Talk:Basic reproduction number/Archive 1

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Disambiguation

We need to clarify the triad of epidemiological R's:

1. R_0 = τ/γ, a quantity describing the dynamics of disease progression under the SIR model, specifically the ratio of the frequency of infection events (τ) to the frequency of recovery events (γ). This is not the transmissibilty of the disease - it is a model parameter.
2. R = R_0*χ, the transmissibility of the disease. χ is the susceptible proportion of the population.
3. R(0) is the number of recovered at time t=0 in the SIR model.

Thoughts on new sections on the page? — Preceding unsigned comment added by Ekalosak (talk • contribs) 06:44, 31 January 2020 (UTC)

Smiley Face

Patient zero in this image is happy. I understand that patient zero for each disease is in fact sick and so would probably be unhappy like all the other people who caught it. I recommend making the circles blue with a sad face. Robert Brockway (talk) 00:41, 15 February 2020 (UTC)

Initial comments

A comment on: "In particular, the proportion of the population that needs to be vaccinated to provide herd immunity and prevent sustained spread of the infection is given by 1 − 1/R0." Herd-immunity is not something that appears at a certain level of vaccine coverage. Infectious disease dynamics are non-linear, to varying degrees depending on R0, at all levels of vaccination. "to provide herd immunity and" should be omitted as it misleads the reader into thinking that vaccination only benefits vaccinated individuals until a certain threshold is crossed, and past that we start observing aggregate-level protection. 72.65.139.162 (talk) 15:57, 17 May 2010 (UTC)

some patent nonsense corrected: R0 is defined in the absence of internventions; R0 for SARS can't have been less than 1, otherwise epidemics would never have started (I think the *effective* or *net* repro number, Rt, is meant - would be worth mentioning that, but I haven't) Also R0>1 doesn't guarantee a major epidemic , it just makes it possible. We also need the word "mean".

This bit is also nonsense "If :R₀ = 1, then the infection will become endemic in the population." In reality R will never be 1, so it is irrelevant. In a stochastic model if R is one extinction will occur within a short time. Deterministic models are fine as approximations to stochstic models for large populations, but a population of 1 case is not large, and the deterministic result is irrelevant.

>>Just learning about this at university now. The way we're taught Ro<1 disease dies off/fails to spread, Ro=1 disease stays constant/endemic, Ro>1 disease spreads/epidemic. I understand how Ro will never realistically be 1, but mathematically it's a very valid concept. Shouldn't the wikipedia entry properly convey the mathematical concept that validly explains Ro=1 as well as (if not more so than) anecdotal data which suggests Ro can't be 1? --98.192.35.52 (talk) 00:00, 16 March 2011 (UTC)

Also, remember this stuff applies equally to animals and plant, so I cut out words like "people".

Generally true that higher R0 makes control harder (though ohter factors come into play) and stuff about herd immunity threshold added.

Stuff on course of epidemics is irrelevant (why stop at SIR - what about SIS, SIRS etc. R0 applies to all).

Should we list sources for R0 estimates? If so, It's Mills, Nature 2004 for R0 for flu, Wallinga, Am J Epidemiol 2004 for SARS (though not strictly R0..more Rt). Anderson & May quote numbers for AIDS and mealses (though not sure if they are the same as those quoted. Generally people quote different values for subsaharan AFrica and elsewhere, adn of course wide variation possible in different groups - so best to say what population R0 is quoted for. Worth adding, maybe).

What about history? Shouldn't we mention history of R0? Kermack Mckendrick, Ross et al? And what about the Heesterbeck formulation for heterogeneous populations (the reason we need the word "typical"). Heesterbeck (who's PhD was titled R0) gives a great lecture including history of R0 (and the other names it's gone by, before Anderson & May popularized R0), and should probably mention his work here. All worth doing if anyone's got the time...

Other points: Why there is a separate definition for basic reproduction number and basic reproductive rate (http://en.wikipedia.org/wiki/Basic_reproductive_rate)? As far as I know they are just different names for the same concept. Now, if there is a difference between terms that I am not aware of, the reader can not tell by reading both articles. I agree that it would also be nice having a definition for Rt. That's my 2 cents anyway!

"By definition R0 cannot be modified through vaccination campaigns." Can this be explained more clearly in terms of what it means for a pandemic virus? — Preceding unsigned comment added by Carusus (talk • contribs) 18:09, 12 March 2020 (UTC)

Inconsistent math

Section "Reproductive number as it relates to contact rate and infectious period" reads "This simple formula suggests different ways of reducing R0" .. "It is also possible to decrease the infectious period γ by finding and then isolating, treating or eliminating".

PROBLEM: With $\gamma$ as the denominator, decreasing it will increase $R_{0}$, which is not desirable. As a previous comment made, Frequency and Period are inversely related. Which one is being used here? --RSH (talk) 18:44, 19 March 2020 (UTC)

Graphs in embedded video...

Two comments:

1. The first plot at 9:10 minutes, "# OF CASES vs TIME SINCE FIRST CASE" makes the point about not exceeding the Healthcare System Capacity, but implies that there are about 9% more "Events" (deaths, total cases, ???) with the protective measures, because the area under the red curve is about 0.91 x the area under the blue curve. May need a discussion to qualify a little better, perhaps some real data from several pathogens addressing this.

2. The second graph at 9:32 minutes, "DEATH RATE/100,000 POPULATION vs EXCESS ... MORTALITY" can't be compared with the first graph because its ordinate is "RATIO OF DEATHS," while the ordinate of the first graph is total events. The area under the red curve is about 2.3 x the area under the blue curve, but I'm not sure integrating ratio of population over time is even meaningful. And it certainly can't be compared to the first plot, because one is deaths by fraction of population and the other is total (but undefined) events.

SixtyRedDevil (talk) 00:48, 25 March 2020 (UTC)

Link to a graph in the embedded video

There is a chart/graph in the video, shown at about 7:30 till 7:43, with a link to London School of Hygiene & Tropical Medicine's Centre for the Mathematical Modelling of Infectious Disease (CMMID)'s GitHub page with that same chart and more data, probably more recent:

 https://cmmid.github.io/topics/covid19/current-patterns-transmission/global-time-varying-transmission.html#summary-of-latest-reproduction-number-and-case-count-estimates

(The tab titled "Summary of latest reproduction number and case count estimates" needs to be clicked to reveal the chart; doesn't seem to scroll/show automatically.)

Shall this be somehow added in the article?

(Don't know how, or if, the video is related to the CMMID page.) — Preceding unsigned comment added by 31.183.172.66 (talk) 03:46, 25 March 2020 (UTC)

"Disease reproduction number"

I've seen the phrase "disease reproduction number", apparently with the same meaning, e.g. here [1]. Should we set up a redirect from Disease reproduction number to this page? Lfh (talk) 08:33, 25 March 2020 (UTC)

Given that it is not in the glossary of terms[2], I would say no.. BiologicalMe (talk) 15:04, 25 March 2020 (UTC)

Inconsistent use of γ

Here γ is used as infectious period (which leads to $R_{0}=\beta \gamma$), but in Compartmental models in epidemiology it is used as the inverse of that period (which leads to $R_{0}={\frac {\beta }{\gamma }}$). Is one of them wrong? Or do different authors use different definitions? Anyway, this is a confusing situation. --mfb (talk) 01:23, 11 March 2020 (UTC)

Even the description is inconsistent. The denominator is described as the infectious period, and that by decreasing it the R0 value will be lower. This is patently impossible. In section 3 The basic reproduction number the equation is described as a product of the two values. Pekkapihlajasaari (talk) 13:02, 28 March 2020 (UTC)

Merge with Net reproduction rate

This is exactly the same principle as net reproduction rate (also abbreviated as R₀), just applied to a different field. The basic reproduction number article already contains information about population ecology that actually corresponds to net reproduction rate. Although the "net reproduction rate" article is written as if it only applies to human demography, this term is widely used in population ecology, e.g. at Brittanica: [3] Bueller 007 (talk) 04:21, 2 February 2020 (UTC)

This may not be such a good idea as infection and reproduction use quite different concept once you go into details - see the above remarks about Disambiguation. In reproduction of non-parasites, there is no incubation period, no disease progression etc. — Preceding unsigned comment added by Gerard Widerot (talk • contribs) 22:30, 3 February 2020 (UTC)

You're missing the point. This article is currently a confused mess of ecology and epidemiology. The ecological use of R0 exactly matches that of https://en.wikipedia.org/wiki/Net_reproduction_rate I've removed the ecological sense. Bueller 007 (talk) 23:13, 30 March 2020 (UTC)

Reproduction numbers conflicting with data on Herd immunity#Mechanics

There's a table with R₀ on the Herd immunity#Mechanics page as well, but the data there is different. It might be good to get them in line. Specifically Influenza shows a very different R₀, but a few others too. The source over there seems like a good source, but I'm not sure how to deal with conflicting sources. M.qrius (talk) 18:25, 17 April 2020 (UTC)

Number for SARS on R0 table

I am looking at ways of importing R0 values to Wikidata.

For SARS, the number currently reported here R0 is 2-5. However is not stated anywhere in the referenced paper.

The paper estimate of effective R0 before containment measures was "R = 3.6, 95 percent CI: 3.1, 4.2". After measures were taken, R0 decreased to "R = 0.7, 95 percent CI: 0.7, 0.8".

I will add 3.1 - 4.2 instead, as we are usually interested in R0 in normal situations.

If there are better solutions, I would love to discuss more about it. TiagoLubiana (talk) 19:19, 6 May 2020 (UTC)

Wikilink to effective reproduction number .. redirects back here

The overview section links to effective reproduction number which comes back here. Either there should be a stub article for that or the wikilink should be removed or made just in-page? Richiez (talk) 20:30, 30 March 2020 (UTC)

Also, the text which links to effective reproduction number says to not confuse it with the basic variant so there must be a distinction which is not explained in the article. ― Ralph Corderoy (talk) 11:17, 16 May 2020 (UTC)

Uncertain reference

I doubt the reference cited for the reproduction range of HIV/AIDS as 2-5. The reference cited is from 1979, and transmission studies by Koopman and others were published at least 10 years later. 66.229.140.215 (talk) 13:35, 29 December 2007 (UTC)

The R0 for HIV/AIDS has been popping up in media coverage (e.g. here and here) and other online discussion re COVID-19, and the source for these figures seems to be this Wikipedia article. I was also curious about the HIV/AIDS R0 info in the chart, sans references, which brought me to this topic on the talk page. I have three comments and a question:

(1) Current references that explicitly state the R0 for HIV/AIDS are sparse, I’ve found none, perhaps it’s buried somewhere in science journals, but I haven’t been able to tease it out, this is not my field either. All I’ve found are articles that discuss how to calculate R0 for HIV/AIDS without actually specifying what the R0 for HIV/AIDS is. However, another reason seems to be that calculating an exact R0 is quite challenging hence the title of the following article: Delamater, P. et al. 2019. Complexity of the Basic Reproduction Number (R0). Emerging Infectious Diseases. 25.1. 1-4.; further, the concluding remarks in the following article state: “During the last twenty years Ro has emerged as a basic concept in infectious disease epidemiology, but it has also become apparent how difficult it is to apply in actual field situations” Dietz, K. 1993. The estimation of the basic reproduction number for infectious diseases. Statistical Methods in Medical Research. 2. 23-41.

(2) Noting above comments from 66.229.140.215 and others, references such as Koopman and Anderson & May (which are mentioned in the talk section, not cited in the main article) certainly are decades old; however, Koopman and Anderson & May are cited with approval in more recent scholarship, including: Delamater 2019 (above), Nah, K., et al. 2017. Test-and-treat approach to HIV/AIDS: a primer for mathematical modeling. Theoretical Biology and Medical Modelling. 14. 1-11.; Bonacci, R. and Holtgrave, D. 2016. Evaluating the Impact of the US National HIV/AIDS Strategy, 2010–2015. AIDS and Behavior. 20. 1383-1389; Holtgrave, D. 2010. Is the Elimination of HIV Infection Within Reach in the United States? Lessons from an Epidemiologic Transmission Model. Public Health Reports. 125. 372-376.; Déirdre Hollingsworth, T., et al. 2008. HIV-1 Transmission, by Stage of Infection. Journal of Infectious Diseases. 198. 5. 687–693.

(3) The closest I’ve come to an answer to current figures for HIV/AIDS R0 is from personal correspondence with Holtgrave (03/02/2020) where he states, based on estimates from his 2010 article (above): “At the current transmission rate of roughly 3.5 in the US, this would imply Rsub0 values in the range of 1.01 to 1.31 (for a variety of estimates of infectiousness duration). This indicates that the HIV Rsub0 is likely still over unity in the US, but getting closer to the bright line of Rsub0 equaling unity.” Of course, without a forthcoming article, this doesn't work as a reference for HIV/AIDS R0 in the chart on the R0 Wikipedia article.

(4) Is it possible to include in the Wikipedia R0 article how R0 relates to attack rate, infection rate, and transmission rate? — Preceding unsigned comment added by 2602:302:D154:79A0:5844:C284:64F3:D46F (talk) 10:17, 3 March 2020 (UTC)

I see that the original citation for the HIV R number must have been deleted. I would simply add that common sense dictates that the R number for HIV/AIDS should be lower than for COVID-19, because the former did not spread very rapidly, even when no treatments were available, and there were no lockdowns. Also, the methods of transmission are much more limited. Nigelrg (talk) 18:17, 7 June 2020 (UTC)

Effective Reproduction Number

The current version of the article is misleading in that it includes only herd immunity in the definition of the effective reproduction number, whereas it should refer to "individuals available for infection". In the current Covid-19 pandemic, a number of countries have been successful in damping the exponential growth by means of quarantines, contact tracing and social distancing. This even though there is some evidence that immunity to the disease is short lived , and therefore herd immunity may be unachievable.

It would also be useful to present the methodology for calculating the effective reproduction number. In its simplest version R is simply the ratio of two successive days' confirmed cases; but this version is unsatisfactory because of the inherent noise in the process and inconsistencies in reporting cases. A better version is that used by the Robert Koch Institut , in which eight days' data are employed, with R then being defined as the ratio of the past four days' total new cases divided by the prior four days' total new cases.

While it is true that a value of R less than unity is indicative of a declining rate of infection, R becomes increasingly ineffective as the number of new cases per day becomes small. This is as a result of the inherent randomness of the arrival of new cases, which can be modeled as a Poisson process. Since the standard deviation of a Poisson process is equal to the square root of the mean, one sees immediately that if the number of new cases per day is averaging 10 000, the standard deviation is but 1% of the mean, but if the number of new cases per day is averaging 100, the standard deviation is 10% of the mean. Also, the form of the equation informs us that if the average number of cases is small, a few new cases can substantially perturb R, while if the average number of cases is large, even a large increase in cases will have little effect. Examples of this, in the context of the Covid-19 pandemic, are the June outbreak in Germany (Nordrhein-Westfallen) where an outbreak of less than 2000 caused R to spike to approximately 2.5, while the near doubling of new cases in the US a week later caused R to increase only to 1.23. — Preceding unsigned comment added by SCS137 (talk • contribs) 14:35, 25 June 2020 (UTC)

References

Reproductive rate

Reproductive rate lands one here, but this article doesn't explain it. Could s.o. start a separate page (for all organisms please, not just for humans or pathogens!) or put a paragraph here. THANKS76.97.245.5 (talk) 16:40, 6 April 2009 (UTC)

The terms "basic reproduction number," "basic reproductive ratio," and "basic reproductive rate" are interchangeable. The dictionary definition of the word "rate" does not absolutely require that a rate have time as denominator, so the claims that it is "incorrect" to call this number a "rate" are themselves incorrect. However, the usage of "rate" is unusual and possibly misleading, so personally I would avoid using the term "rate." Here are the current usage statistics from Wikipedia.

• basic reproduction number: 264,000 results
• basic reproduction ratio: 34,200
• basic reproductive number: 99,700
• basic reproductive ratio: 33,700
• basic reproductive rate: 28,200
• basic reproduction rate: 16,200

Several learned, published, and/or academic sources allow the use of rate, so it should not be called incorrect - just unusual. Jaredroach (talk) 04:20, 3 July 2020 (UTC)

References

Spelling of R naught?

Does anyone happen to know why it's R naught instead of R nought? I always understood that "naught" means nothing and "nought" means zero. It's not a mistake in the article, because I have seen (and been puzzled by) the same spelling elsewhere.79.103.108.162 (talk) 20:28, 1 February 2016 (UTC)

According to many dictionaries, 'naught' and 'nought' are variant spellings of each other. They are synonymous when the meaning is "zero". Crossing the pond changes usage. Jaredroach (talk) 04:21, 3 July 2020 (UTC)

Dispersion parameter k for COVID-19

These days, the media are talking about the "k" parameter that goes along with the R0 value, for example here. There is a mathematical description in Endo et al, Estimating the overdispersion in COVID-19... (estimating k=0.1 for COVID-19) but my statistics undestanding is not enough to understand what it means. Maybe someone who does could add a paragraph to this article or add it to index of dispersion (and refer to it)? Han-Kwang (t) 21:23, 2 June 2020 (UTC)

R is the average number of people that get the disease from each infected case. The dispersion parameter is a measure of how much variation there is around that average. So if everyone gives the disease to exactly R people, then the dispersion is zero. If the majority of people give the disease to zero other people, but a few people are superspreaders, then dispersion is very high. It is a fairly complex discussion, and would require considerable expansion of the article from its current state. Might be worth it, as different dispersion parameters can result in very different predictions. Even the dispersion parameter concept itself is limited, as it cannot cover extreme (non-smooth) distributions of transmissibilty across a population. Which may actually be the case for COVID-19. Jaredroach (talk) 04:28, 3 July 2020 (UTC)

Reproduction ratio

The correct term, according to Diekmann et al (1990) is the basic reproduction ratio, since R0 does not have a dimension (it is cases per case), so it is certainly not a rate. Should/can we change the page title? SpaceLem (talk) 12:50, 12 September 2012 (UTC)

A Google search as of May 2020 shows that the term "Reproduction ratio" is mostly about photography. So although Diekmann et al (1990) might have been correct about usage in 1990, they may not be correct today. Jaredroach (talk) 04:32, 3 July 2020 (UTC)

How about R0 of COVID-19 ?

CDC now estimates COVID-19 R0 to be 2.5 based on data through the end of April 2020. Can we change the 5.7 value (an earlier estimate) to this newer, more thought out value? https://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scenarios.html?fbclid=IwAR1GekgfCKmoY53Jr5Od098ho_RR6uNkUbh91TbTZiJLyVP-R5ZNdjEpF6Y

At present it is writen '1.4–3.8', but a week and some day ago it was '1.4–6.6'. It is not stablized. --Kyuri1449 (talk) 19:25, 1 March 2020 (UTC)

I noticed this too and while I'm in virtually no mood to go seek out some sources, there was one paper that the "6.6" value linked to that said :that the value was between 4 and 6 roughly. It isn't 100% stabilized and most of the information is just unknown/not researched enough yet. --Prezi2 (talk) 09:33, 4 March 2020 (UTC)

"THE" Ro of COVID-19, or of any desease for that matter, does not exist. The Ro locally measured in Wuhan at the beginning of the epedimic was averaged at 2.2 within a 90% rate of certainty between 1.4 and 3.8. The Ro will differ between localities, and thus between countries. It all has to do with how close people make contact, with how many different people they make close contact and how mobile they are. It might well be that the Ro in a develeoped county like the Netherlands can run up to 3.5. especially at the start of the epedemic. Since the Netherlands are highly populated, highly mobile, highly international and often greet kissing each other. Special measures may temporarily lower the Ro, but since all special measures will (have to) end finally, so will this temporal reduction of the Ro77.60.121.89 (talk) 12:47, 18 March 2020 (UTC)

Median R0 value of 5.7 at https://wwwnc.cdc.gov/eid/article/26/7/20-0282_article — Preceding unsigned comment added by 184.98.48.118 (talk) 22:03, 8 April 2020 (UTC)

The lower end of the possible R0 is stated as 1.4 (which represents the lower end found in studies) but the higher end is stated as 5.7. However, the higher end found in the CDC study mentioned above is 8,9. Why this inconsistency? Shouldn't it be stated as "1,4 to 8,9"? — Preceding unsigned comment added by 83.163.205.75 (talk) 10:15, 3 May 2020 (UTC)

It is misleading to just blank state that COVID-19 (and maybe other of these diseases) has a R0 value within this range. It should be noted on the article that this number is rapidly changing and is not official. See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7073717/Jackalakalaka (talk) 17:47, 22 May 2020 (UTC)

These concerns about variability and uncertainty and unreliablity of R are already addressed in the Delameter reference, which is already cited, and the content summarized in the main article. Basically, Delameter agrees with you; everything about R is misleading. Also note that there never will be an "official" R. This metric is not the sort of thing the Bureau of Standards opines on. Or any other official source. Jaredroach (talk) 04:22, 3 July 2020 (UTC)

I have edited this yet again as it said "3.48" which is needlessly precise, to reflect the cited source from LANL, which lists a range of values from 2ish to 6ish. We really shouldn't include a single value here (as it was before). As others have said, a "true" R0 is elusive, as there are so many local factors; additionally there's some mounting evidence that the G variant has a higher R0 than the D variant, which could partially, but certainly not entirely, account for the seeming increase between earlier and later estimates. (https://www.cogconsortium.uk/news/commentary-cog-uk-report-9-25th-june-2020/).

I'm not wedded to 2-6, but in general, we should not include more than 1 or 2 significant digits or and should always report a range, not a single value. Mvolz (talk) 17:09, 24 July 2020 (UTC)

Several places have R0 values for COVID19 lower than 3.8. Putting 3.8 as the lowest estimate is outright false and should be changed. 01:17, 13 September 2020 (UTC)

OK, done. There's lots of references for this in PubMed. I grabbed one of many reliable ones. Jaredroach (talk) 21:43, 18 September 2020 (UTC)

Mvolz, thanks for your edit. As you point out, R0 does in deed change with interventions. That is why there are ranges for R0 in the table. We could create another table for a new concept, like "R0 without interventions". But that isn't this table. I have reverted your change. Please feel free to discuss. The main article does cover this concept that R0 is not a constant; perhaps we can make it more clear? Jaredroach (talk) 15:41, 20 September 2020 (UTC)

R0 with interventions is definitionally NOT R0, it's Re - R effective. From the paper itself: "assuming a basic reproduction number (R0) of 2.2 for no intervention, and subsequent effective reproduction numbers (R) of 0.8 and 0.5." It doesn't call .5 R0, it calls it R. In fact, this paper doesn't even estimate R0 at all, it's a modelling paper that assumes R0 IS 2.2, so the paper itself is actually entirely irrelevant. (https://pubmed.ncbi.nlm.nih.gov/32430840/) I'm going to revert the change and delete the reference to the paper this time as well. Cheers! Mvolz (talk) 18:26, 28 September 2020 (UTC)

I agree that is tempting to create a new definition of some quantity like R0 that is invariant to interventions. However, that is not the definition of R0 that is defined in textbooks and the literature and review articles. In other words, until such a new concept becomes standard, we cannot push it into Wikipedia. The only intervention that cannot alter R0 is vaccination or some other approach that produces herd immunity. So any physical distancing (aka, social distancing) measure will change R0. In these cases R0 equals Re unless there is also a change in the fraction of the population that is immune (e.g., through acquisition of natural immunity via infection and recovery, vaccination, births, or deaths). Rather than engage in an editing back-and-forth over the exact range of numbers for R0, I will add a note that these R0s in the table represent initial R0s before any physical distancing interventions. Jaredroach (talk) 17:27, 29 September 2020 (UTC)

Which is the R being used by the press and governments?

I think that is is not immediately clear from this article what the R number being used in the press ( example: https://www.bbc.co.uk/news/health-54025713 ) and in government announcements ( example: https://www.gov.uk/guidance/the-r-number-in-the-uk ) means; in other words which of the R numbers in this article, if any, is the one being used by press & govts.

It could be the Basic reproduction number, R zero, as this article states in the lead: "R_{0}.. is also affected by other factors such as environmental conditions and the behaviour of the infected population." and this sounds like the R that people are talking about, as we are being asked to modify our behavior to affect it, and there is talk of the seasons/weather being relevant (ie "environmental conditions").

It could be the effective reproduction number, Re. The name "effective" suggests that this is the one that is the final result of all the relevant factors, and "is the number of cases generated in the current state of a population" as it says in the lead. But the section on this version of R says "In reality, varying proportions of the population are immune to any given disease at any given time." suggesting it is only immunity effects that are been taken into account.

The lead also refers to "Some definitions..." (in relation to R zero) which means to me that not everyone is using the same definition of this particular variable.

I propose that something about how the various R's mentioned in this article relate to the R in general use is included in this article.

FrankSier (talk) 19:42, 4 September 2020 (UTC)

I have noticed that the section "Limitations of R_{0}" starts with "Use of R_{0} in the popular press..." implying that is indeed it is R zero, not Re that is used. Is this actually true? FrankSier (talk) 07:48, 6 September 2020 (UTC)

This news item https://www.bbc.co.uk/news/av/health-52494495 explicitly says "R0". FrankSier (talk) 19:47, 9 September 2020 (UTC)

FrankSier, It doesn't (yet) matter for COVID-19 because there has been no vaccination and no substantial herd immunity. So to date, R_{0} and Re have always been identical in any COVID-19 article written for the popular press. That has led a lot of popular-press writers to drop the subscript. Because to them, it doesn't matter what subscript the reader implicitly adds, as the meaning is the same. Jaredroach (talk) 00:16, 20 September 2020 (UTC) Quick update - obviously as of January 2021, Re is probably the better term to use in most instances... Jaredroach (talk) 07:19, 19 January 2021 (UTC)

The effect of an R number of 1 on the number of infections

EDIT: I now think that most of what I earlier wrote in the in the rest of this section is mistaken. (The first 2 lines below are okay I think; they are just quotes from web pages; they may be worth adding to the article.) The part "I am not convinced that..." onwards is mistaken I now think, so I am inserting this bit at the beginning of the section to save people the trouble of reading further, unless you want to of course. I cannot see now how I can to make such a bad error, and also I may be wrong about being wrong, so I will leave this issue alone for the moment. FrankSier (talk) 20:20, 9 September 2020 (UTC)

Both of these web pages https://www.bbc.co.uk/news/health-54025713 and https://www.gov.uk/guidance/the-r-number-in-the-uk contain the statement:

"An R number of 1 means that on average every person who is infected will infect one other person, meaning the total number of infections is stable."

I am not convinced that "on average every person who is infected will infect one other person" leads to "the total number of infections is stable"?

(In this discussion, I am assuming that "total number of infections" means the total number of people who are infected at any given point in time, as opposed to, for example, the total that have been infected so far, including those are no longer infected (total so far can only go up).)

It seems to me that there is also a time factor involved. For example: if this next infection happens in just one day (between a person becoming infected and that person infecting someone else), this is very different from that next infection happening in 7 days.

Taking the case when each person remains infected for 14 days.

Looking at this in more detail:

Simplifying to it being exactly either one day or 7 days to passing on infection.

If there is one day to next infection:

• Day 1: 1 person infected.
• Day 2: 2 persons infected.
• Day 3: 3 + 1 persons infected (3 infected by person 1, plus 1 infected by person 2)
• Day 4: 4 + 1 + 1 persons infected (4 infected by person 1, plus 1 infected by person 2, plus 1 infected by person 3)
• ...
• Day 7: I've not calculated up to Day 7 but I will approximate to "quite a lot".

If there is 7 days to next infection:

• Day 1: 1 person infected.
• Day 2: still 1 person infected.
• ...
• Day 6: still 1 person infected.
• Day 7: 2 persons infected. This is considerably less than the "quite a lot" for Day 7 with the "one day" scenario.

I have only calculated for a case where the time to pass on infection is less than the time remaining infectious, and I have calculated where these are exact times, rather than averages, as those calculations using averages properly would be much more difficult.

My calculations are fairly rough but I think that the overall conclusions are clear:

• If one person infects one other, then there are at least some cases where the number of infections goes up, and does not remain stable.
• The rate at which the number of infections changes is greatly dependant on the time taken to infect another person.

If my calculations or conclusions are wrong, please correct and explain.

There are some calculations in the existing article, but they are pretty technical, and mostly beyond me. As I giving my opinion here, I think I should say what my level of technical understanding is: I think I am above the average of the general population. In more detail: IQ at some stage of my life measured as a bit above 120 (I don't remember exactly. I have a fairly good university degree in physics (which involved quite a lot of maths) but that was back in 1973 and I no longer understand some of my notes from back then. I taught fairly basic computing in adult education. So I think that it would be good to have some calculations and explanations that would be understandable by the average general public, if possible.

The relevance to WP in general and this article in particular: if these influential sources are getting it wrong, or have explained something in a unclear way, then I think it would be appropriate for WP to point this out, having found the relevant, generally-agreed-to-be-correct, sources, and clarify the issue.

FrankSier (talk) 09:36, 6 September 2020 (UTC)

Frank: you are basically correct. There are plenty of nuances of special situations which you illustrate. These typically involve changing parameters related to the length of incubation and the length of disease. Obviously, if the disease never goes away (e.g., it lasts forever) in a very long lived populations, an R of 1 will allow the disease to keep going up. But the folks who simplify in the context of COVID-19 (and most other diseases) are not wrong either. The incubation time is pretty short, as is the length of the disease, so the quotes along the lines of "on average every person who is infected will infect one other person" leads to "the total number of infections is stable"? are fine and unobjectionable.Jaredroach (talk) 21:58, 18 September 2020 (UTC)

If I understand correctly, R=1 means one infection overall, not daily. That is, a contagious individual will infect 1 other individual for the duration of their infection. Thus, the number of new cases (and of active cases) remains constant, and the cumulative number of infections grows linearly. In the scenario FrankSier suggested, an infected individual was infecting 1 other person daily, but more than 1 person overall. Bottom line: R and R0 do not provide any information of how the disease evolves with time, such as a rate of daily growth; only about whether the growth will be constant, exponential, or decreasing. --Cousteau (talk) 11:33, 22 December 2020 (UTC)

Cousteau, the number of active cases does not necessarily remain constant. A case may remain active for many years, but may transmit to contacts over a period of days. The total number of active cases will grow, even with R=1. Also, to be clear, we are talking about the degenerate case where the population is infinite or time=0 (i.e., R0 = Re). By "active", you might not mean "symptomatic" or "conceivably transmissible". You might mean "the effective period of time over which a case is infectious for purposes of this model". In which case the back-of-the-envelope math works. Kinda like a relay race: only one person can hold the baton at a time. Note, I am talking generally about all possible diseases, not a specific disease such as COVID-19. Jaredroach (talk) 02:15, 9 February 2021 (UTC)

Are there two R0's?

If I take the R0 from the SIR model and put it at time=0 in a growth model , I get different results

I asked at reddit If I understand well we have to distinguish the former rho ('relative removal rate' by Baily, 1957) which is used in the SIR-model and the former z ('basic reproduction rate' by Bartlett, 1955) which is used in a growth model instead of two times R0

What are your thoughts about it? --Rcsmit (talk) 11:56, 9 February 2021 (UTC)

Arithmatic and HIT

The current article text contains (the values in) the first table next to or below this text.

Values of R₀ and herd immunity thresholds (HITs) of well-known infectious diseases prior to intervention

Disease	Transmission	R0	HIT
Measles	Aerosol	12–18	92–94%
Chickenpox (varicella)	Aerosol	10–12	90–92%
Mumps	Respiratory droplets	10–12	90–92%
Rubella	Respiratory droplets	6–7	83–86%
Polio	Fecal–oral route	5–7	80–86%
COVID-19 (Variants)	Respiratory droplets and aerosol	3–8	+80%
Pertussis	Respiratory droplets	5.5	82%
Smallpox	Respiratory droplets	3.5–6.0	71–83%
COVID-19 (wild type)	Respiratory droplets and aerosol	2.9 (2.4–3.4)	65% (58–70%)
HIV/AIDS	Body fluids	2–5	50–80%
SARS	Respiratory droplets	2–4	50–75%
Common cold	Respiratory droplets	2–3	50–67%
Diphtheria	Saliva	2.6 (1.7–4.3)	62% (41–77%)
Ebola (2014 Ebola outbreak)	Body fluids	1.8 (1.4–1.8)	44% (31–44%)
Influenza (2009 pandemic strain)	Respiratory droplets	1.6 (1.3–2.0)	37% (25–51%)
Influenza (seasonal strains)	Respiratory droplets	1.3 (1.2–1.4)	23% (17–29%)
Nipah virus	Body fluids	0.48	0%
MERS	Respiratory droplets	0.47 (0.29–0.80)	0%

The note with respect to the HIT column says:

"Calculated using p = 1 - 1 / R₀."

As things stand, the HIT values and R0 values given for quite a few of the diseases do not match in an arithmatic sense. Starting from the R0 values given (which I am not checking or verifying) and using the formula mentioned, I arrive at the HIT values (rounded to the nearest integer percent value) given in the second table next to or below this text.

Now, for certain HIT values in the current article table it might have been the case that the difference between them and my results was caused because they had been arrived at by using more decimals in the R0 values than are in the table. For two of the lines / diseases I have checked the sources mentioned for the R0 values, to see whether any of the HIT values in the current article table turned up there. That was not the case.

Based on this and on rounding to the nearest integer percent value, and certainly if none of the HIT values in the current article table have been directly copied from a publication, I hold that several of those HIT values are arithmetically incorrect and ought to be replaced.

Values of R₀ and herd immunity thresholds (HITs) of well-known infectious diseases prior to intervention

Disease	Transmission	R0	HIT
Measles	Aerosol	12–18	92–94%
Chickenpox (varicella)	Aerosol	10–12	90–92%
Mumps	Respiratory droplets	10–12	90–92%
Rubella	Respiratory droplets	6–7	83–86%
Polio	Fecal–oral route	5–7	80–86%
COVID-19 (Variants)	Respiratory droplets and aerosol	3–8	67-88%
Pertussis	Respiratory droplets	5.5	82%
Smallpox	Respiratory droplets	3.5–6.0	71–83%
COVID-19 (wild type)	Respiratory droplets and aerosol	2.9 (2.4–3.4)	66% (58–71%)
HIV/AIDS	Body fluids	2–5	50–80%
SARS	Respiratory droplets	2–4	50–75%
Common cold	Respiratory droplets	2–3	50–67%
Diphtheria	Saliva	2.6 (1.7–4.3)	62% (41–77%)
Ebola (2014 Ebola outbreak)	Body fluids	1.8 (1.4–1.8)	44% (28–44%)
Influenza (2009 pandemic strain)	Respiratory droplets	1.6 (1.3–2.0)	38% (23–50%)
Influenza (seasonal strains)	Respiratory droplets	1.3 (1.2–1.4)	23% (17–29%)
Nipah virus	Body fluids	0.48	0%
MERS	Respiratory droplets	0.47 (0.29–0.80)	0%

Would anyone like to comment on this or even object to substituting the values I calculated?Redav (talk) 00:56, 13 July 2021 (UTC)

@Redav: I added the note to the HIT column. If you check the edit history, for many years this column has been unsourced and has not been clearly defined. In fact, it is calculated from the most accurate R0 values from the sources. It might be nice to set R0 to full precision, but as you'll see, not all sources use the same precision. Several editors have tried to balance the accuracy differences so that the numbers look "pretty" next to each other, which seems to stabilize edits. Perhaps calculations using the most accurate values can be added as a hidden comment to editors attempting to verify them. --Fernando Trebien (talk) 17:56, 11 August 2021 (UTC)

I edit the reported R values to no more than tenths precision when I have a chance to edit this article. There is no way anyone can measure R values with more precision (it is not possible to do so in a real-world situation with humans), so it is not to make these values look pretty that I round them, but rather to avoid conveying false accuracy. Jaredroach (talk) 22:36, 16 August 2021 (UTC)

Ah ok. I actually don't mind the rounding. But it may cause confusion when trying to verify the calculated HIT values. Maybe one way to avoid this is to use Template:Round, then whoever edits the article will see the number with full precision while those reading it will see only the rounded value. --Fernando Trebien (talk) 13:03, 18 August 2021 (UTC)

I just restored the more precise values. Only three entries were changed: COVID-19 (ancestral strain), Ebola and Influenza (2009 pandemic). I also added expressions to calculate all HIT values from R0 instead of doing the calculation by hand. There was a rounding error for COVID-19 (ancestral strain), and the calculation was incorrect for the Andes hantavirus (added recently by someone else). --Fernando Trebien (talk) 18:24, 11 August 2021 (UTC)

References