Mathematician Arni SR Srinivasa Rao has been trying to understand the real magnitude of the pandemic. To do this, Rao, Director of the Laboratory for Theory and Mathematical Modelling at August University’s Medical College of Georgia, in the US, along with his peer Steven Krantz, professor of mathematics at Washington University, has been trying to mathematically ascertain the number of unreported cases of Covid-19 in many countries, including China, Italy, Spain and the US, where the infection wreaked havoc.

In their study, recently published in the journal Infection Control and Hospital Epidemiology, they visualised the disparities between reported cases and what they projected using what is called a Meyer wavelet. The higher the wave, the higher the under-reporting; lowering the wave means improved reporting.

Accordingly, they projected that the number of cases reported in China was anywhere between 1 in 149 and 1 in 1,104, whereas that in Spain was 1 in 53. In South Korea and Italy 1 in 4 cases come to the fore while in Germany and in the US they detected 1 out of 3 and 2 out of 3 cases, respectively. In a separate correspondence published in the April 10 issue of the Current Science journal, Rao and Krantz, along with others, projected India may also be detecting 1 out of 4. Rao had earlier developed mathematical models to understand the spread of HIV, avian flu and swine flu in India. Excerpts from an email interview:

Many groups have been trying to mathematically model the pandemic. But the projections using such modelling exercises do not often agree with one another...

Different mathematical techniques and tools can derive different results from the same set of data. We see a variation in predictions because either two models might have expressed it differently with different variables or applied a different set of parameters, or it could be both. The models I built in collaboration with Prof Steven Krantz involved differential equations and wavelets.

What are the major challenges faced by mathematicians trying to model a pandemic like this?

The deeper we understand the data the better we can build accurate models. Sometimes with very little data as well we should be able to build models, but in such cases, the predictions will have a wider range of windows within which the true cases could fall.

You seem to have tried to understand the under-reporting that has been happening in different countries. Why do you think this is important?

Unless we know how many more people are in the population who have the virus but not diagnosed yet, we do not know the true magnitude of the epidemic. We studied several countries' data because the wave in each country could be different and the speed of the spread could be different based on the composition of the population and behaviours. The wavelets we drawn for each country demonstrated these differences.

For the paper, you considered the cases till March 9. Did you have a chance to analyse the numbers at a later date? If so, what were your findings?

Yes, we did look after March 9. For example, in the US, the reporting was around 1 in 6 sometime around the middle of March, and the reporting as of April 6 was estimated at 2 out of 3. This happened due to a rapid rise in the number tested. During the last 15 days, the severity of Covid-19 in terms of cumulative positivity rate among the tested went down in Italy, whereas this rate increased in Canada, France, Germany, Japan, the UK, and the US.

What is your reading on the situation in India? Are the steps being taken by the country good enough to contain the pandemic?

Based on the model-based predictions, when we look at the data from India, by the end of March almost 1 in 4 Covid-19 cases were identified. But the other three who were not been tested could be spreading or might be taking precautions — we never know accurately. The steps taken by the Central and State Governments and Indian Council of Medical Research (ICMR) were in the right direction. People have to cooperate with the guidelines, and I guess they are doing it to a large extend. If the spread among 60+ is somehow controlled — especially those who have prior medical conditions like hypertension, cardiovascular, lung diseases and diabetes — a large number of ICU admissions can be avoided. This will also reduce a large number of deaths. The youth may not see that many hospitalisations but they could be carriers to their elders in their homes if proper precautions are ignored.

The testing strategy by ICMR is in the correct direction because unnecessarily increasing the number tested randomly will have no gains if ICMR has pieces of evidence that they are predominantly negative people. Statistically, it is good to show that the positivity rate is low by testing many, but that could lead to a waste of resources. However, the advantage of large scale random testing is that it could catch people who are asymptomatic. But in any large country like India, it is not easy to conduct random testing in such a short time.

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