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Home >Science >Health >Opinion | Analysing corona via differential equations

Not that it’s surprising, but there’s so much making the rounds about corona that it’s hard to keep up. Or to make sense of it all.

India’s numbers, or at least the ones we have, are rising, but the growth rate has been generally declining through April. It’s important to consider the growth rate, for the absolute growth numbers continue to trend upwards. Does the decline suggest the success of the lockdown? Or have we not tested enough people? Is wider testing effective? How and whom should we test? What effect will an order of magnitude more testing—if we can achieve it —have on the number of cases detected?

Plenty of questions, and testing is just one aspect of these strange times. Apart from those battling the disease in our hospitals, there are journalists and doctors, mathematicians and drug researchers, searching for answers.

For instance, a team of four doctors in our armed forces: Three from the Armed Forces Medical College, Pune, and one from INHS Ashvini, Mumbai. Some weeks ago, I read a paper they had written on a mathematical model for the pandemic. They started with the data then available—in late March—made certain assumptions and worked out various projections about India’s covid-19 situation by the end of May. It says something about the way the pandemic has grown that only a few weeks later, some of their assumptions are not the ones we would make today. And yet, who’s brave enough to predict what they will be in another few weeks from now?

Yet too, their paper is both a useful snapshot of a particular moment in this pandemic, and a guide to how we should be analysing all the data.

In an earlier column, I wrote about the SIR model, which splits up a population affected by a virus into three groups: susceptible, infected and recovered. The authors used this model too—well, a modification of it. They had three aims: To determine the magnitude of the pandemic, assess its impact on healthcare resources in the country, and study what impact some “non-pharmacological interventions" (NPI) would have.

What is the point of these categories? If we divide the population this way, then modelling the disease is a matter of defining how the numbers in each group change over time, and in relation to each other. The effort to get that right is what keeps epidemiologists busy and, if they manage it, we can learn plenty about how we should be facing the disease.

A model like this is just a set of equations that describe how the numbers change. Because they describe difference, or change, they are called “differential equations". The term sounds intimidating, but it doesn’t have to be. Consider this initial cut at analysing data from a pandemic.

Remember: We want to know how each of those categories change. What can we say about the way S changes? Well, we know that as the disease spreads and ever more people become infected, the number of susceptible people in the population decreases. So the change in S is negative, first of all.

And what is the magnitude of the change? If you think about it, it is simply the number of people who get infected. And that depends on three things: The number of susceptible people (the pool available to infect) which is S, the number of already infected people (the pool from where our new infections come), which is I, and an infection rate that measures how fast the virus is transmitted, call that n (for iNfection). So we have the following equation:

S’ = - (n x S x I)

The term “S’" (read it as “S prime") is standard mathematical notation for the rate of change of S. (In calculus, S’ is the derivative of S with respect to time). There’s our first differential equation.

Now look at I. That number starts near zero and increases, so the change in I is positive. By how much does it increase? Clearly, the number of people who get infected, which is precisely the number from the first equation above, except it’s positive in this case. We must also account for a drain in that pool of infected people: Some of them recover, meaning they move from I to R. How much of a drain? That depends on the number of infected people and a recovery rate that measures how fast people recover— call that e (for rEcovery). So we have a second differential equation, this one spelling out the rate of change of I:

I’ = (n x S x I) - (e x I)

Finally, what about R? It increases too, so the change is positive. But that increase is exactly the number of people who recover, moving from I to R. Which is what we calculated, the second term in the equation above. So, we have a third differential equation:

R’ = e x I

There you have it: Three differential equations that together form a first cut at an SIR model for a pandemic like corona. We know S starts from pretty much the whole population and ends near-zero. I starts at zero, will reach a peak of some magnitude and then peter out as the pandemic plays out. R starts at zero too, and eventually equals the whole population as well. What about those factors, n and e? In some sense, those are the heart of this model, and it is by giving them values and then tweaking them that we begin to understand the spread of the disease, and the measures that can be effective in fighting it.

Mathematicians have confronted and solved differential equations for centuries, evolving techniques to do so. But as a buddy and once-colleague from my software days told me the other day: “Differential equations are so 1900s!" Computers now can use the so-called “Monte-Carlo method" (one example in my recent column, A French count and his needles, p in the sky) to quickly solve differential equations like these, letting you easily change the parameters e and n to see what effect that has.

This is essentially what the four doctors did, though they modified the SIR model to add three more categories: “Exposed" (E), “Quarantined" (Q) and “Dead" (D). E is especially relevant to a virus like corona, because some people exposed to it have not yet, or may never, become infected. That is, the disease has a period of incubation. The doctors also needed to account for people who were quarantined (Q), removing them from the I group.

And finally, their D applies an assumed death rate to both I and Q. This gave them a model of six differential equations. They had a number of assumptions to round out the model, like the death rate, the infectious period, the growth rate of infections at the time (mid-March) and so on. Putting it all together, they ran their model using the Monte Carlo method.

Their paper is Healthcare impact of covid-19 epidemic in India: A stochastic mathematical model (Drs. Kaustuv Chatterjee, Kaushik Chatterjee, Arun Kumar Yadav and Shankar Subramanian, Medical Journal Armed Forces India, 2 April 2020). It is worth looking at in full, for there are plenty of interesting projections their model throws up. Unfortunately, I have the space here for only this sample:

* The total number of cases will reach 3 million by 25 May and “India’s healthcare system would be completely overwhelmed". However, this assumes a daily growth rate of 15%, which was a reasonable assumption in mid-March. But at least, according to the data we have, the growth rate has slowed: The average since then is 14%, but just under 10% over the last two weeks (as I write this), and showing signs of falling further.

* For every 1,000 people 80 years and older who are infected with corona, we can expect 273 will need hospitalization and 93 will die. For the 20-29 age bracket, the corresponding numbers are 12 and 0.3. These and other age-specific numbers are broadly known, but it is still revealing to see them laid out.

* A quarantining rate below 50% is “ineffective", so the aim of NPI measures (lockdown, distancing, hygiene, etc.) should be to take that rate above 50%. The infection numbers drop dramatically if we can reach that level, so this is one of the team’s key recommendations. In fact, the model suggests that with this, we will reach a peak in the number of active infections (I - R - D) by mid-April. That we haven’t yet reached that peak is a sign of both the slowing growth rate and that we have probably not managed quarantining at that level.

Do we have all the answers? Of course not. Yet the feverish pace at which so many are pursuing different lines of inquiry into corona is, for me, the assurance that we will beat this thing.

P.S.: This column owes much to an excellent “Numberphile" video explaining the SIR model.

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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