Epidemiological Models: A Simple Explanation of How they Work

There is a certain logic in the way the universe works and so it is not surprising that the same models can describe phenomena that seem to be completely different. Here, I'll show you how the same equations describe chain reactions that govern such different phenomena as the spread of an epidemic, the cycle of extraction of crude oil, and even the nuclear reaction that creates atomic explosions. All these phenomena depend on the efficiency of energy transfer, the parameter that's known in energy studies as EROI (energy return on energy invested), related to the "transmission factor" (R) of epidemiological models. Above, a classic clip from Walt Disney's 1957 movie, "Our friend, the atom." 

You may be surprised to discover that epidemiological models share the same basic core of peak oil models. And it is not just about peak oil, the same models are used to describe chemical reactions, resource depletion, the fishing industry, the diffusion of memes on the Web, and even the nuclear chain reaction that leads to nuclear explosions. It is always the same idea: reinforcing feedbacks lead the system to grow in a frenzy of exploitation of an available resource: oil, fish, atomic nuclei, or people to be infected. In the end, it is perhaps the most typical way the universe uses dissipate potentials. As always, entropy rules everything!

Modeling these phenomena has a story that starts with the model developed in the 1920s by Vito Volterra and Alfred Lotka. They go under the name of "Lotka-Volterra" models or, sometimes, "Prey-Predator" models. This heritage is not normally recognized by people in the field of epidemiology, but the model is the same: the virus is a predator and we are the prey. The only difference is that an epidemic cycle is so short, typically a few months, that the prey, people, don't reproduce during the cycle. Then, if you think that oil companies are predators and oil fields are the prey, then we have again the same model. Finally, you can see the atomic chain reaction that takes place during fission as generated by neutrons acting as predators and atomic nuclei acting as prey. In the Walt Disney interpretation, shown in the clip above, ping-pong balls are the predator and mousetraps are the prey.

 

To describe the model, let's focus on epidemiology. These models are called "SIR," with the acronym standing for "Susceptible, Infected, Recovered." The idea is that the Infected stock grows proportionally to both the Susceptible and the Infected stocks -- it is a feedback loop. No feedback, no growth, this is how these models work. Then, of course, the virus will gradually run out of susceptible people, growth will slow down and, eventually, the infected stock will start declining. Then, the epidemic will be over.

So, let's see what the model produces in its simplest version. I made it using the Vensim (TM) system dynamics package (see at the end of the post for the details *)

Note how the number of susceptible people (blue curve) gradually declines. Instead, the number of cases per unit time (green curve) and the total infected people (red curve) show a cycle of growth and decline. Finally, the recovered people (gray curve) grow and then stabilize. (they might also die, the equations won't change.)

Let's compare with peak oil models: the names of the variables change, but the model is the same

Susceptible  --> Oil Resources

Infection rate --> oil production
Infected --> Extracted Oil
Recovered --> Pollution

Note the green curve in the figure. It is symmetric and bell-shaped: it is the typical "Peak Oil" curve. In the case of oil, the curve describes the production in barrels per day. In the case of an epidemic, it describes the number of new cases of infections per day. The curve for the victims should be the same, but (hopefully) smaller and shifted forward in time to take into account that you die after having contracted the virus. The red curve in the figure is proportional to the amount of oil extracted and not yet burned. It is the "capital" of the oil industry. As oil is burned, it becomes pollution and disappears from the model 

You can play the same game with other phenomena. For instance, in the case of the "mousetrap model" developed by Disney studios, the one shown in the clip at the beginning of this post, you have that

Susceptible --> trapped balls
Infection rate --> number of traps springing per unit time.
Infected --> number of flying balls
Recovered --> balls on the ground

In general, epidemiological models are normally much more complicated than the basic SIR model that I showed above. That is, in my opinion, a weakness of these models. Attempting to evaluate such parameters as how many people will contact each other per day, and from that estimating the infection rate is nearly hopeless and, indeed, these models have a poor record in terms of quantitative forecasting. Even peak oil models, although not so bad, turned out to be unsuccessful in estimating the data of the peak, at least in terms of volumes of liquids produced.

But this is a long story and I won't get into it, here. Let me just say that, in general, models may be useful even (and perhaps especially) when you don't ask them to make exact predictions. Often, a correct warning may be much more useful than an incorrect prediction. That's true when the models are well-grounded in physics and can tell you what will happen, even though not necessarily when.

Something that you can learn from these models is how the behavior of the system is determined by an efficiency parameter called R in epidemiology and EROI (energy return on energy invested) in peak oil studies. Yes, these two parameters are the same -- apart from some details. They share the property that they need to have a minimum value in order for the chain reaction (the epidemics or a cycle of extraction) to start. In epidemiology, you can show that R must be >1 for the infection to grow. As the epidemic proceeds, R becomes smaller. When R=1, you have the "peak virus" and the number of infected people starts declining. That's called "herd immunity."

Things are not so simple for the peak oil curves, but the story is the same. You can show that an energy-producing resource cannot be produced with a positive energy yield unless you have EROI=1/η at the beginning of the extraction cycle, with η the efficiency of the transformation of the energy of the extracted resource into useful energy (exergy). For crude oil, we may probably take η as equal to 0.1-0.2. The implication is that oil extraction is not viable for EROI<5-10, which is consistent with the current situation. We are close to EROI values that correspond to an unavoidable decline of the industry. (Note that this condition is for "peak capital" -- "peak oil" comes for even larger values of the EROI)

Ah... by the way, these limits of the EROI values are valid only for exhaustible resources such as crude oil. They do not hold for renewable energy sources such as solar energy -- of course, you can't run out of sunlight!
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The R and the EROI parameters of chain reaction models -- a quick explanation


Ro is defined as the expected number of cases generated by one case in a population where all individuals are susceptible to infection, that is, at the initial stages of the epidemic. As the epidemic proceeds, varying proportions of the population become immune. To account for this, the "effective reproduction number" is used, written as Rt or simply R. It is the average number of new infections caused by a single infected individual at time t. When the fraction of the population that is immune increases so much that R drops below 1, it is said that "herd immunity" has been achieved. It means that the number of infected people does not grow any longer and gradually decreases toward zero.

The EROI (or EROEI) (energy return of energy investment) factor in oil extraction is defined as the number of barrels of oil produced using the energy obtained from one barrel. It is more general than that, but let's remain with crude oil. Obviously, when the EROEI goes below one, the whole enterprise of oil extraction becomes useless in terms of producing useful energy. But "peaking" of oil production starts well before the EROI goes below one, as we'll see in the following.

R and EROI look similar and, indeed, they are the same thing. To say something more about this matter, we need to write down the equations of the model. Here they are for the SIR system, with S=susceptible, I=infected, and R= recovered

dS/dt = - k₁SI

dI/dt = k₁SI  - k₂I

Note that the coefficient k₁ is the same in both equations because the number of people who become infected is equal to the number of those who cease being susceptible -- these two coefficients won't be the same in the equivalent equations for oil extraction. The other coefficient, k₂, is the frequency of recovery of the infected people. There is a third equation describing the growth of the "recovered" stock, but it is simply equal to k₂I and we can neglect it here. 

Now, from the equations above, we can say that the R factor is equal to the number of new infections divided by the number of infected people. We need to take also into account the recovery frequency: the gradual disappearance of people from the "infected" stock. So that the result is:

R=  Sk₁/k₂

Note that the variables in this model are usually expressed in terms of fractions. So, the number of susceptible people at the very start of the epidemic is supposed to be 100% of the population, that is, unity. There follows that

Ro= k₁/k₂

Now we can determine the value of R needed for attaining "herd immunity." the value needed for stopping the growth of the infection. For this, we take the second equation of the two of the model. We want to know when the number of infected people, I, starts to decline. That means to find when dI/dt <0. That is:


k₂R-k₂ <0

Or, R<1.  

This is the condition for herd immunity. It explains the attention dedicated to this number for the current coronavirus epidemic. 

Measuring R may be a good idea but, in reality, it is not a very useful way to forecast the trajectory of an epidemic. To measure R you need to know S, but normally you don't know who is susceptible and who is not unless you try to infect them. So, saying that "R has become smaller than one" is the same thing as saying that "the number of infected people in the population has started declining." And the latter term is what you can actually measure or, at least, estimate. As someone said, "Models are accurate only when they become irrelevant."

How about the EROI? The equations are the same, but with a small difference. Whereas people move quantitatively from the "Susceptible" to the "Immune" stock, transforming a unit of energy embedded in underground oil into a unit of usable energy cannot be 100% efficient. So, you need another coefficient in the equations, a "transformation efficiency", η as a coefficient of k1 in the second equation. Obviously, it must be that η<1 because of the 2nd law of thermodynamics.

We go through the same mathematical tricks and we find the condition for the amount of stored energy (the "capital" of the industry) starts declining. It has to be:

EROI < 1/η

For the transformation of crude oil into useful energy into a thermal engine, we can roughly estimate a life cycle efficiency of the order of 10%-20%. There follow that oil extraction is not thermodynamically viable for an EROI < 5-10, which agrees with independent estimates of EROI for crude oil. It was probably around 30 during the early stages of exploitation and therefore allowed the industry to grow. Currently, the average EROI for oil extraction is probably around 10-15, so that we are close to the start of the irreversible decline of the industrial system that exploits it. Or, it may have already started. 

Note that the condition EROI < 1/η does NOT correspond to "peak oil" as it is normally defined. It is, rather, "peak capital". Peak oil refers to oil production, which is not the same thing. But it is not possible to find an equivalent simple expression that correlates the EROI of the system with the occurrence of the peak. We can only say that it occurs earlier and, therefore, for larger values of the EROI.


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(*) Here is the Vensim model I used for the graph shown in this post. If you want the code, just write to me. 

The Great EROEI Scam: Are renewables a good idea?

The cheetah in the figure knows very well that it cannot spend more energy in chasing the impala than the impala can provide once eaten (in other words, the cheetah needs an energy return on the investment (EROEI) >1). Carnivores make no calculations about that question, they only know that, if they want to survive, they have to run. And this is our destiny, too. If we want to survive, we need energy and we need to move away from fossil fuels before depletion or climate change (or both) destroy us. But, unlike lions and cheetahs, we tend to discuss a lot on the subject and, sometimes, to get it wrong. This is the problem with the recent movie "Planet of the Humans" and its wrong evaluation of renewable energy (image by Nick Farnhill, creative commons license)

 
Years ago, when I discovered the concept of "EROEI" (or EROI), energy return for energy invested, I was both delighted and elated. "Here is," I thought, "an objective way to evaluate and compare the efficiency of energy technologies. No more shaky financial calculations, no more ideology, no more politics, only facts. And everyone has to agree on the facts." And the beauty of the concept was that if the EROEI is smaller than one for a certain technology, then it is an energy sink, not an energy production system.

I was wrong more than I could have imagined. From when the idea of EROEI was first proposed, in the 1980s by Charles Hall, the concept was stretched, squashed, squeezed, twisted, and shaken until it became a useless mongrel. Ideology took over from physics and EROEI became a support for preconceived ideas rather than an evaluation tool.

I should have known better, but I hadn't realized how difficult it is for most people, including opinion leaders, to think in terms of data. One of the problems with EROEI is that many people just can't understand it, but the main one is that it is not easy to quantify its value. You can say, "this apple is red" just by looking at it, but not "this plant has an EROEI of 12.5." That is unless you perform a complex calculation based on the rules of LCA (life cycle analysis). Impossible if you are not trained in this specific field of work.

In practice, it is not difficult to find people with sufficient knowledge of the rules to be able to bend them in such a way to obtain the results they (or their sponsors) want. The result is a jungle: for every conceivable energy technology, you can find a range of values of the EROEIs so large that you can pick one according to what you like (2+2=5).

But it is not so much a question of cheating, it is just that science and politics talk two different languages. If you are a scientist, you reason in terms of data and models. If you are like most people, you reason in narrative terms. And narrative, as we all know, is all about a conflict between bad guys and good guys. So, that's the way people tend to evaluate the EROEI issue: they judge the messenger rather than the message. And they tend to like messengers who bring them messages they like. It is called "selective exposure" or "confirmation bias."

That's the reason for the success of a movie such as "Planet of the Humans." It is a clever movie because it passes a simple message that resonates with many people in the environmentalist movement: "big capital bad." That may even be true: there are plenty of scams embedded in concepts such as "green growth." But, in the movie, the real target is not big, bad capitalism, but renewable energy. It is presented as being unable to support itself because it has EROEI<1  or that, at least, the value of the EROEI is too small to support a complex civilization.

Then, most environmentalists know nearly nothing about the most elementary technologies of communication and many of them reacted by criticizing the many factual errors of the movie. But they didn't realize that most people are unable to evaluate the details of the takedown of the movie that many experts engaged in. And the result was a remarkable success for the movie and for its sponsors.

In any case, you can't change the way people think, not in a short time at least. So, my personal opinion is that it is useless to engage in infinite discussions about the EROEI of renewables or of other technology. And it is especially useless to discuss at length about a bad movie such as "Planet of the Humans." The human mind is such that it can demonstrate just about anything on the basis of apparently logical arguments. You probably know the story of how in 1903 the American physicist Simon Newcomb "demonstrated" on the basis of physics that human-made machines couldn't fly -- that was just five years before the first flight of the Wright brothers.

I am sure that with a little work it would be possible to "demonstrate" on the basis of physical principles that cheetahs cannot catch impalas. But cheetahs don't need theories telling them that they cannot feed on impalas. They run and they catch their prey. So, we can demonstrate that renewables work by installing renewable energy plants, as many as possible. And show that they work -- they do produce energy. And the more plants we install, the better the world will be for us and for those who'll come after us. It will help climate to stabilize, too. 

On this matter, below you may read a post that I wrote three years ago. See also a post I published in 2016: But what's the REAL energy return of photovoltaic energy?

Why EROEI matters: the role of net energy in the survival of civilization 

Published on Cassandra's Legacy, March 13,, 2017  (slightly modified in this version of May 15, 2020)

The image above was shown by Charlie Hall in a recent presentation that he gave in Princeton. It seems logical that the more net energy is available for a civilization, the more that civilization can do. Say, build cathedrals, create art, explore space, and more. But what's needed, exactly, for a civilization to exist? Maybe very high values of the EROEI (energy return on energy invested) are not necessary.

A lively debate is ongoing on what should be the minimum energy return for energy invested (EROEI) in order to sustain a civilization. Clearly, one always wants the best returns for one's investments. And, of course, investing in something that provides a return smaller than the investment is a bad idea. So, a civilization grows and prospers on the net energy it receives, that is the energy produced minus the energy required to sustain production. The question is whether the transition from fossil fuels to renewables could provide enough energy to keep civilization alive in a form not too different from the present one.

It is often said that the prosperity of our society is the result of the high EROEI of crude oil as it was in the mid 20th century. Values as high as 100 are often cited, but these are probably widely off the mark. The data reported in a 2014 study by Dave Murphy indicate that the average EROEI of crude oil worldwide could have been around 35 in the past, declining to around 20 at present. Dale et al. estimate (2011) that the average EROEI of crude oil could have been, at most, around 45 in the 1960s Data for the US production indicate an EROEI around 20 in the 1950s; down to about 10 today.

We see that the EROEI of oil is not easy to estimate but we can say at least two things: 1) our civilization was built on an energy source with an EROEI around 30-40. 2) the EROEI of oil has been going down, owing to the depletion of the most profitable (high EROEI) wells. Today, we may be producing crude oil at EROEIs between 10 and 20 on the average, and the net energy yield keeps going down.

Let's move to renewables. Here, the debate often becomes dominated by emotional or political factors that seem to bring people to try to disparage renewables as much as possible. Some evidently wrong assessments claim EROEIs smaller than one for the most promising renewable technology, photovoltaics (PV). In other cases, the game consists of enlarging the boundaries of the calculation, adding costs not directly related to the exploitation of the resource. That's why we should compare what's comparable; that is, use the same rules for evaluating the EROEI of fossil fuels and of renewable energy. If we do that, we find that, for instance, photovoltaics has an EROEI around 10. Wind energy does better than that, with an average EROEI around 20. Not bad, but not as large as crude oil in the good old days.

Now, for the mother of all questions: on the basis of these data, can renewables replace the increasing energy-expensive oil and sustain civilization? Here, we venture into a difficult field: what do we mean exactly as a "civilization"? What kind of civilization? Could it build cathedrals? Would it include driving SUVs? How about plane trips to Hawaii?

Here, some people are very pessimistic and not just about SUVs and plane trips. On the basis of the fact that the EROEI of renewables is smaller than that of crude oil, considering also the expense of the infrastructure needed to adapt our society to the kind of energy produced by renewables, they conclude that "renewables cannot sustain a civilization that can sustain renewables." (a little like Groucho Marx's joke, "I wouldn't want to belong to a club that accepts people like me as members.").

Maybe, but I beg to differ. Let me explain with an example. Suppose, just for the sake of argument, that the energy source that powers society has an EROEI equal to 2. You would think that this is an abysmally low value and that it couldn't support anything more than a society of mountain shepherds, or not even that. But think about what an EROEI of 2 implies: for each energy-producing plant in operation there must be a second one of the same size that only produces the energy that will be used to replace both plants after that they have gone through their lifetime. And the energy produced by the first plant is net energy fully available to society for all the needed uses, including cathedrals if needed. Now, consider a power source that has an EROEI= infinity; then you don't need the second plant or, if you have it, you can make twice as many cathedrals. In the end, the difference between two and infinity in terms of the investments necessary to maintain the energy-producing system is only a factor of two.

It is like that: the EROEI is a strongly non-linear measurement. You can see that in the well-known diagram below (here in a simplified version, some people trace a vertical line in the graph indicating the "minimum EROEI needed for civilization", which I think is unjustified)):

You see that oil, wind, coal, and solar are all in the same range. As long as the EROEI is higher than about 5-10, the energy return is reasonably good, at most you have to re-invest 10%-20% of the production to keep the system going. It is only when the EROEI becomes smaller than ca. 2 that things become awkward. So, it doesn't seem to be so difficult to support a complex civilization with the technologies we have. Maybe trips to Hawaii and SUVs wouldn't be included in a PV-based society, but about art, science, health care, and the like, well, what's the problem?

Actually, there is a problem. It has to do with growth. Let me go back to the example I made before, that of a hypothetical energy technology that has an EROEI = 2. If this energy return is calculated over a lifetime of 25 years, it means that the best that can be done in terms of growth is to double the number of plants over 25 years, a yearly growth rate of less than 3%. And that in the hypothesis that all the energy produced by the plants would go to make more plants which, of course, makes no sense. If we assume that, say, 10% of the energy produced is invested in new plants then, with EROEI=2, growth can be at most of the order of 0.3%. Even with an EROEI =10, we can't reasonably expect renewables to push their own growth at rates higher than 1%-2%(*). Things were different in the good old days, up to about 1970, when, with an EROEI around 40, crude oil production grew at a yearly rate of 7%. It seemed normal, at that time, but it was the result of very special conditions.

Our society is fixated on growth and people seem to be unable to conceive that it could be otherwise. But it is clear that renewables, with the present values of the EROEI, cannot support a fast-growing society. But is that a bad thing? I wouldn't say so. We have grown enough with crude oil, actually way too much. Slowing down, and even going back a little, can only improve the situation.



(*) The present problem is not to keep the unsustainable growth rates that society is accustomed to. It is how to grow renewable energy fast enough to replace fossil fuels before depletion or climate change (or both) destroy us. This is a difficult but not impossible task. The current fraction of energy produced by wind and solar combined is less than 2% of the final consumption (see p. 28 of the REN21 report), so we need a yearly growth of more than 10% to replace fossils by 2050. Right now, both solar and wind are growing at more than a 20% yearly rate, but this high rate is obtained using energy from fossil fuels. The calculations indicate that it is possible to keep these growth rates while gradually phasing out fossil fuels by 2050, as described in a 2016 paper by Sgouridis, Bardi, and Csala

Climate Change Mitigation: Is it a Good Idea to Sweep the Carbon Under the Carpet?

Above: our paper recently published in Nature Energy. Our conclusion is that, in terms of energy returns, renewable energy in the form of solar or wind is much better than carbon capture and storage for mitigating of climate change. Sweeping the carbon underground is not a good idea.

We have a little problem: for more than thirty years, the climate scientists of the International Panel on Climate Change (IPCC) have been telling us that if we don't stop emitting greenhouse gases into the atmosphere -- mainly CO2 -- we are in dire trouble. And we have done very little, nearly nothing. As predicted, we ARE in dire trouble.

There is some element showing that things may change: the polls indicate that more and more people are starting to understand the mess we are in and the action of the young Swedish activist Greta Thunberg is making waves in the memesphere. We may be awakening from a 30 years slumber to discover that we have to hurry up and do something. But what?

Not that we lack plans: every IPCC report released includes plans on what we could or should do to avoid the worse. We have to follow a steep trajectory of de-carbonization while, at the same time, maintaining a vital minimum supply of energy to society. But how to do that?

The most common idea floated in these discussions is to use Carbon Capture and Sequestration (CCS). It is straightforward: instead of releasing into the atmosphere the CO2 emitted by a power plant, you pump it underground, sequestering it in a porous reservoir, maybe one that, earlier on, had contained gas or petroleum.

Is that a good idea? Maybe, but it is also a way of sweeping the problem under the carpet. If our problem is the use of fossil fuels, and it is, then by using CCS we are striving to keep alive the technology we should strive to get rid of.

But what if CCS were the only possible solution to the problem? Then, yes, if we really had no choice we would have to settle for the least bad idea. But is it true that CCS is the only way forward, or is it a failure of the imagination? Wouldn't renewable energy be a better idea?

The way to decide this point is to make a quantitative calculation based on real-world data. CCS doesn't come for free: it is a complex technology with an energy cost. This cost must be factored in comparison with alternatives: it is the concept of energy return for energy invested (EROI). The better the value of the EROI, the better the technology. In the long run, EROI factors trump monetary cost factors: you cannot get energy by printing money.

This is exactly what we did, myself and my colleagues Sgouridis, Csala, Dale, and Chiesa, in a paper we published this week on Nature Energy. We compared the EROI of CCS and renewable energy using the latest available data and a broad range of assumptions, including the need for energy storage of renewables.

The result? Renewables are by now a sufficiently mature technology that for most reasonable assumptions they have a better energy return than fossil-based CCS. You can read our paper at this link. Below you can read some of our conclusions. Here sRE stands for "scalable Renewable Energy", while the "sower's strategy" is the concept that we need to invest some fossil energy in order to build up the renewable infrastructure that will replace the fossil one.

The energy return of using fossil resources with CCS in power generation is lower than the EROEI of most current deployment of sRE. Therefore it would be preferable to direct these fossil resources towards building a self-sustaining renewable energy infrastructure, an approach previously termed “the sower’s strategy. Even when sRE adoption reaches or exceeds 80% our calculations indicate that the system EROEI may be equal to the better EROEI CCS without the additional issues related to the reliance on depleting resources and non-energetic biophysical complications

You may discuss the details of this result and argue that, in some special conditions, CCS may still have a better EROEI than renewables. But the point is that CCS has additional negative points that renewables don't have. CCS is still a largely untested technology on a large scale but the main problem, as I was mentioning at the beginning, is that by adopting CCS we give new life to the presently agonizing fossil fuel technologies. But fossil fuels are doomed by depletion in any case, so what sense does it make investing the few resouces we still have in a technology that doesn't have a future? In the end, CCS is mainly a failure of the imagination: we can and we should do much better than sweeping the carbon underground.

As a final note, our study deals only with CCS as a way to remove carbon from the emissions from the combustion of fossil fuels in power plants. That doesn't mean that we may not need CCS as a way for removing carbon dioxide from the atmosphere. This is called "direct atmospheric capture" (DAC) or "Negative Emission Technologies" (NET). Given the situation, it may be the only way to go back to a CO2 concentration low enough to avoid the worst and it might work! But DAC can work only if it is powered by renewables and it is nice to know that we have the technologies we need. If only we'll ever decide to use them.




Link to the paper on Nature Energy.

The real Energy Return of Crude Oil: smaller than you would have imagined

 

A simple but important study by Luciano Celi shows what is the real energy return that oil companies manage to attain. Much smaller than you would have believed, in several cases it is today well below 10. Which means that renewable energies already produce a larger EROI than oil and gas. No more excuses for not switching to renewables as fast as possible!And we have to do it fast because, as Celi shows, we are on the edge of the Net Energy Cliff of fossil fuels.

By Luciano Celi

Researchers involved in the energy sector know very well what the EROI (acronym of Energy Return on Investment) is and how frequently is cited in scientific publications, in spite of differences of definition among researchers.

Well aware of the fact that I arrived last in the middle of a dispute that has engaged researchers for decades, I accepted the challenge and tried to understand, with Claudio Della Volpe, Luca Pardi, and Stefano Siboni, if there was a way to know something that is difficult know: the EROI of oil companies.

The method we have implemented is quite simple, even if with some useful simplification that we have discussed in details in our paper.

The denominator in the EROI value is the Energy Invested, while the numerator is the Energy Return (how much energy is gained with respect to the investment). Knowing the numerator it is quite simple (we are talking of oil companies) because this value corresponds to the production of a day (or of a year). The issue in different cases was knowing the value of energy cost of that production. It is difficult to know how much energy they use to produce what they produce in a year (or in a day), sitting at a desk without wandering the world knocking on the doors of companies. However... we have found an indirect way to have these data.

The oil companies are requested to compile a Sustainability Report (SR) yearly. Even if not mandatory many companies have accepted to prepare them probably under the pressure of the public opinion or/and the business branch named CSR (Corporate Social Responsibility), especially after the alarms launched at the last world conferences on climate.

In these reports, it is possible to find the emissions of their up- and downstream activities expressed in CO2 equivalent. This is good news because we can try to do a mental experiment: to burn all oil that a company produces and convert it in CO2 equivalent, with a simple stoichiometric ratio. In this way, we have the equivalent value in CO2 of production (numerator) and the equivalent value in CO2 of emissions (denominator).

As already mentioned this is an approximation - first we have many kinds of oil, moreover the petroleum production almost always includes a quota of natural gas that has a different composition (mostly methane) and a different HHV (Higher Heating Value) compared to oil - but we have corrected our estimation taking into account these variables. The interesting side of the issue is that the data are above criticism because they come directly from the oil companies reports (Sustainability, Annual, Commercial, and so on).

The choice of companies

I focused my research activity on SR 2015 and looked at the companies with the highest revenues in the world. I have found this list (on worldatlas.com or Wikipedia: both these sources - the second one takes the data from the previous one - are updated to 2017) and I have chosen to make a cut off for a market share above 1%. According to this criterium the companies selected are 30:

#	Company	Billions $	Market share %
1.	Saudi Aramco	478,00	8,31	34,77
2.	Sinopec	455,50	7,92
3.	China National Petroleum Co.	428,62	7,45
4.	PetroChina	367,98	6,40
5.	Exxon Mobil	268,90	4,68
6.	Royal Dutch Shell	265,00	4,61
7.	Kuwait Petroleum Corporation	251,94	4,38
8.	BP	222,80	3,88
9.	Total SA	212,00	3,69
10.	Lukoil	144,17	2,51
11.	Eni	131,82	2,29
12.	Valero Energy	130,84	2,28
13.	Petrobras	130,00	2,26
14.	Chevron Corporation	129,90	2,26
15.	PDVSA	128,44	2,23
16.	Pemex	117,50	2,04
17.	National Iranian Oil	110,00	1,91
18.	Gazprom	106,30	1,85
19.	Petronas	100,74	1,75
20.	China National Offshore Oil	98,53	1,71
21.	Marathon Petroleum	97,81	1,70
22.	PTT	93,55	1,63
23.	Rosneft	91,72	1,60
24.	JX Holdings	90,67	1,58
25.	Engie	89,64	1,56
26.	Statoil	82,48	1,43
27.	Indian Oil Corporation	81,55	1,42
28.	Sonatrach	76,10	1,32
29.	Reliance Industries	73,10	1,27
30.	Pertamina	70,65	1,23	< 1%
	Others	623,27	10,84
	Total amount >>>>	5.749,52	100,00

Highlighted in green is found the total amount of the market share for the first five companies: this value is quite high considering that the first thirty companies represent almost the 90% of the market share, and only the 16,6% of them hold the 34,7% of the market share.

The EROI values

Applying the method briefly described above, the estimated values of EROI are represented in the graph below:

If we consider that the total contribution of oil and gas to the world primary energy demand in 2016 was 57.41% of the total (as shown in the graph below), the weighted mean of the EROI is rather low. Why does it seem that nothing happens?

 
The explanation is found in the relation between EROI and Net Energy.

The net energy and the Seneca Cliff

Net energy is defined as:

Net Energy (NE) = Energy Return - Energy Investment.

If we divide all terms for a single quantity ER, the result is:

NE/ER = 1-(1/EROI)

and, under the hypothesis of the ER always equal 100, the NE value can be expressed in percentage. 

So, the equation is:

NE(%)= [1-(1/EROI)]*100.

If we graph NE vs. EROEI the result is the following:

The industrial society experiences Net Energy, not EROI, decline. As shown in the graph above, there is a strong non-linear relation between Net Energy and EROI. For a long range of EROI values, say from 100 to 10, the Net Energy is declining very slowly. Presently although the EROI decline is quite clear the Net Energy is still well above 90%. The society feels pretty safe. The problem is that we are walking along a cliff and it is increasingly urgent to make an energy transition, before finally ending up in the abyss.