In 1920, after examining data from the recent bubonic plague outbreaks in India, two epidemiologists named McKendrick and Kermack published a mathematical model of how infectious diseases spread. Their model, known as S-I-R, is still the basis for modern mathematical epidemiology.

The mathematical principle that makes epidemics so dangerous is known as "exponential growth". Most of the familiar processes we see every day are linear: they proceed smoothly and uniformly, the mathematical equivalent of adding 1, and then another 1, and then another, and so on. Exponential growth is different: it is not linear. It is the equivalent of adding 1 and 1 to get 2, then adding another 1 to EACH of the resuting 2, then another 1 to each of the resulting 4, then 8, then 16 .. and so on. Things begin very slowly, but then they balloon rapidly and massively.

The concept of "exponential growth" is usually illustrated with this example:

Suppose I have a lake and I introduce an invasive plant that doubles in size each day--every day, each plant produces one new plant. Suppose we determine that at the rate the plant is growing, it will cover the entire lake in 30 days.

The question is: on which day will the plant cover half of the lake?

Most people will answer "day 15", because they are thinking in linear terms and not exponential. On day 15, the plant will actually cover far less than 1% of the lake. Even on day 27, it will cover less than 13%. On day 28 it will cover 25%, and it is not until day 29 that it covers 50%. It will then double in size again to cover the entire lake the next day. Of the plant's total growth, three-fourths of it happens in just the last two generations. That is the power of exponential growth.

The S-I-R mathematical model studies the factors behind the growth of a disease by dividing the target population into three groups. The "Susceptibles" are those people who have not yet been infected with the disease. Those who became infected and are therefore capable of spreading the disease are "Infectives". And those who are no longer capable of infecting others--either because they have recovered and gained immunity or because they are dead--are "Removals". It is the interplay of these three groups that determine the course of an epidemic.

Almost immediately, the S-I-R model was used to answer one of the longest-running questions in medical science: why do epidemics end? The standard answer was that an epidemic came to a halt when it had entirely run its course and infected everyone who was capable of being infected, and everyone was now either immune or dead. But the math showed that this is not the case: no matter how infectious the disease is, there is always a segment of Susceptibles who have not been infected. This happens because as the proportion of Removals grows (because people have recovered and are immune), it becomes harder and harder for the Infectives to come into contact with fresh Susceptibles, whose proportion of the population keeps getting smaller. Eventually the epidemic will reach the point where the pathogens cannot find enough new Susceptibles to keep reproducing themselves before they become Removals (by either recovering or by dying). The number of Infectives therefore dwindles, and the epidemic ends.

This, in turn, also indicates that in many cases the best way to stop the spread of an epidemic is to increase the proportion of Removals, thereby preventing the pathogen from reproducing.

One way to do this is with vaccines. By conferring immunity to people, vaccination takes large numbers of people out of the Susceptibles category and immediately moves them to the Removals, without the intervening Infective period. So even if the pathogen is introduced, this robs it of the opportunity to reproduce itself, and quickly brings the epidemic to a halt (or prevents it from being started in the first place).

Another method is to treat the Infectives and cure the disease, or at least to prevent its transmission. Successfully curing the disease in effect moves the treated individual from "Infective" to "Removal", and deprives the pathogen of the chance to move to another host.

And the third way to prevent the Infectives from accessing the Susceptibles is through quarantines. This uses physical separation to keep the Infectives somewhere isolated until they become Removals, thereby preventing the pathogens from reaching any new Susceptibles. In effect, quarantine also transfers all of the Susceptibles into the category of Removals without any intervening Infective period. The pathogen is unable to reproduce and dies out as the quarantined Infectives recover or die and become Removals.

One of the first modifications that was made to the model, however, was to include the idea of "Carriers"--people who have been infected but do not know it because they do not show any overt symptoms. In the S-C-I-R model, then, there is a period of time, known as the "incubation period", during which a portion of the population is spreading the disease as a Carrier before they themselves show symptoms and can be moved to the Infective category. Vaccination still works well under this model, because the Carriers have trouble finding any still-Susceptible individuals amongst the vaccinated population.

But the Carriers complicate the process of treatment, or of quarantine if it should become necessary (if for instance there is no effective vaccine or treatment against the pathogen). Even if people are immediately quarantined once they exhibit symptoms and are moved from the Carrier to the Infective category, they have already done their part to spread the pathogen to new hosts amongst the Susceptibles.

The strategy that is used to combat the role of the Carriers, then, is called "contact tracing". This allows medical investigators to reconstruct the recent movements of Carriers, identify Susceptibles that they have been in contact with (and which may now be possible Carriers themselves), and quarantine them (perhaps after a diagnostic test to determine if they actually are Carriers). If this is done diligently and if it happens early enough in the exponential growth curve to catch most of the Carriers and prevent them from producing any more new Carriers and Infectives, it is often enough to halt the epidemic.

There are other factors which may enter into the model. In some cases, such as most sexually-transmitted diseases, recovering from the disease does not confer immunity, and thus there is no "Removal" category; every Infective who recovers goes straight back into the Susceptible group. This is known as an S-I-S model.

In general, human diseases tend to evolve to become either highly infectious or highly lethal, but not both. If a pathogen kills its host too quickly, it greaty reduces the chances that it can infect another person before the original host dies: on the other hand, if the pathogen produces only mild but chronic symptoms, the host can live a long time and thus facilitate the infection of a large number of new hosts.

Taking these and many other factors and considering them overall as a whole, mathematical epidemiologists are able to calculate two numbers which are crucial to understanding how a particular outbreak is working and how well it is being combatted. The math here is pretty simple. The first number is the "basic reproduction rate", called R0 (pronounced "R-zero"). It is the average number of Susceptibles that any given Infective is typically able to spread the pathogen to before becoming a Removal. Some diseases, like Ebola, have low R0 numbers: they typically are able to infect only 2 or 2.5 Susceptibles on average. Other highly-infectious diseases like measles can have an R0 of 10 or 12 or even more.

The R0 value itself is dependent on three chatacteristics, which are important in understanding the development of any particular epidemic. The first is the size of the potential population of Susceptibles--the more potential targets the pathogen has, the more quickly it can spread through them. The second is the rate at which the pathogen is able to spread from Infectives or Carriers to new Susceptibles--the faster it can spread, the more new hosts it can infect. And the third is the speed at which Infectives become Removals, either through recovery or through death--the longer the infected person is able to spread the pathogen, the more targets it will be able to reach.

All of these factors modify the "basic reproduction rate" into a different number, the "effective reproduction rate", called Re (and pronounced "are-eeh"). This is the actual number of Susceptibles that each Infective or Carrier is in fact infecting on average: this number will fluctuate from one outbreak to another, and also varies over time within each particular outbreak. As long as the Re is above 1, the disease will continue to propagate, as each host is able to produce at least one other host before it is removed. The goal of the medical community then is to bring the Re value below 1, insuring that Infectives in the population are being lost (through Removals) faster than they are being made (through new Carriers or Infectives). And this is the purpose of vaccinations, treatments, and quarantines.

So everything we see going on around us--quarantines, self-isolations, travel restrictions, public-space closures, social-distancing--is, mathematically speaking, an attempt to separate the Covid-19 Carriers/Infectives from the Susceptibles and reduce the Re value below 1, to the point where the virus is no longer able to move from one host to another often enough to maintain itelf and it dies out as people recover and gain immunity.

Looking around, we can also see that some countries, such as South Korea, have been pretty effective at reducing the Re value of their outbreak by intervening early in the exponential growth curve through a vigorous program of contact tracing and quarantine. Other countries, such as Italy and the United States, have been less effective. The math is relentless, and those societies will pay the price in a higher exponential growth curve, which will translate into more sicknesses and, ultimately, more deaths.