7 min read.Updated: 09 Apr 2020, 10:52 PM ISTDilip D'Souza

Mathematicians study differential equations between the ‘S’ for susceptible, ‘I’ for infected and ‘R’ for removed to understand the spread of the disease

Lots of discussions, two weeks into a 21-day lockdown, on how it will end. Suddenly, we’re familiar with a phrase like “staggered exit", and there are whispers of it continuing beyond 21 days, and serious worries about what it has already done to our economy and what it will do and whether these should figure in the decision.

Lots of discussions, too, about what a good strategy is to handle a possible flood of detected cases, especially if they all need hospitalization. Plenty of weighing of alternatives and trying to determine which is best.

One dilemma that faces us all is, what happens when the lockdown ends? Whether it is 21 days or 45 days or somewhere in between or longer, the virus will still be out there when we emerge from the lockdown. We might have flattened that curve, but we’ll still be at least many months away from access to a proven vaccine. Put another way, there’s a risk to returning to “normal", whatever that may mean in a post-corona world. And yet none of us can stay locked down indefinitely, until there’s a vaccine available to protect us. What will happen to our economies? What will we do if there are second and third waves of infections—which some evidence shows that Japan and Singapore are facing?

To start answering these and other similar questions, it’s important to know and understand who among us are more susceptible to the infection. That will help decide how healthcare systems must react to this emergency. But it may be even more important to estimate how many among us are likely to be affected to some degree by covid-19. For, that is how we can make plans for the kinds of facilities we will need, and get some idea of how long we will be fighting this battle. Every country that’s been hit by the virus has attempted such estimates. For the most part, they are driven by something you have probably heard by now: SEIR, for Susceptible-Exposed-Infected-Removed.

SEIR is a model, a technique, fundamental to the science of epidemiology—the branch of medicine that investigates the start, spread and control of diseases. It’s medicine, but not a clinical pursuit like surgery or ophthalmology; instead, it delves into the data about a disease and looks for patterns. Epidemiologists do this by mathematically modelling the disease. One way to do so is to “compartmentalize" the population that’s been affected by the disease. The idea is that each compartment has similar characteristics as far as the disease is concerned, and we can draw conclusions about each one, as well as about how they relate to the others. Epidemiologists have used compartment models for years to make vital predictions about epidemics like measles and polio. This is how we can estimate how many people will be infected, how quickly the infections will spread through a population, how long the pandemic will last, what effect various measures — vaccination, hospitalization, etc — will have on the disease, and more. Understanding the disease through such data and models helps clarify the best way we can use always limited resources to fight the disease.

Now SEIR is really a modification of a simpler model, SIR. SIR divides the population into three compartments. “S", for those who are susceptible to the disease, likely a significant chunk of the population. “I" refers to people who are infectious; meaning they can pass on the infection to someone else. “R" is the set of people who have either recovered from the disease or have died, and thus can no longer spread the disease like their compatriots in “I" can. (Thus “R" is sometimes said to stand for “removed", meaning people who have been removed from the infectious population). The SEIR model adds the “E", for with some diseases there are people who have been exposed, but may not actually be infectious yet. This applies to coronavirus in particular, because we know that the virus has a significant incubation period.

Of course, these are not fixed numbers. Instead, they vary as the disease progresses. It’s that very variation that the model seeks to simulate, though we can say some things about it simply by observing and inferring. In fact, that’s exactly how we flesh out the mathematical model from its simple beginnings.

Goes something like this: before a disease breaks out in a country, “S" is the whole population, because we can assume everyone is susceptible. Then one person gets infected. If she in turn infects every person she meets as she moves about, you can imagine how quickly the disease spreads — because each of those infected people will also move about and infect others. What this means is, as the disease spreads, “S" decreases rapidly, because susceptible people get infected and move from the “S" basket to the “I" basket. Thus “I"’s numbers rise dramatically. But of course this cannot carry on indefinitely, because those who get infected will eventually recover. (And some will probably die). If they have developed an immunity to the disease, they cannot return to the “I" basket. So “I" eventually reaches a peak and starts decreasing. In tandem, “R"’s numbers grow until most of the population is in that compartment. At that point, the disease has run its course.

Mathematicians consider this basic description of an epidemic’s spread and set out to model it. They will seek to capture the change in the numbers in each compartment as a function of elapsed time, but also as it relates to the change in other compartments. After all, the decrease in “S" is a direct consequence of the increase in “I"; and the rise in “R" has consequences for both “I" and “S". Taking all this into account produces a set of what mathematicians call “differential equations". This set is the SIR model.

Once they have the equations worked out, the fun—such as it is, of course, and with pandemics, let’s never forget that death ends fun—begins. Because now, they can tweak their assumptions to see what effect that will have on the course of the disease.

For example, certainly an infected person moves about infecting others. But how many others does she meet in a day? If she’s a housewife who doesn’t go out much, that number is probably just four or five. But if she’s a vegetable vendor in a busy market, she probably meets hundreds of customers in a day. Which of these we assume will change how the model functions. Or try this: what’s the chance that an infectious person actually passes on the infection to someone she meets? We could assume that it is certain she will pass it on, in which case the disease will spread like wildfire. But what if it’s only a 50% chance? Or 10%? After all, diseases vary in how contagious they are. So clearly, getting that probability right is important in understanding how fast a disease spreads. But even more telling, we can think of these lower probabilities as the result of measures like washing hands, social distancing, wearing masks and so on. If we know a disease spreads by touching, for instance, and people are careful not to shake hands when they meet, undoubtedly, the disease will spread more slowly. Then the mathematical model can show us just how effective such measures are.

There are other tweaks we can make. Consider that few of us spend our days simply wandering about without much of a purpose. Take me, on any given day from that time before the virus that’s rapidly fading from our memories. It usually went like this: I work mostly at home. I might make one or two trips to nearby stores for supplies—you know, important things like rum or chocolates. Perhaps a trip across town to attend a meeting. Walk to the nearby tennis court in the evening for a game.

Notice that my outings are to places where there are, typically, plenty of others also present. I’m sure that applies to you as well. Mathematicians can capture such a phenomenon in the SIR model and what it invariably shows is that this kind of routine behaviour makes the disease spread much faster. This is exactly why parks, tennis courts, swimming pools and gyms have shut down. This is also why grocery stores, which cannot shut down, have started insisting that their customers take turns shopping so that only a few are inside at any one time.

As you can imagine, there’s plenty more to say about these epidemiological models. In a future column, for example, I hope to explain the differential equations that make up SEIR.

The point is, if we understand the model, we give ourselves a better chance to know how to fight the pandemic. That simple.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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