Base reproduction number

The basic reproduction number (pronounced “R-zero”), also incorrectly called the basic reproduction rate, and the net reproduction number are terms from the epidemiology of infections . ${\displaystyle R_{0}}$ ${\displaystyle R}$

The base reproduction number indicates how many people on average are infected by an infectious person if no member of the population is immune to the pathogen ( susceptible population).

${\displaystyle R_{0}}$is not a biological constant for a pathogen, as it is also significantly influenced by other factors such as environmental conditions and the behavior of the infected population. In addition, values are typically estimated using mathematical models, and the estimated values ​​then depend on the model used and the values ​​of other parameters. It makes a difference whether the values ​​are collected for the entire population of a country and thus in some cases very rough average figures are determined or only an outbreak is considered on a smaller scale, whether warnings have been issued and are being followed by the population, distance or quarantine rules in Force were set. In addition to the mean values, other parameters of the statistical distribution of the reproduction number also play a role, particularly in the case of superspreading . ${\displaystyle R_{0}}$

Schematic representation of the number of infected people in an epidemic for the reproduction numbers R = 2, R = 1 and R = 0.5 with an assumed generation time of 4 days and an initial number of 1000 infected

Using the base reproduction number

With the help of the base reproduction number one can estimate how the spread of a communicable disease will proceed at the beginning of an epidemic and what proportion of the population must be immune or immunized by vaccination in order to prevent an epidemic. In commonly used infection models, the infection can spread in a population if is but not if is. In general, the greater the value of , the more difficult it is to keep the epidemic under control. The base reproduction number relates to a population in which all people are susceptible to the infection, i.e. in particular no people are resistant. It is determined by the contagiousness , the population density and the degree of mixing of the population. Mixing is a measure of how homogeneous the interactions are within the population; she is z. B. smaller when people form groups and preferentially interact with people in their own group. The base reproduction number can therefore be very different for the same pathogen in different populations. From the base reproduction number it can be calculated how high the immunized proportion of the population must be in order to achieve sufficient herd immunity for the disease to die out in the given population in the long term (see also: the mathematics of vaccinations ). In simple models, herd immunity is achieved when the population is immunized. ${\displaystyle R_{0}}$ ${\displaystyle R_{0}>1}$ ${\displaystyle R_{0}<1}$ ${\displaystyle R_{0}}$ ${\displaystyle R_{0}}$ ${\displaystyle 1-{\frac {1}{R_{0}}}}$

Calculation of the basic reproduction number

The base reproduction number can be broken down further for a simple infection model:

${\displaystyle R_{0}=\kappa \cdot q\cdot D}$

with , the number of contacts of an infected person per unit of time,, the mean duration of the infectiousness and , the probability of infection upon contact. ${\textstyle \kappa }$ ${\textstyle D}$ ${\textstyle q}$

For more mathematical backgrounds and models see:

• Mathematical modeling of epidemiology
  • SI model (contagion without recovery)
  • SIS model (spread of infectious diseases without building up immunity)
  • SIR model (spread of infectious diseases with immunity formation)
  • SEIR model (spread of contagious diseases with immunity development, in which infected people are not immediately infectious)

Estimates for can be obtained from the models . For example, if you look at the SIR model with initially exponential growth of the infected (growth exponent with the doubling time ), you have the equation: ${\displaystyle R_{0}}$ ${\displaystyle r={\frac {\ln 2}{t_{\rm {d}}}}}$ ${\displaystyle t_{\rm {d}}}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta {\frac {SI}{N}}-\gamma I=r\cdot I}$

where for the beginning of the epidemic can be set, and thus the estimator: ${\displaystyle S\approx N}$

${\displaystyle R_{0}={\frac {\beta }{\gamma }}=1+{\frac {r}{\gamma }}=1+{\frac {\ln 2}{\gamma \cdot t_{\rm {d}}}}}$

It is the average time in which an infected is contagious. ${\displaystyle {\frac {1}{\gamma }}}$

Another estimator for is based on the generation time : ${\displaystyle R_{0}}$ ${\displaystyle T_{\rm {G}}}$

${\displaystyle R_{0}^{\frac {t}{T_{\rm {G}}}}=e^{r\cdot t}}$

what on the appraiser

${\displaystyle R_{0}=e^{r\cdot T_{\rm {G}}}}$

leads, for little ones this can be approached through . ${\displaystyle r\cdot T_{\rm {G}}}$ ${\displaystyle R_{0}\approx 1+r\cdot T_{\rm {G}}}$

Estimates in which the generation time is not a constant but obeys a distribution function are based on the Euler-Lotka equation , which provides a relationship between the base reproduction number and the growth rate. ${\displaystyle R_{0}}$

Net reproduction number

Other terms for the net reproduction number are the net reproduction number at a certain time and the effective reproduction number , which is based on the English term effective reproduction number . The net reproduction number is derived from the base reproduction number and indicates the average number of people infected by an infected person if a certain part of the population is immune or certain measures have been taken as part of a prescribed mass quarantine that are intended to serve as containment. If no control measures are taken, is , where the number of "susceptible" (susceptible to infection) persons is and the total number of persons in a population; is the probability of encountering an infectable person during contact. With control measures - such as hygiene and distancing measures to reduce the transmission rate per contact, a reduction in the number and duration of contacts and / or the limitation of interactions to smaller groups - the effective number of reproductions continues to decrease. ${\displaystyle R}$ ${\displaystyle R_{t}}$ ${\displaystyle R_{\rm {eff}}}$ ${\textstyle R_{\rm {eff}}=R_{0}\cdot {\frac {S}{N}}}$ ${\textstyle S}$ ${\textstyle N}$ ${\textstyle {\frac {S}{N}}}$

Since portions of the population are often immune to a disease while effective countermeasures are taken while it is spreading, or if immunity to the disease is subsequently developed, the net reproductive number becomes increasingly important as it spreads. The aim of containment measures is usually to keep the net reproduction figure below 1. Because only when the net reproduction number is less than 1 does the number of infected people fall and the disease disappears completely at some point.

Net reproduction number using the example of the COVID-19 pandemic in Germany

For the estimation of different estimators are used reproductive numbers. The procedure of the Robert Koch Institute (RKI) in the COVID-19 pandemic in Germany in March and April 2020 is an example . These are averaged figures for the whole of Germany, with large regional differences. The starting point is the daily cases of new illnesses reported to the RKI due to the reporting obligation. From this, a correction is made ( nowcasting ), taking into account the delay in diagnosis, reporting and transmission , which estimates the number of cases according to the days on which the illness began. The generation time was estimated by the RKI to be 4 days (if a distribution is used for the generation time, the formulas are a bit more complicated, see below). In a generation time, the number of new infections changes by the factor R (reproduction factor); R is determined as the quotient of the new infections in two successive time periods of 4 days each. Since the values ​​of the last three days are not yet final (late registrations, corrections, etc.), the RKI did not use these last three days for the R calculation according to a notification in May 2020. A point in time is therefore assigned an R that was determined from the course of the eight days that are four to eleven days ago (the days 1 to 3 before the respective day are therefore not taken into account, the quotient is calculated from the sum of the numbers of days 4 to 7 before the current day by adding the numbers of days 8 to 11). The current R-value provides information about the illnesses (onset of illness) that were on average seven days ago. The associated infection process was also an incubation time ago (with COVID-19 this is an average of 5 days). The R-value published by the RKI was slightly above 2 in Germany at the beginning of March, had its maximum of around 3.5 around March 10, and then fell. Around March 20, R reached a value below 1 and then stayed at around 0.9 (with a brief increase above 1.0). A minimum of 0.7 was reached on April 16; but the value rose again to 1.0 (April 27) to 0.9 and fell on 29/30. April to 0.75; The usual fluctuations in statistical values ​​must be taken into account in the assessment.

See also : COVID-19-Pandemie_in_Deutschland # Reproduction number

After a person is infected, their infectivity takes a course over a number of days (typically: after a latency period of 2 days, it rises steeply, then falls gently), so the generation time is a distribution. The somewhat more complicated formula with a distribution of the generation time can be found below. a. in (1st formula, page 2) or in formula (1). The quality of the R-estimate depends heavily on the quality of the estimate used for this distribution. See section “discussion”: “Since Inference of the value of the reproductive number depends crucially on the generation interval distribution, it is surprising. . . ".

Instead of the distribution of generation time, the RKI uses a single scalar constant to simplify matters; In terms of distribution, this corresponds to a single impulse on day 4 after the infection: If the simplified distribution is inserted into the above formula in p. 2 or in formula (1) and the new infections are averaged over 4 days for smoothing one arrives at the estimation formula used by the RKI: quotient of the new infections in two consecutive periods of 4 days each (two consecutive periods of 4 days each have a time interval of 4 days). If one averages over 7 days (while maintaining the single impulse on day 4 as a distribution), one arrives at the so-called 7-day R value. For the inaccuracy resulting from the simplification see

Sample values ​​for various infectious diseases

Example values ​​for the base reproduction number are 6 for smallpox and poliomyelitis , 15 for measles , 7 for diphtheria , and 14 for whooping cough . In the 1918 flu pandemic , the base reproduction number was estimated at 2 to 3. The basic reproduction number of COVID-19 is estimated by the Robert Koch Institute to be 2.4 to 3.3 (before the countermeasures come into effect) . The WHO- China Joint Mission Report gave the base reproduction number for China - i.e. when no measures such as curfew had yet been taken - as 2 to 2.5. In April 2020, the CDC gave it a significantly higher rating , namely 5.7 (95% - CI 3.8–8.9). ( see also basic reproduction number of COVID-19 )

The following table provides an overview of the basic reproduction numbers for some infectious diseases and pandemics . Some of the values ​​vary considerably. As with their individual vaccination history or their measures against the spread of the disease, such as curfews or social distancing , on the other uncertainties in the historical review.

Values ​​of R _{0 for} some infectious diseases

illness	Route of infection	R 0
measles	droplet	12-18
chickenpox	droplet	10-12
polio	fecal-oral	5-7
rubella	droplet	5-7
mumps	droplet	4-7
whooping cough	droplet	5.5, 14
smallpox	droplet	3.5-6
COVID-19	droplet	1.4-5.7
AIDS	Body fluids	2-5
SARS	droplet	2-5
cold	droplet	2-3
diphtheria	saliva	1.7-4.3
Spanish flu (1918)	droplet	1.4-2.0, 2-3
Ebola ( 2014-2016 )	Body fluids	1.5-2.5
Swine flu ( H1N1 )	droplet	1.4-1.6
Influenza	droplet	0.9-2.1
MERS	droplet	0.3-0.8

Over-dispersion parameters and superspreading

${\displaystyle R_{0}}$often spreads strongly among individual individuals and during individual events. Diseases such as measles, Sars, Mers and Covid-19 are particularly noticeable because superspreading plays an important role here . According to a publication by J. Lloyd-Smith and colleagues from 2005, a group of 20 percent of those infected are responsible for over 80 percent of infections in sars and measles. Most infected people do not transmit the disease afterwards. Adam Kucharski even assumes that 10 percent of those infected contribute to 80 percent of other infections with Covid-19. The individual differences are likely to be due to the immune system or the number of receptors of the individual infected for ingestion of the virus and the characteristics of the superspreading events, with close contact, loud talking or singing, and indoor spaces having a negative effect. This is measured by an over- dispersion parameter of the distribution of , which measures irregularities in the distribution (over- dispersion ). This is 0.16 for Sars (Lloyd-Smith et al.), 0.25 for Mers, and around 1 for the influenza pandemic of 1918. Various estimates are given for Covid-19, with an average of 0.54, thus slightly higher than with Sars and Mers ( Gabriel Leung also spoke out in favor of this ), up to 0.1 (Kucharski and colleagues, assuming a negative binomial distribution ). ${\displaystyle k}$ ${\displaystyle R_{0}}$

See also

• Contagion index
• Section Reproduction Count in the Epidemiology Article
• Mathematical modeling of epidemiology

literature

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Individual evidence

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12. ^ For example, Odo Diekmann, Hans Heesterbeek, Tom Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton UP 2013
13. ↑ Estimation of the current development of the SARS-CoV-2 epidemic in Germany - Nowcasting , Epidemiologisches Bulletin 17/2020, Robert Koch Institute. April 23, 2020, p. 14, evaluation from R until April 9. The graphic above on page 14, "Estimation of the effective reproductive number R ...", clearly shows that R does not fall below 1 until March 21st, while the number of new patients already reaches its maximum on March 18th; This is caused by the fact that the R for March 21st (if one carries out the explanation in the text) was determined from the sum for the days March 18 to 21 divided by the sum of the days March 14 to 17.
14. ↑ As the Vice-President of the RKI Lars Schaade explained in a press briefing on May 12, 2020, the number of reproductions announced daily was an additional three days longer ago (i.e. with a total of around one and a half weeks of incubation), as the new infections occurred in the last three days too great uncertainties would not be included in the calculation of the reproduction number. See also the message from RKI President Wieler to the press on April 28, 2020: ntv: RKI boss explains key number. Which period describes the R-value? , about 1:26 min. In the future, a smoothed number of reproductions will also be specified in order to compensate for daily fluctuations, for example due to local outbreaks, which would have greater effects with an absolutely smaller number of new infections.
15. ↑ Daily situation reports, RKI
16. ↑ ^{a b} Explanation of the estimate of the number of reproductions RKI May 15, 2020
17. ↑ ^{a b} Thomas Hotz et al Monitoring. . . by estimating reproduction numbers over time TU Ilmenau 04/18/2020
18. ↑ Wallinga1, Lipsitch how generation Intervals shape. . . Proc. R. Soc. B 2007
19. ↑ R-calculation: generation time as distribution vs. constant
20. ↑ ^{a b} Klaus Krickeberg, Pham Thy My Hanh, Pham Van Trong: Epidemiology. Springer, 2012, p. 45.
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22. ↑ ^{a b} SARS-CoV-2 Profile on Coronavirus Disease-2019 (COVID-19) , Robert Koch Institute, March 13, 2020
23. ↑ World Health Organization (ed.): Report of the WHO-China Joint Mission on Coronavirus Disease 2019 (COVID-19) . February 2020, p. 10 (English, who.int [PDF]).
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26. ↑ Fiona M. Guerra, Shelly Bolotin, Gillian Lim, Jane Heffernan, Shelley L. Deeks, Ye Li, Natasha S. Crowcroft: The basic reproduction number (R0) of measles: a systematic review . In: The Lancet Infectious Diseases . 17, No. 12, December 1, 2017, ISSN  1473-3099 , pp. E420 – e428. doi : 10.1016 / S1473-3099 (17) 30307-9 . Retrieved March 18, 2020.
27. ↑ Ireland's Health Services: Health Care Worker Information (accessed March 27, 2020).
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Remarks

1. ↑ R is a dimensionless number and therefore formally not a rate.
2. ↑ In the source used, Marc Lipsitch et al. The equivalent information is found with the number of contacts of each infected person per unit of time, the probability of transmission per contact between an infected person and a “susceptible” person and the mean duration of the infectiousness. ${\textstyle R_{0}=kbD}$ ${\textstyle k}$ ${\textstyle b}$ ${\textstyle D}$
3. ↑ In the source used, Marc Lipsitch et al. there is the information , whereby the effective reproduction number and the proportion of the "susceptible" in the total population is. Because of the described identity applies . ${\textstyle R=R_{0}x}$ ${\textstyle R}$ ${\textstyle x}$ ${\textstyle x={\frac {S}{N}}}$ ${\textstyle R_{\rm {eff}}=R_{0}\cdot {\frac {S}{N}}}$