*In
these notes, I do not intend to replace the epidemiology specialists,
my purpose is informative and tries to provide
some data and some useful information to everyone in this situation,
where the pandemic has become more a political issue than a scientific
one. So, if we are to make
informed decisions, we need to have the tools to understand what we are
talking about, very difficult in the current cacophony of
data and reasoning. Here, I
have done my best to clear up the Rt factor issue using as an example a
hypothetical epidemic, "bluite", which causes you to turn blue like the
characters in the movie "Avatar". *

*Note, this post was translated and adapted from my Italian blog Medio Evo Elettrico." It still contains references to the Italian situation. But I think most of it is of general interest.*

You surely noticed how in the discussions about the pandemic, the "*R* factor"
is very popular. This factor, expressed as "*Ro*" or "*Rt*", seems to give us useful information in a simple form, and we all know that politicians are always looking for simple solutions to complicated problems. And it is also based on the *Rt* factor that many governments decide on their restriction policies.

However,
I bet that neither the politicians nor many of the tv-virologists who populate the media really understand what exactly this *Rt* factor is. In the real world, things are never simple and the *R* factor is not an exception to the rule. As
Professor Antonello Maruotti (1) noted the use of the Rt factor could result in a "persistent blindness on the part of political decision-makers."

So what exactly is this *Rt*? How is it determined? How useful is it? And is it really a parameter on which it is worth basing all the restrictions policy that the government is doing? Let's try to understand how things stand.

A definition you can easily find all over the Web is that the *Rt* factor is “ *the average number of people infected by an already infected person over a certain period of time."*

There is a big problem, here. If we take the definition literally, it means that the epidemic can never go down. If there is at least one infected person, it will always infect someone else, and so the epidemic will grow forever. Clearly, the definition above is incomplete. We also need to take into account the people who recover (or die) in the time interval considered.

The matter is made more complex by the fact that in epidemiology there are two similar terms, one is called *Ro* and the other *Rt*. To give some idea of the confusion, read the Wikipedia article on *Ro*, where you'll find that the definition of *Ro* "is not universally shared" and that "The inconsistency in the name and definition of the parameter *Ro* was potentially a cause of misunderstanding of its meaning." In short, a nice mess, to say the least.* (this is from the Italian version of Wikipedia. The English one is better, but confusion reigns anyway)*

Now, I understand that those who are specialists in a certain field tend to keep in the dark those who are not. But it seems to me that they are a little exaggerating, here. So, let's try to extricate ourselves from the various involved and misleading definitions, the best way to understand this story is to consider that a virus is a living creature so that biological laws and definitions apply. So, *Rt* in epidemiology is nothing else but the **net reproduction rate** parameter in biological populations.

This is an easily understood concept: take a population (say, rabbits). Consider the number of bunnies born for each generation: that's the "reproduction rate." Then, consider the number of rabbits that die in the same period of time because they are old, or they are eaten by foxes. Take the ratio of births to deaths and you have the *net* reproduction rate: if more rabbits are born than die, it must be* Rt> 1*. It is the opposite if *Rt <1*. A virus population is no different from a rabbit population in terms of growth or decline. Viruses multiply when someone infects someone else, but they die when someone is healed (or dies).

How about *Ro*? It is simply the net reproduction rate at* t=0*, that is at the very beginning of the epidemic when there are no recovered and immunized individuals.

These are the basic points. Then, it is always easier to understand something when it is expressed in terms of a
concrete example, so let me propose an explanation based on a hypothetical epidemic that I call "*bluite*." There is some math in the following, but if you are willing to spend some time on that, you can develop a good "mental model" of how this *Rt* factor works.

**The "bluite": a simplified epidemic example**

Let's imagine a hypothetical infectious disease that is transmitted by contact, let's call it "

*bluite*" because it makes you turn blue. Incidentally, a disease that turns people blue really exists, it's called "

*argyria,*" the result of being exposed to silver salts. Some people ingest silver as an alternative therapy for certain diseases, not a good idea unless you want to find a job as an actor on the set of a science fiction movie. But let's not go into this, in any case, argyria is not infectious.

So, let's imagine that we counted, on a certain day, the number of blue-skinned people passing by on the street. Let's say that we counted 1000 people and that 10 of them had blue faces. If the sample is statistically significant, we can say that 1% of the population is infected. If we extrapolate to the whole population, suppose we are in Italy with 60 millions of inhabitants, it means that there are 600,000 people infected with bluite. This fraction is called "prevalence" in the jargon of epidemiology.

So far, so good, but that doesn't tell us anything about how the epidemic is evolving. For this, we need data measured as a function of time. Let's assume then that we do the same measurement again the next day. We find that there are now 20 blues, again out of 1000 people: this number of new infections in a certain period is called the "incidence." In this particular case, since the infection lasts one day and we make one measurement per day, the incidence is equal to the
prevalence.

Can we now measure the *Rt* factor? Sure. We said that *Rt* is the net reproduction rate of the population. So, over a one day interval, we have 20 newly infected people, but 10 people recovered in the meantime. **It follows that Rt = 20/10 = 2. **Easy, isn't it? (note that I chose the data in such a way as to have a nice round number as the result).

Easy, but you have to be careful when you extrapolate this procedure. At this point, you could say that if in one day the number of infected people have doubled, their number will continue to double every day. That is, 10, 20, 40, 80 ... etc.

This is the mistake made by those who speak of the "exponential growth" of the epidemic; it is an acceptable approximation only in the very early stages of diffusion. Do some math, and you will see that if the number of cases of bluite were to double every day, in a week, there would be more people infected than the whole population. Slightly unlikely, to say the least.

The mistake here is to confuse the *net* reproduction rate (*Rt*) with the (simple) reproduction rate. They are not the same thing: the former is the growth rate of the population, the latter is the probability that a "blue" has to infect a "normal" when an encounter takes place. In general, we cannot directly measure the reproduction rate, we can only estimate it. Just to propose some numbers, let's assume that, on average, everyone in the population encounters 4 people every day at a close enough distance to infect them. Since there were 10 blues in the beginning, and 20 new ones came out, it would seem that the probability of infection at close range was 50% for each encounter. But is not so.

Not all people a blue encounters are "normal," that is susceptible to infection. We said that there were 10 blues in the population when the measurement was made and we may also assume that there were 10 grays (previously infected, now immune). It follows that only 98% of the population are susceptible ("normal") people. So the probability for a blue to infect someone is *not *50%. It is 0.5 / 0.98 = 51%. It's a small difference, but it's the key to the whole story.

To understand this point, first let's estimate the value of *Ro*, when the first blue alien from the planet Pandora landed and began infecting Earthlings. At that time, the whole population (100%) was susceptible to infection. Since we found that the simple reproduction rate is 0.51, it follows that ** Ro = 0.51x4 = 2.4**. This was the initial value of the net reproduction rate when the epidemic had just begun.

But *Ro *has to do with the past, let's instead calculate how things are expected to develop in the future. The next day, the 20 infected people will each interact with 4 people, and a total of 80 people will be exposed to the virus. Not all of them will be susceptible, the number will be equal to 1000 (total number of people) - 20
(the blues of the day) - 20 (the grays of the previous days) divided by the total population. That is 960 people, or a fraction of 96%. It follows that the 20 infected people will generate 20 * 0.51 * 4 * .96 = 39 new infected individuals and not 40, as it would have been the case if the number of infected people had remained constant. At this point, *Rt* has shrunk to 39/20 = 1.96. You can see that *Rt* will shrink a little every day that goes by.

From here, you can have fun doing a calculation with an excel sheet, but I did it for you. Here are the results, the red curve is a fitting with an asymmetric sigmoid curve:

Note how the curve of the daily infections (red) has the typical “bell shape" of epidemic curves (mathematically, it is the same as the "Hubbert Curve" in petroleum extraction). Note also that we didn't assume that the infection was cured or that there were precautionary measures in place: distances, face masks, nothing like that. Infections go to zero simply because fewer and fewer people remain susceptible.

In this particular case, the number of people who contracted the infection stabilizes at around 74% of the total at the end of the epidemic cycle. The rest will never be infected. Do you see how “herd immunity” works? **Over a quarter of the people in the population do not become infected**, even though the virus was highly infectious at the beginning and no one took precautions of any kind. It is an intrinsic property of the spread of an epidemic.

Notice also how the curve for *Rt *always goes down, at least in this simplified case. You see that when the epidemic is at its peak, *Rt* is equal to one. Eventually, it stabilizes around 0.5. Depending on the various parameters, it can stabilize on different values, but always less than 1.

** **

**Effect of restrictions on bluite**

Now let's have a little fun using this model to see the effects of restrictions. The idea of things such as "social distancing" or face masks is that they reduce the likelihood that the virus will be transferred from one person to another. This is sometimes called "crushing the curve".

First, let's plot again the results we obtained above without assuming any restrictions.

Now let's try to reduce the likelihood of the infection by 25% by some unspecified method. Here are the results

You see that the curve is indeed "crushed". But also note that the duration of the outbreak is longer and that the final value of Rt, contrary to what one might expect, increases slightly instead of decreasing. As for the total number of infected people, the restrictions have reduced it from 74% to about 58% of the population. If we assume that the effect of the restrictions is even greater, say to 50%, we can squeeze the curve even further and reduce cases to about 15% of the population. By further reducing the likelihood of infection, the epidemic just doesn't develop. Finally, note that this is the result of having imposed the restrictions from the start of the epidemic cycle and of maintaining them for the whole cycle.

Let's now try to see what happens if, as it is more likely, the restrictions start at some moment after the epidemic has already started and they are maintained for a limited time window. In the graph below, restrictions with a 25% reduction effect are assumed to have been put in place on the third day, and reopening occurs on the ninth day.

Notice that the contagion curve more or less retains the "bell shape," although it is now a bit skewed. Instead, the

*Rt*factor shows fairly sharp discontinuities. Note also that the infection lasts longer.

**We have reduced the intensity of the outbreak in exchange for a longer duration**. In these assumptions, the total number of cases is intermediate compared to the two previous examples: the number of infected people stands at 67%.

You can have fun by changing the parameters, but the results can be summarized by noting that using restrictions to bring the infection curve to zero is almost impossible. The effect of the restrictions is seen as a discontinuity in the Rt factor curve better than in the contagion curve.

**The real world**

All this applies to a hypothetical epidemic that we have called bluite and to a simplified model. In the case of a real epidemic, the situation is more complex, but the results are not very different. The basic prediction of the model, that of the "bell" shape of the contagion curve, is confirmed by real-world data. In the figure, we see an example, a recent cholera epidemic in Kinshasa, Congo.

In this, as in many other real cases, we see a "bell-shaped" curve. Note how the number of cases never really goes to zero, contrary to what the model predicts. The pathogen becomes "endemic", ready to return to the scene when it finds favorable conditions to start over.

What can we say about *Rt* in the real world? Here, the calculation is much more complex than for the hypothetical bluite. The infection does not have a fixed duration and it is also possible to get re-infected. Then there are the various uncertainties in determining the number of infected people, the delays with the availability of data, the effects of mutations, and more.

The result is that calculating *Rt* for an ongoing epidemic is a complex matter that is left to specialists. With these methods, the prediction that *Rt* should fall with time during each epidemic cycle is generally
verified, but it is also true that many epidemics have multiple cycles,
so the *Rt* factor can also reverse its trend and restart growing for a certain period.

*for Italy*) from Maurizio Rainisio's FB site (2). Here, you see an equivalent of

*Rt*(which Rainisio calls the "Weekly Growth Rate"). The epidemic had two phases, probably due to seasonal factors, or perhaps also to the effect of the "variants" of the virus. Notice how the peak of the most recent phase corresponds to

*Rt = 1*.

Here, it is very difficult to see an effect of the various red, orange, yellow, etc. zones (*as they were created in Italy*). For example, *Rt *showed a steep rise at the beginning of February 2021,
while it started to decline around February 20. Is there a correlation with any specific action taken by the government that can be seen in the curve? Maybe, but it is certainly weak.

**Conclusion: is Rt any good?**

The usefulness of something always depends on the context. A submachine gun can be very useful in certain circumstances, but it's a bad idea if it's in the hands of a Taliban,
especially if there's a tv shop nearby. **This also applies to statistical models if they end up in the hands of people who don't understand them.**

Thus, in the first place, the calculation of the * Rt* factor

**does not give you, and could never give you, any more information than what is already present in the curve of the trend of the epidemic.**We saw that epidemic curves tend to have a "bell" shape so that it is possible to qualitatively understand whether the epidemic increases or decreases simply by the shape of the curve. The calculation of the Rt factor may be more sensitive to the trend, but it adds no more information.

Then
there is the problem that the value of Rt can tell us if the epidemic grows or declines, but nothing about the number of infected people. Clearly,
there is a big difference if we have 100 infected people out of 1000 or if we only have 10, but the value of *Rt* could be the same. And this is not a detail: depending on the absolute value of the number of infections, hospitals may or may not risk becoming saturated. But **the Rt factor, alone, tells us nothing on this point.**

Above all, when the infected are few, the importance of the inevitable measurement errors and approximations changes (3). If you have 100 cases out of 1000, an error of a few units has little effect: whether they are 101 or 99, nothing changes. But if you have two cases on a certain day, while you had just one the day before, you would think that *Rt* is much larger than 1, and you should sound the alarm**. In this case, the sensationalism of the media is a big problem. **And so you could find yourself shutting down an entire country because of a statistical fluctuation.

But the biggest problem is precisely in the concept. As I said before, many people don't understand how an epidemic mechanism works and **truly believe that an epidemic grows exponentially until everyone is infected. **And,
consequently, they are convinced that if we see that the contagion curve decreases, this is due solely and only to the restrictions. You find it explicitly written, sometimes: "the Rt factor measures the effect of the containment measures". But this is absolutely not the case!

Not that there is no way to slow down an ongoing epidemic! Vaccines, for example, force the achievement of immunity in individuals and cause herd immunity to be achieved more quickly. But if you see the epidemic waning or rising, you don't necessarily have to relate it to restrictions or vaccines alone. **The epidemic has its own cycle, you can slow it down, but you have to take that into account.**

Unfortunately,
the debate has arrived at the conclusion that the only thing
(aside from vaccines) that can stop the epidemic are restrictions. **And the restrictions have a huge cost** not only on the economy but also on the health of citizens. But until we think about it we will continue to insist on measures that may be exaggerated and not justified in comparison to the costs.

In essence, the problem is that **many people, even among policymakers, cannot read a Cartesian graph** and have no idea how an epidemic cycle works. So, they tend to rely on a single magic number, "*Rt*" for simplicity. But the situation does not lend itself to extreme simplifications and, as always, ignorance pays only negative dividends.

References

1. https://www.romatoday.it/attualita/coronavirus-professore-lumsa-sbagliate-decisioni-su-rt.html

2. https://www.facebook.com/La-Peste-111172767208456

3. http://www.radiocora.it/post?pst=39381&cat=news