008). "Further Notes on the Basic Reproduction Number". Mathematical Epidemiology. Lecture Notes in Mathematics. Vol. 1945. pp. 159–178. doi:10.1007/978-3-540-78911-6_6. ISBN 978-3-540-78910-9.
• Jones, James Holland. "Notes on R₀" (PDF). Retrieved November 6, 2018.