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R0 and Rt
A clarification on their meaning
2: Meaning of R0
3. Meaning of Rt
4. Widespread misunderstanding on R0 and Rt
5. Clarifications on a public mistake
"Phase 3" has started in Italy a while ago and, in order to better understand the reason for the precautionary behaviors that continue to be required in order not to regress to previous phases, it may be useful to clarify the real meaning of the epidemiological index R0 and its "younger brother" Rt, indices that for several weeks have been systematically quoted - sometimes improperly - as shown by the unfortunate example that came to mind to a regional council member to "explain the Rt to the people".
Incidentally, I think that the mistake made by the council member about what Rt = 0.51 means is less serious than what his critics report to test the competence of which, it would be interesting to ask them to explain "in their own words" why this is a mistake ... (and under no circumstances it’s comparable with Gelmini’s - former Italian minister some years ago - mistake about the "700 km tunnel" going from Gran Sasso to Geneva!)
Once the meaning of the indices in question has been clarified, it should be easy enough to understand why Rt = 0.5 does not mean that - as asserted: “To infect a person, if the index is a 0.5 it is necessary that there are two infected people at the same time... on the contrary when the index is at 1 it is enough to meet an infected person to become infected ".
The following considerations are not (nor do they want to be) virological or epidemiological in the strict sense, but derive from some basic statistical concepts applicable to the epidemic field such as the industrial, demographic, electoral fields, and so on...
2. MEANING OF R0
Without going deep into too many details, R0 (Basic Reproduction Number) can be defined as the average number of "secondary" infections, that is to say, directly caused, produced by a single infected individual during its infectious period in a population which is"entirely susceptible" (i.e. in which practically everyone can become infect).
Assuming that we have established that the Ro of an infectious disease in its initial stage is exactly 2, this means that a single infected person (whether asymptomatic, pauci-symptomatic, symptomatic in home care or in hospital ...) will directly infect two other people as an overall result of the period of time in which it was contagious, regardless of the specific moment in which it infected them: this last clarification is fundamental to understand that Ro is not an infection rate (if it were, it would grow over time during the period of infectivity), but instead it is a statistical index (i.e. a pure number "to things done"). However, the theoretically considerable difference between rate and index is of marginal importance in practice because what matters is knowing that the higher the Ro value is, the faster the infection will spread: the total number of infected since the outbreak of the epidemic grows exponentially at the start, but the speed with which it grows depends on the value of Ro.
- if Ro = 2, from 1 initial infect person, 2 will be directly infected, each of which will infect 2 other people, and therefore there will be 4 infect people, everyone of which will infect 2 for a total of 8, after three serial intervals (better explained further on; for now it is enough to know that it is a period of time of a few days that depends on the type of virus and the "average response" of the infected individuals)
- if Ro = 3, from 1 initial infect person, 3 will be directly infected, each will infect 3 other people and therefore there will be 9 infect people, everyone of which will infect 3 people for a total of 27, also in this case after three serial intervals
For Ro = 4 there will be 64, for Ro = 5 there will be 125 (also in this case, from one initial infected and always after three serial intervals) and so on ...
With these numbers at stake, it should be evident why we are talking about the "outbreak" of an epidemic: because hundreds of new infected people can be generated from a single outbreak in a few days and because the speed in the identification of the "patient zero" is so important , and also to trace its contacts by isolating them in order to "cut" the rate of growth, which otherwise would remain exponential.
After some time from the initial outbreak, the growth begins to slow down for some reasons explained further, and the value of the "R" index progressively decreases compared to the initial Ro: it is therefore appropriate to change its name by calling it Rt (Effective Reproduction Number, function of time) in order to distinguish it from the initial index which was Ro.
These are not two different indices, but the same index calculated at different times: Ro is calculated at the beginning of the epidemic, during the pure expansion phase when it maintains a value that is virtually constant (it is a static index) , while Rt is calculated during the development of the epidemic, variable over time according to the trend of the new infections (it is a dynamic index). Graphically the curve of the "total infected" tends to flatten, changing trend (the Ro index switches to Rt) and passing from approximately exponential (Rt> 1) to almost linear (Rt ~ 1), to finally become almost horizontal (Rt ~ 0 ): this means that the total number of infected does not increase virtually anymore as there are nearly no new infected people.
Note that what decreases is not the total number of infected people from the beginning of the epidemic, but the number of current infected as they heal. In epidemiological slang, they pass from the Infected category to the Recovered category, or they die, passing from Infected to Removed (the set of Recovered and Removed cases is sometimes called "Closed").
The value of R0 essentially depends on two factors:
I = Infectivity. It is the probability that, in a single contact between an infected and a susceptible, the latter becomes infected (this usually takes a certain time, which is called the incubation period and varies from individual to individual). This probability is extremely variable, going from very low to very high, as it strongly depends on the modalities (proximity and mutual orientation) and on the duration (seconds, minutes, hours) of the single contact, but it also depends on the "viral load" of the infected person (measured by the number of viruses present in 1 ml of blood, normally proportional to the quantity of viruses present in his/her body) and by the "infectious dose" (number of viral particles inhaled by a susceptible person, on a quantity necessary to trigger the infection in a person who does not have specific previous pathologies) the order of magnitude of which - estimated by some studies - seems to be about thousand units of the coronavirus.
It may be useful in this regard to mention the risk of a considerable difference between the effective "intrinsic capacity" of a person not to get infected by a certain amount of viruses coming into contact with his respiratory tract, capacity similar to a sort of "viral resilience" (non-technical terminology!) and his belief that he can be easily infected or, vice versa, particularly resistant; these beliefs that can lead to excessively precautionary behaviour measures or, on the contrary, the carelessness of even normal precautions!
NC = Overall Number of single contacts that the infected person had had during the entire period in which it was contagious (it strongly depends on the habits of social interaction of the population where the epidemic is developing) assuming that each single contact occurs with a susceptible as there is not yet a significant number of immune people (healed or vaccinated).
It is intuitive that, without considering the other factors mentioned below, Ro is proportional to their product (Ro ∝ I x NC) as the "expected number" of people infected directly by a single infected person (regardless of the transmission mode of the contagion) is given by the product of probability I to infect in a single contact a susceptible for the total NC number had with infectious people, initially coinciding with the entire population.
As an example, if a coin is tossed twenty times, the number of "heads" that will be obtained, except for rather rare cases, will be approximately ten, given by the product of the 50% probability of getting heads in each toss for the number of throws made, that is 0.5 x 20 = 10 (which is the "expected number" of heads). Similarly, if at the beginning of an epidemic an infected person encounters, during the period when it is infected, about twenty be infected people and, if in each meeting the probability of infecting is 10%, then the expected number of infected persons infected directly by him will be 0.1 x 20 = 2 (which is none other than the Ro)
There are, however, other factors (virological, epidemiological, environmental/climatic) that contribute to the value of Ro, such as:
- the incubation period, which is the interval of time between the moment of infection and the appearance of the first symptoms (for asymptomatics this cannot be determined!) and during which the infected people can be more or less contagious (for SARS-CoV-2 this period goes - in nearly all cases - from 2 to 11 days, with an average of about 5 days)
- the serial interval, i.e. the average time between the appearance of symptoms in an infected person and the appearance of symptoms in a susceptible person infected directly by him (for SARS-CoV-2 this average time is estimated in about 7 days)
- the "viral resilience" of the population concerned, highly variable depending on the age group and other less known factors
- the temperature, humidity and the degree of air pollution (there is no consensus on the degree of influence of these environmental/climatic factors)
It should be clear at this point that, unlike a rather widespread belief, the value of Ro is not exclusively related to the type of virus, but is also determined by other factors and therefore depends both on the geographical location and on the population in which the epidemic is developing. The value of 2.5 is the one most frequently quoted in the epidemiological calculations on SARS-CoV-2, but some estimates calculated for
give values between 1.5 and 3, while for
between 2 and 3.5 (maybe the Lombardy
Region was around 4). The
estimates differ according to the calculation methods adopted and the input
data used. Italy
NOTE 1 (quite technical and can be skipped at a first reading)
Without going into statistical details, it is appropriate to clarify that it doesn’t make much sense to ask for (and it’s misleading to give ...) the "exact value" of Ro because, as on many statistical parameters detected on samples, it is only possible to make an “estimation by interval" or identify two values (min and max) which determine the so-called "Confidence Interval" generally - but not necessarily - centered on the average of the values derived from the experimental data (with all the uncertainties deriving from their suboptimal collection and transmission to the analysis centers) which has a high probability of containing the "true value" (still unknown) of this parameter.
Just to fix the ideas, a typical estimate of Ro can be given by an expression such as: "AV = 2.6 (95% CI 1.4-3.8)" which is a slang way used in statistics to say simply that the Average Value of R0 on the sample examined is 2.6 and there is a 95% probability that the Confidence Interval between 1.4 and 3.8 (centered on 2.6) contains the "true value" (still unknown) of Ro, which also implies that there is a 5% probability that Ro is out of interval, unknown exactly where but yet not too far ...
3. MEANING OF Rt
As far as Rt is concerned, or "Effective Reproduction Number", it is defined analogously to Ro (average number of secondary infections etc.), but with the substantial difference that its value is affected by the fact, intrinsic to any epidemic, that after a certain time since its outbreak, not all the population has remained Susceptible (i.e. not all are infected), but a progressively increasing number of susceptible has changed to Recovered i.e. healed (no longer infectable - provided the fact that healing gives full immunity, what still not ensured for the virus in question). This fact causes, for purely statistical reasons, a progressive decrease in the value of "R" from the initial Ro to the current Rt (but this does not mean that Ro has decreased: it is simply no longer applicable to the current situation), as the average number of "secondary infects" produced by each infected person during its infectious period is less than the initial number, due to the fact that - since the % of susceptible people in the total population has decreased - with the same number of contacts during his/her infectious period, each infected person met - on average - fewer infectable people because some of the encountered people were immune: the overall result is that they have become globally less infected. By indicating with S the % of susceptible (virtually between 0 and 1) it is possible to assume that - all other conditions remaining the same - Rt = Ro x S and, if S <1, then Rt <Ro.
If specific measures for risk containment are added to this intrinsic and progressive decrease over time, such as:
1. surgical masks, spacing and restraint of droplets (in increasing order of potentially infectious dose: speaking, singing, coughing/sneezing) which reduce infectivity (impact on 1st factor)
2. reduction in the number of total contacts (impact on 2nd factor) by decreasing the number of people met and the frequency of meetings, for example with smart working and the scarcity of social exchanges.
3. isolation of the infected (impact on 2nd factor) or at least of the ones found positive to the swab,
it is obvious that Rt the more decreases, the more incisive the containment measures adopted are.
In conclusion, Rt decreases:
Ø spontaneously over time, as the% of immune (susceptible gone to healed) increases
Ø forcibly, following the containment measures listed above
Considering this, it should be clear that the effect of the containment measures is not "all or nothing", but is function of their intensity: if I halve the number of people met, I will have a reduction in Rt of approximately half, if I halve it again, Rt will decrease further and so on. It is therefore missleading to argue that if ever single person doen’t follow the rules doing exactly as required then it serves no purpose: the empirical criterion is trivially that, if many people follow the assigned rules, all the more unlikely to erupt outbreaks of infection difficult to switch off.
NOTE 2 (also quite technical and can be skipped at a first reading)
The same clarifications made in NOTE 1 on the intrinsic uncertainty of the estimate of Ro also apply to Rt with the aggravating circumstance that the latter varies over time and is therefore even more elusive ... Considering this, hearing expert claims with conviction that "in the region A, the value of Rt has been stable at 0.23 for two days and therefore everything is ok; instead in region B, the value of Rt was 0.34 yesterday, while today is 0.42 and therefore the situation is getting worse" confirms that these so called "experts" do not know what they are talking about ... (or that they know it, but they consider useless to explain that these are estimates only more or less reliable)
A more realistic way of presenting the Rt index is to analyze its trend over a few days to check if it remains substantially steady in this period, even with random fluctuations around a certain average value, or if it shows a stable increasing or decreasing trend, which is the only cases when it is correct to speak of worsening or improvement.
A final clarification is that the Rt is "local", that is it relates to the area in which the data used to calculate it was collected: if I have a couple of outbreaks in a certain area, while in another area I don’t have even one, it is obvious that the Rt in the first zone will be considerably higher than in the second, but it is completely misleading to average it and present it as "the overall Rt"!
Considerations on "herd immunity":
- without adopting any containment measures (i.e. without changing the behavior of the population)
- if the virological, epidemiological, environmental and climatic factors do not change substantially
- if there is still no vaccine
then the number of "effective contacts" (i.e. those with a not negligible probability of transmitting the infection) varies approximately proportionally to the % of susceptibles present in the total number of contacts: if it is 50% (i.e. half of the people encountered are immune because healed) the number of contacts in which the infection can be transmitted is halved, if it is 25% (i.e. 75% is immune) it becomes 1/4, etc.
Indicating with "V" the % of immune persons, the % of susceptible (expressed in decimals and ranging from 0 to 1) is S = 1-V and therefore Rt = Ro x S = Ro (1-V), so Rt / Ro = (1-V) and therefore V = 1 - Rt / Ro.
Without restrictions of any kind and if the virus does not mutate (essential conditions for these algebraic passages to have an effective meaning!) this formula allows to obtain approximately the V percentage of immune through the ratio between Rt and Ro (obviously with the final uncertainty deriving from the individual uncertainties of the Ro and Rt estimates).
If we assume that Rt is equal to 1 (which means that the infection has "stabilized", each person infects a new one as an overall outcome of his "infectious state") then the number of "current infected" remains constant: a "stationary replacement" is created: for each infected that has become recovered / removed, there is a new infected: that is, a susceptible which, infected by him, becomes infected himself.
In these conditions V = 1 - 1 / Ro and, if we assume Ro = 2.5, it results that V = 1 - 0.4 = 0.6 (i.e. 60%). For the epidemic to become "stationary" (Rt = 1) the % of immune persons must be 60%.
Since, however, there are still new contagions, even in a non-expansive way, after some time the % of immune will rise more than 60% with the consequence that Rt will be less than 1 and the epidemic will gradually decrease, yet over a not particularly short period of time, unless Rt is stably very low.
It is in this way that we can understand the otherwise incomprehensible value of 60-70% given in February in the UK and DE as the minimum % of infected / healed (and therefore probably immune) necessary to obtain herd immunity, without imposing any lockdown...
4. WIDESPREAD MISUNDERSTANDING ON R0 AND Rt
It seems that understanding the real meaning of R0 and Rt presents insurmountable difficulties for a large part of journalists, politicians, regional governors, regional council members and public figures in general, which continue to crowd the talk shows making a considerable confusion, both terminological - which is marginal - and conceptual - which is more serious - because it makes it difficult to realize how the trend of contagion over time is actually determined by the intensity of the containment measures through the reduction of the factors "Infectivity" and "Number of Contacts".
An exemplary case of this misunderstanding is stigmatized on the Gimbe Foundation website (at the following link https://coronavirus.gimbe.org/press/comunicati.it-IT.html ) in which there is an interesting criticism of the improper use of Rt that, in case the episode happened exactly as reported, it would be absolutely disconcerting (euphemism ...).
I quote from the website: "The Regions, among the amendments of the Relaunch Decree, had asked to exclude the Rt parameter to measure the spread of the virus from the 21 (!) indices of evaluation of the epidemic situation" (Note: so far one could even agree, given the intrinsic uncertainty with which Rt can be estimated) but then it goes on: "Replacing it with the parameter Ro, which represents the average number of secondary cases of an index case".
For anyone who has read the previous lines, I think the amazing "not knowledge" is clear, which has made it possible to write the phrase shown in bold in an official document: to ask to replace the Rt index (which gives an idea of the current situation even if approximate) with the Ro index that no longer has any relevance because it refers to an initial situation that has been overcome for some time, demonstrates - at least in the case in question - the disconnect between the proposed decision criteria (absurdly based on the Ro) and reality (described even roughly from the Rt) that the "decision makers" should consider!
Another example is the presence in an article of an online newspaper (and not of the lesser known ones) of the following bizarre concept: "When the index Ro is less than zero, then the epidemic is over!" Apart from the fact that the Ro index does not change from its initial value (and that if it is not > 1, no epidemic can begin ...), the Rt index cannot be "less than zero" because it would mean that the total number of infected would decrease over time, but it is clear that if one has been infected, he cannot go back to "not infected", but becomes "closed" (recovered or removed), that is, he leaves the list of “current infected”, but not that of the infected by the beginning of the epidemic, the number of which can only increase or - at best - remain constant (which means that there are no more new infections): Rt becomes zero and only if it remains zero for a sufficiently long time, the epidemic can be said to be actually over...
5. CLARIFICATIONS ON A PUBLIC MISTAKE
It seems plausible that the mistake above mentioned in “Abstract” was caused by having interpreted Rt as an index linked to the "infectious dose" which, as explained when we talked about infectivity, is the one necessary to trigger the infection in a susceptible individual from an infected individual, and therefore carrier of a certain viral load.
In this case, then it would be conceivable that the simultaneous presence of a susceptible and two infected people - each of whom emitting during the meeting a little more than half of the infectious dose - would make Rt go from 0.51 to 1.02, but this would imply a complete redefinition of its meaning which would change its nature because - as illustrated above - Rt is not an "instant virological index", but instead is a "statistical index in the aftermath” (which is also affected by the viral load of an infected person, but only indirectly and in some ways that are very complex to determine).
Finally, it would be extremely appropriate, before making in public an example of a not very simple topic, to ask an expert whether it is also fitting.
Ing. G. Zurlo - 06/30/2020
Ref: CDC (Centers for Disease Control and Prevention) - High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2