Monday, May 31, 2021

The Long Term Perspectives of Nuclear Energy: Revisiting the Fermi Paradox

This is a revisitation of a post that I published in 2011, with the title "The Hubbert hurdle: revisiting the Fermi Paradox" Here, I am expanding the calculations of the previous post and emphasizing the relevance of the paradox on the availability of energy for planetary civilizations and, in particular on the possibility of developing controlled nuclear fusion. Of course, we can't prove that nuclear fusion is impossible simply because we have not been invaded by aliens, so far. But these considerations give us a certain feeling on the orders of magnitude involved in the complex relationship between energy use and civilization. Despite the hype, nuclear energy of any kind may remain forever a marginal source of energy. (Above, an "Orion" spaceship, being pushed onward by the detonation of nuclear bombs at the back).
 
 

Post revised and readapted from "The Hubbert hurdle: revisiting the Fermi Paradox" -- Published on "Cassandra's Legacy" in May 2011
 

The discovery of thousands of extrasolar planets is revolutionizing our views of the universe. It seems clear that planets are common around stars and, with about 100 billion stars in our galaxy, organic life cannot be that rare. Of course, "organic life" doesn't mean "intelligent life," and the latter doesn't mean "technologically advanced civilization." But, with  so many planets, the galaxy may well be teeming with alien civilizations, some of them technologically as advanced as us, possibly much more.

The next step in this line of reasoning is called the "Fermi Paradox," said to have been proposed for the first time by the physicist Enrico Fermi in the 1950s. It goes as, "if aliens exist, why aren't they here?" Even at speeds slower than light, nothing physical prevents a spaceship from crossing the galaxy from end to end in a million years or even less. Since our galaxy is more than 10 billion years old, intelligent aliens would have had plenty of time to explore and colonize every star in the galaxy. But we don't see aliens around and that's the paradox. 

One possible interpretation of the paradox is that we are alone as sentient beings in the galaxy, perhaps in the whole universe. There may be a bottleneck, also known as the "Great Filter," that stops organic life from developing into the kind of civilization that engages in spacefaring. 

Paradoxes are often extremely useful scientific tools. They state that two contrasting beliefs cannot be both true and that's usually powerful evidence that some of our assumptions are not correct. The Fermi paradox is not so much about whether alien civilizations are common or not, but about the idea that interstellar travel is possible. It may simply be telling us is that traveling from a star to another is very difficult, perhaps impossible. It is not enough to say that a future civilization will know things we can't even imagine. Any technology must obey the laws of physics. And that puts limits of what it can achieve. 

The problem of interstellar travel is not so much about how to build an interstellar spaceship. Already in the 1950s, some designs had been proposed that could do the job. An "Orion" starship would move riding nuclear explosions at its back and it was calculated that it could reach the nearest stars in a century or so. Of course, it would be a daunting task to build one, but there is no reason to think that it would be impossible. More advanced versions might use more exotic energy sources: antimatter or even black holes.

The real problem is not technology, it is cost. Building a fleet of interstellar spaceships requires a huge expenditure of resources that should be maintained for a time sufficiently long to carry out an interstellar exploration program - thousands of years at least. An estimate of the minimum power that a civilization needs in order to engage in sustained interstellar travel is of the order of 1000 terawatts (TW). It is just a guess, but it has some logic. The power installed today on our planet is approximately 18 TW and the most we could do with that was to explore the planets of our system, and even that rather sporadically. Clearly, to explore the stars we need much more.

Of course, we are not getting close, and we may well soon start moving in the opposite direction. John Greer and Tim O'Reilly may have been the first to note that the "great filter" that generates the Fermi paradox could be explained in terms of the limitations of fossil fuels. Because of the "bell-shaped" production curve of a limited resource, a civilization flares up and then collapses. I dubbed this phenomenon the "Hubbert Hurdle" in 2011. The hurdle may be especially difficult to overcome if the Seneca effect kicks in, making the decline even faster, a true collapse.

But let's imagine that an alien civilization, or our own in the future, avoids an irreversible collapse and that it moves to nuclear energy. Let's assume it can avoid the risk of nuclear annihilation. Can nuclear energy provide enough energy for interstellar travel? There are many technological problems with nuclear energy, but a fundamental one is the availability of nuclear fuel. Without fuel, not even the most advanced spaceship can go anywhere.
 
Let's start with the technology we know: nuclear fission. Fissile elements (more exactly, "nuclides") are those that can create the kind of chain reaction that can be harnessed as an energy source. Only one of these nuclides occurs naturally in substantial amounts in the universe: the 235 isotope of uranium. It is a curious quirk of the laws of physics that this nuclide exists, alone. It is created in the explosions of supernova stars and also in the merging of neutron stars. It has accumulated on Earth's surface in amounts sufficient for humans to exploit to build tens of thousands of nuclear warheads and to currently produce about 0.3 TW of power. Fission could power a simple version of the Orion spaceship, but could it power a civilization able to explore the galaxy? Probably not. 
 
The uranium reserves on Earth are estimated at about 6 million tons. Currently, we burn some 60.000 tons of uranium per year to produce 0.3 TW of energy.  It means we would need 200 million tons per year (600,000 tons per day) to stay at the 1000 TW level estimated as needed for interstellar travel. At this rate, and with the current technology, the reserves would last for about 10 days (!)
 
This is no surprise: it was already known in the 1950s that the uranium reserves would not have been sufficient even to keep our current civilization going using the fission of U(235) nuclides. Imagine engaging in the colonization of the galaxy! But, of course, we know that we are not limited to U(235) for fission energy. There also exist "fissionable" nuclides that cannot sustain a chain reaction, but that can be turned ("bred") into fissile nuclides when bombarded with neutrons (usually generated by fissile isotopes). We never deployed this technology, but we know that it can work at the level of prototypes. So, in principle, it could be expanded and become the main source of energy for a civilization.
 
The naturally occurring fissionable nuclides are isotopes of uranium and thorium: U(338) and Th(232), both much more abundant than U(235). Let's say that, using these nuclides, the efficiency of energy production could be increased by a factor of 100 or 1,000 in comparison to what we can do now. But, even in the most optimistic estimate, at an output of 1000 TW, we would simply pass from 10 days of supply to a few decades. No way!

We can think of ways to find more uranium and thorium, but it is hard to think that bodies in the solar system could be a source. You need an active plate tectonic condition in order for geological forces to accumulate ores and, in on bodies such as the Moon and the asteroids, there are no uranium ores. Only extremely tiny amounts, of the order of parts per billion. And that makes extracting it an impossible task. We also know that there are some 4 billion tons of uranium dissolved in seawater, an amount that would change the game, at least in principle. But the hurdles are enormous: uranium is so diluted that you are thinking of filtering quintillions (10^18) of tons of water to get at those huge amounts. Would a planetary civilization destroy its oceans in order to build interstellar spaceships? 

Maybe we can stretch things in more optimistic ways, but within reasonable hypothesis we remain at least a couple of orders of magnitude short of what is needed. Fission is not something that can sustain an interstellar civilization. At most, it can sustain a few interstellar probes, just like fossil fuels have been able to create a limited number of interplanetary probes. (BTW, the Oamuamua object might be one of these probes sent by an alien civilization). But, sorry, no fission-based galactic empire

There is one more possibility: nuclear fusion, the poster child of the Atomic Age.  The idea that was common in the 1950s is that nuclear fusion was the obvious next step after fission. We would have had energy "too cheap to meter." And not only that: fusion can use hydrogen isotopes and hydrogen is the most abundant element of the universe. A hydrogen-powered starship could refuel almost anywhere in the galaxy. Hopping from a star to another, a fusion-based galactic empire would be perfectly possible. 
 
But controlled nuclear fusion turned out to be much more difficult than expected. In more than half a century of attempts we have never been able to get more energy from a fusion process than we pumped into it. And, as time goes by, the task starts looking steeper and steeper.
 
Maybe there is some trick that we can't see now to get nuclear fusion working; maybe we are just dumber than the average galactic civilization. But we may have arrived at a fundamental point: the Fermi paradox may be telling us that controlled nuclear fusion is NOT possible.

All this is very speculative, but we arrived at a concept completely different from the one that is at the basis of the Fermi paradox: the idea, typical of the 1950s, that a civilization keeps always expanding and that it rapidly arrives to master energy flows several orders of magnitude larger than what we can do now (sometimes called the "Kardashev Scale."). 

Maybe we'll arrive to exploit solar energy so well that we'll be able to use it to build interstellar spaceships, but we are talking of a future so remote that we can't say much about it. For the time being, we don't have to think that the Fermi Paradox is telling us that we are alone in the universe. It just tells us that we shouldn't expect miracles from nuclear technology.
 

Monday, May 24, 2021

The Great Turning Point for Humankind: What if Nuclear Energy had not been Abandoned in the 1970s?

  The Italian translation of Walt Disney's book, "Our Friend, the Atom," originally published in 1956. It was a powerful pitch of the nuclear industry to sell a completely new energy system to the world. It could have been a turning point for humankind, but it didn't work: nuclear energy was abandoned in the 1960s-1970s. It was probably unavoidable: too many factors were staked against the nuclear industry. But we may wonder about what could have happened if it had been decided to pursue nuclear energy and abandon fossil energy. (In the background: a completely different concept, that of "holobionts,")


I remember having read Walt Disney's book, "Our Friend, the Atom," (1957) in the 1960s when I was, maybe, 10 years old. That book left a powerful impression on me. Still today, when I visualize protons and electrons in my mind, I see them in the colors they were represented in the book: protons are red, electrons are blue or green. And I think that one of the reasons why I decided to study chemistry at the university was because of the fascinating images of the atomic structure I had seen in the book.

More than 60 years after its publication, "Our Friend the Atom" remains a milestone in the history of nuclear energy. You can easily find on the Web the Disneyland TV episode from which the book was derived. It is still stunning today in terms of imagery and sheer mastery of the art of presentation. The nuclear industry was in rapid expansion and it saw itself as able to grow more. Hence, a pitch for the "Atomic Age" that would have brought cheap and abundant energy for everyone, perhaps even energy that was "too cheap to meter." 

It didn't work. You see in the figure the number of new reactors installed worldwide. It peaked around 1970, and plans to build new reactors must have been declining earlier than that. Already in the 1960s, the enthusiasm for nuclear energy was falling, a trend that would last until now, despite some recent signs of a possible restart. (image from Univ. Texas)

What went wrong? Today, the whole story is usually dismissed as the result of the machinations of those evil Greens who had opposed nuclear energy for ideological reasons. Yet, the popular "smiling sun" campaign didn't become widespread before the late 1970s, when the nuclear industry was already in free fall. Never in their history, the Greens had been able to stop an industrial field that was making money. Why should they have been so successful with the nuclear industry? (by the way, with a campaign that started at least a decade after that the intended target had begun its decline. Those damn Greens even had time machines!)

Reviewing this old story, we see that the smiling sun campaign was not the cause, but a symptom of the troubles that the nuclear industry was facing. Up until the 1950s, the industry had prospered almost exclusively in the military market, producing mainly nuclear warheads. The production of electric power for the civilian market was a side job, just like the production of isotopes for research and for medical applications. The problem was that warheads were being stockpiled in absurd numbers, well beyond the reasonable needs (if we want to use that term) of national defense. 

It must have been clear already in the 1950s that the industry was saturating its market. The only solution to stimulate the demand was to start an actual nuclear war. Surely, it must have been considered but, fortunately, not everyone agreed on that idea. 

But where to find new markets for the nuclear industry? With already so many nuclear weapons around, a possible solution was to move into the civilian market and to expand outside the US national boundaries. In the 1950s, the US engaged in a program that started with the speech by President Eisenhower known as "Atoms for Peace" in 1953. The idea was to disseminate nuclear technology all over the world as a way to produce energy and other useful products. Walt Disney's 1956 movie was an offshoot of this program.

Seen in retrospective, the "Atoms for Peace" program couldn't possibly have worked, and it didn't. The nuclear industry faced a series of hurdles, each one sufficient to stop its growth, alone. All together, they were truly too much. Here is a list.

1. A mineral resource problem. In the 1950s, it was already known (*) that the mineral reserves of fissile uranium, the 235 isotope, were insufficient for nuclear plants to take over the task of energy production worldwide. That could have been possible only by means of the new and scarcely tested technology of "breeding." A few attempts were made to build commercial breeding reactors, but they were victims of the general rule that everything always costs more and takes more time. Gradually, the funds needed to keep developing the technology dried out and the efforts stopped. The best known of these reactors, the French "Superphenix" was closed in 1996, but it was clear much earlier that it had not been a success. No breeders, no atomic age.

2. A pollution problem. In the 1950s, nuclear waste was not seen as a major problem, but it was also clear that a substantial increase in the number of nuclear reactors would have created the necessity of doing something with the radioactive waste. And it started to be understood that dismantling the old nuclear reactors after the end of their life was a long and expensive task. Some of the waste would require centuries or millennia to become inoffensive. And, in all cases, the costs involved were huge and who was going to pay? The question was never answered at that time, and it remains unanswered today.

3. A commercial problem. Electrical energy from nuclear reactors always remained more expensive than the energy produced by gas or coal. So, the production of energy for the civilian market needed to be subsidized to be competitive. Up to 1977, subsidies were provided indirectly by the military industry with the purchase of the plutonium produced by the reactors, used to make nuclear warheads. These subsidies were abolished by president Carter in part because the US had already too many warheads, and in part to avoid the proliferation of fissile material. At this point, the industry was not any more competitive and who would invest money in a non-competitive industry? 

4. A competition problem. In the 1960s, the concept of "hydrogen economy" started becoming popular. For the nuclear industry, it seemed to be a good idea to claim that they could produce not only electric power, but also a fuel that could power vehicles. Unsurprisingly, that put the nuclear industry in direct competition with the fossil fuel industry. We know that everyone tends to defend their turf when it is threatened and we can't imagine that the fossil industry would supinely accept to be superseded. By the late 1970s, an aggressive public relations campaign based on the "smiling sun" symbol had turned nuclear power into everyone's bugaboo. Probably we will never know who financed that campaign, but we know who benefited from it.

5. A strategic problem. The idea of "atoms for peace" was complete nonsense in strategic terms. It just put the US in an impossible strategic quandary: how to stop nuclear proliferation while at the same time disseminating nuclear technologies all over the world? The solution was to quietly forget about atoms for peace while aggressively stopping the construction of nuclear reactors everywhere, especially in countries believed to be strategically unreliable. In 1981, the "Tammuz" reactor under construction in Iraq, near Baghdad, was destroyed by the Israeli air force. In 1987, a referendum against nuclear energy was held in Italy, a country believed to be at risk as an ally of the US because of the presence of a large Communist Party. The referendum forced the Italian government to dismantle four already built reactors and never to engage again in nuclear energy production. Iran continued the nuclear program that had been started with the "atoms for peace" program, but it was sabotaged at every step. From the 1980s onward, it became clear that not only nuclear weapons but also nuclear energy was something that belonged only to a selected club. 

 

You see that, as usual, when something must happen, you cannot stop it from happening. That the nuclear industry was to fail was written on the wall of the reactors because of a series of factual circumstances, surely not because a bunch of long-haired Greens were protesting in the streets. Yet, it is not impossible to think that history could have followed a different path. 

Imagine that the US military leaders had stomped their feet on the ground and said, "we are going to have breeders in America." Imagine that sufficient funds could have been funneled into the task. Finally, imagine that the technological problems of breeders could have been solved. At that point, the US and the whole Western World could have switched to a largely nuclearized energy system, possibly including a hydrogen-powered transportation system. It is unlikely that China and the Soviet Union would not have followed along the same path. And it would have been difficult to stop nuclear technology from diffusing in other regions of the world. It would have been the "Atomic Age" that was dreamed of in the 1950s.

What kind of world would that be, today? Theoretically, we would much more energy than we have today, at least for the elite countries that had embarked on the nuclearization of their economies. And this energy could be produced without emitting greenhouse gases into the atmosphere, so that Earth's climate would not have been affected, at least not directly.

But we would have faced a completely different range of problems. With the Atomic Age, the amount of fissile material available in the world would have been multiplied by one or two orders of magnitude and it is almost unthinkable that it would stay forever out of the hands of the many petty tyrants, fanatical religious leaders, and assorted psychopaths who tend to crave for that kind of things. 

Consider also that nuclear plants (especially breeders) offer a delicious target for military and terrorist attacks not just for their strategic value but also for the possibility of spreading radioactive material around and making large areas of the targeted territory uninhabitable. So, you may imagine what kind of problems we could have today. Even for a limited nuclear exchange, the "nuclear winter" scenario, proposed in the 1990s, implied a cooling period sufficiently long to exterminate most of humankind. The idea was heavily criticized, but never really debunked. And that without mentioning the possibility of the mismanagement of the nuclear wastes and the fact that plutonium is among the most poisonous substances known to humans.

Consider also another problem, much bigger, that lurks unrecognized in the shadows for the atomic age scenario. In the 1950s, Marion King Hubbert was working on oil depletion and in 1956 he proposed his famous "bell shaped" production curve, later known as "peak oil." Hubbert also proposed that nuclear energy would replace fossil fuels. But note in the figure below how, in his view, nuclear energy would not have prevented "peak oil" from taking place at about the same time that was foreseen without nuclear energy.  Hubbert understood very well that the enormous effort needed to build the new nuclear infrastructure would have had to be based on fossil fuels, and so would not have reduced their production.


Now, note something in the image: whereas fossil fuels follow a bell-shaped production curve, nuclear energy reaches a plateau and remains there for thousands of years. Why?

Hubbert must have been well aware that the "thousands of years of supply" that the nuclear industry often claimed for the mineral reserves of uranium were possible only if production were not to increase over a certain rate. But what would have stopped people from increasing energy production even more? You think that people would have been thinking, "now we have enough" and then spend their time relaxing? One world: pyramids. 

Why wouldn't Plutonium follow the same trajectory of oil, a "bell-shaped" curve, peaking and starting to decline afterward? (Want to mention thorium? Sure, but it is another finite resource, it doesn't change the concept). So, it would grow, peak, and then decline.

It is impossible to calculate when "peak plutonium" could take place in a fully nuclearized world. It would depend on many factors, the available resources, the efficiency of the breeding technology, the energy return on investment, the cost of waste management, and more. In a previous post, I made some very rough estimates: if the plutonium-based economy were to be run on the known laws of the economy, it is hard to imagine that the reserves of fissionable materials would last for more than a few centuries, possibly even less than a century. (Fusion? Sure, let's wait 50 more years and....).

And here we stand. Playing the "what if" game is a lot of fun, but we should remember that we are talking about the dream expressed by Walt Disney's "Our friend, the Atom," A dream that, likely, had the same chances to turn into reality as others proposed by Walt Disney, such as for a poor country girl to marry a prince. And it is not at all guaranteed that the country girl would have a happy marriage!

We don't know if a plutonium-based economy ever was something more than a dream. Today, it is too late to turn back to a moment in history that is past and gone, although it is not impossible that someone will want to try to resurrect a dream that could easily turn into a nightmare. 

What we know is that, as always, we stand at the intersection of past and future, in that fleeting moment we call "present." From now on, infinite possibilities branch out. Those leading to a peaceful and prosperous future are few, maybe there are none. But we must plod on. It is a journey that will lead us somewhere, even though we can't say where.


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(*) The story of the assessment of the uranium reserves is fascinating in itself. Palmer Putnam published in 1953 the book "Energy in the Future" where he carried out one of the first quantitative assessments of the potential of fission energy in terms of mineral reserves of uranium  See below the relevant paragraph
 
 

 
 
Note the key words: "assuming breeding." That is, the assumption is that energy can be extracted from both isotopes of uranium, the 235 and the 238. The result is 1700 Quads, or about 100 times the energy content of the (then) known oil and gas reserves. 
 
You understand why in the 1950s it became obvious that breeding plutonium was absolutely necessary for a nuclear-based economy. If only U235 were to be "burned," then the resource would be suddenly reduced to 0.72% of the total, that is to 12 Quads. Assuming an optimistic 30% efficiency (but, really, way too optimistic), the total obtainable would be 4 Quads. Earlier on, Putnam had established that the world would require more than 70 Q of energy by the year 2000. No breeding, no atomic age. Simple.


Friday, May 21, 2021

The Rt Factor in the Pandemic: Is it Useful for Anything?


by Ugo Bardi, 

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. “And it is also said that when Rt is greater than 1, the epidemic grows. If it is the other way around, Rt is less than 1, the epidemic decreases (If Rt = 1, the epidemic is stable).  

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 need also 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 everywhere)

Now, I understand that those who are specialists in a certain field tend to keep in the dark those who are not. But, here, it seems to me that they are a little exaggerating. Anyway, 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 -- let's say that they are all eaten by foxes. Take the ratio of the births to the deaths and you have the net reproduction rate: if more rabbits are born than are eaten, 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. 

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. It is a simplified epidemic that allows us to describe it by a simple mathematical model, even though it has the main characteristics of a real epidemical disease.
 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 the bluite arrived on Earth from the blue (indeed!). Let's also assume that bluite is a 100% benign disease. That is, it does not cause unpleasant symptoms and does not kill anyone. Hence, no one takes special precautions against it. Let's also assume that those who have been infected become immune forever, or at least for a long time. But their skin remains slightly grayish for some timeFinally, let's assume that bluite has a very short infection cycle: in one day it passes and this applies to everyone. It would also be fine to consider a different cycle length, for example, a week, but let's keep this short duration as an example.

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 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 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 bluitis 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 that the epidemic has already started and that 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.

Here are some recent data (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