Seneca and the Virus: Why does the Pandemic Grow and Decline?

 

Seneca, the Roman philosopher, knew the term "virus," that for him had the meaning of our term "poison." But of course, he had no idea that a virus, intended in the modern sense, was a microscopic creature reproducing inside host cell. He also lived in a time, the 1st century AD, when major epidemics were virtually unknown. It was only more than one century after his death that a major pandemic, the Antonine Plague, would hit the Roman Empire. 

But Seneca was a fine observer of nature and when he said that "ruin is rapid" he surely had in mind, among many other things, how fast a healthy person could be hit by a disease and die. Of course, Seneca had no mathematical tools that would allow him to propose a quantitative epidemiological theory, but his observation, that I have been calling the "Seneca Effect," remains valid. Not only people can be quickly killed by diseases, but even epidemics often follow the Seneca Curve, growing, peaking, and declining. 

Of course, the concepts of growth and collapse depend on the point of view. In many cases one man's fortune is someone else's ruin. What we see as a good thing, the end of an epidemic, is a collapse seen from the side of the virus (or bacteria, or whatever). But, then, why do epidemics flare up and then subside? It is a fascinating story that has to do with how complex systems behave. To tell it, we have to start from the beginning. 

One thing that you may have noted about the current Covid-19 pandemic is the remarkable ignorance not just of the general public about epidemiology, but also of many of the highly touted experts. Just note how many people said that the epidemic grows "exponentially." Then, they got busy extrapolating the curve to infinity, predicting hundreds of thousands, or even millions, of deaths. But, to paraphrase Kenneth Boulding, "Someone who claims that natural systems grow exponentially has to be either a madman or an economist." It just doesn't work that way!

But how does an epidemic grow, exactly? The basic shape of an epidemiological curve is "bell shaped" (yes, just like the Hubbert curve for petroleum extraction). 

 

The reason for this shape is easy to understand in qualitative terms. Initially, the virus (or the pathogen) has a whole population to infect, so it grows rapidly (nearly, but not exactly, exponentially). Then, as it grows, its number of targets decline. Eventually the virus can't grow any longer for lack of targets. It reaches a peak and starts declining. 

These considerations can be set in a mathematical form: it is the model called "SIR" (susceptible, infected, removed), developed already in 1927. You may be surprised to discover that the SIR equations are exactly the same that describe the growth of the oil industry and the phenomenon of "peak oil." They are also the same equations that describe the behavior of a trophic chain in a biological system. I won't go into the details, here. Let me just tell you that, with my colleagues Perissi and Lavacchi, we are preparing a paper that describes how these and other physical systems are related to each other. 

Of course, modern epidemiological models are much more complicated than the simple "bare bones" SIR model, but it is an approach that tells us what to expect. No epidemic grows forever and even if you do nothing to stop it, it will eventually fade out by itself. After all, pathogens have the same problem we have with crude oil: they are exploiting a limited resource (us).

Now, back to the Seneca Effect, we said it implies that ruin must be faster than growth. In other words, the shape of the "Seneca curve" should be something like this:

 

 

There are such cases in the history of epidemics.

 

 

Let me show you an example: the cholera epidemic that struck London in mid-19th century (data from Wikipedia Commons)

 

And here you clearly see the Seneca shape. The decline of the cholera burst was significantly faster than its growth. The data for more recent cholera epidemics show the same shape. 

Yet, that "Seneca shape" is not common in epidemics. Often, we see the opposite kind of asymmetry. Here is an example: Hepatitis A, with data taken from Wikipedia. You see how the curve declines more slowly than it grows. 

Here is another pre-Covid example: the acute respiratory syndrome of 2003 in Hong Kong. 

There is no fixed rule in these historical cases, let's just say that this asymmetric shape is rather common. So, let's go to the current pandemic, and here are some data for the first cycle of 2020. (Image from "The Economist"). Also here, the trend is clear: decline is slower than growth.

 

 

 

 

It is a common trend all over the world and we could call it the "Anti-Seneca" effect. But, apart from giving it a name, why this shape?

The answer is not univocal: there are several factors that may affect the shape of the curve. In this case, the easiest explanation has to do with the parameter that describes how fast infected people cease to be infected, either because they are healed or because they die. If they heal/die fast, the curve goes down fast, otherwise it is the opposite. It makes sense: cholera may kill affected people in just a few hours, if untreated. Instead, people infected by the Sars-Cov-2 may go through one or two weeks of agony before their demise. That would explain the different shape of the curves.

 

But, be careful! As I said, there are other possible explanations. For instance, if you compare Sweden with Italy, you see that the mortality curve is more asymmetric for the former. Why is that? It is hard to think that sick Swedes would take more time to die than sick Italians. More likely, it is a question of geography. The Swedish population is concentrated in the southern regions, where the pandemic hit first. It took some time for the virus to spread northward and that explains the "tail" in the mortality curve. In Italy, instead, the first pandemic wave was confined to the Northern regions, which are relatively homogeneous in terms of population. Probably, geographical effects account for the commonly observed asymmetric curve shapes of the COVID-19 epidemic in other regions of the world. 

 

With vaccinations, the SIR model shows that we should see the epidemic curves falling down fast, at least if the vaccinations are started before the peak. So far, this effect is not seen anywhere, it may be too early. As vaccinations progress, we should be able to say more on this matter.

As for everything in science, epidemiology takes a little work to be learned, a virtue that's difficult to find in the discussion on social media. Even experts in virology and diseases don't really study epidemiology, their job is to heal people, not to make mathematical models. That's the reason why the behavior of the virus is so widely misunderstood. But, as Einstein said, "The Lord God is subtle, but malicious he is not." Epidemiology may be subtle, but it is not impossible to understand how epidemics grow and spread.