Hold the hurrahs

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The first two COVID-19-related deaths were reported on March 18. Since then, the cumulative numbers at weekly intervals up to April 21 are as follows: 7, 26, 55, 96 and 192. Ignoring the initial turbulence, the number of deaths is doubling roughly every seven days. Staring from 192 deaths on April 21, the sequence would be as follows at one-week intervals: 384, 768, 1,536, 3,072, 6,144, 12,288, 24,576, 49,152, 98,304, 196 and 608. By 23 June, just two months from now, the number of deaths could be in the neighborhood of 200,000.

 By Anjum Altaf

A number of congratulatory articles have lauded the containment of the Covid-19 epidemic in Pakistan based on the relatively smaller number of deaths to date (222) compared to those in the USA (47,973) and Italy (25,085). The authors have also offered a number of explanations for this difference ranging from outstanding management to contextual variations.

I would love these people to be right but would urge caution. The conclusion could be premature owing to a lack of appreciation of the nature of exponential growth. Take a look at the number of Covid-19-related deaths in Pakistan. The first two were reported on March 18. Since then, the cumulative numbers at weekly intervals up to April 21 are as follows: 7, 26, 55, 96 and 192. Ignoring the initial turbulence, the number of deaths is doubling roughly every seven days.

The lockdown along with its associated measures went into effect on March 18 but there has been no impact on the doubling time. This is contrary to the experience in many countries where the doubling time has lengthened considerably. One is forced to conclude that the lockdown in Pakistan has been largely ineffective which should not be a surprise if one has kept one’s eyes open to its enforcement.

The reality is that the curve of infections and therefore of deaths has not flattened at all. Extrapolate the number of deaths to see where the trajectory might be headed. Staring from 192 deaths on April 21, the sequence would be as follows at one-week intervals: 384, 768, 1,536, 3,072, 6,144, 12,288, 24,576, 49,152, 98,304, 196 and 608. By 23 June, just two months from now, the number of deaths could be in the neighborhood of 200,000.

Only a miracle can prevent this outcome because the crisis management has been inadequate. Even stretching the doubling rate from 7 to 14 days would only postpone the inexorable trajectory of exponential growth. What needed to beat the virus are measures that cause the number of new cases to decline as they did in China. The likelihood of that happening in Pakistan is low because preventive measures have been muddled — recall that Hubei province was sealed off for two months and residents in Wuhan were similarly sealed inside their homes.

While uncertainties related to the virus and its transmission remain, a simple explanation can suffice for the seemingly huge difference to date between the death toll in the USA and Italy compared to Pakistan. Just one variable, the starting point, can be central. The number of foreign visitors per day in these countries probably exceeds the total numbers that arrive in Pakistan in a year. For a hypothetical order-of-magnitude comparison, suppose that by mid-February, before any preventive measures were put in place, 5000 infected persons had arrived in the USA, 2500 in Italy, and 10 in Pakistan. Use uniform doubling times of 7-days and mortality rates of 2% to project the number of deaths in the three countries. The pattern should become obvious. Pakistan appears to be doing much better at the moment but it is at most two months behind — that is the amazing arithmetic of exponential growth; once it passes an inflexion point the initial differences become irrelevant.

This is not to deny contextual differences. The virus is disproportionately affecting individuals over 60 years of age and the fraction in this cohort is much smaller in Pakistan. It transmits more in dense locations and Pakistan is over 50% rural. It is more lethal for low-income individuals with co-morbidities and in Pakistan such individuals have a low life-expectancy anyway. But even adjusting for these, the pool of vulnerable people in a country of 220 million remains huge (between 2 to 3 million low-income people over the age of 60 in urban Pakistan). Thus, despite the differences, the potential for the number of deaths to climb into the tens of thousands remains real.

There is great need to be realistic and cautious and to address the crisis with the seriousness it deserves. I would urge everyone to read the story that illustrates exponential growth. A sage impressed a chess-loving king and was promised anything he asked for. He requested the following: One grain of rice to be placed on the first square of the chessboard and the number doubled on each succeeding square. The chess board has 64 squares. It would be worth the reader’s while to go through with this computation.

Now relate this to the Covid-19 crisis. Consider the number of dead to be 1 on March 18 and double this number at the end of each week. Compute the number of dead after 64 weeks if nothing works in the interim to change the trajectory. Of course, this exponential growth cannot continue forever because after a point the population to be infected will get exhausted. But terrible damage can occur well before that point is reached.

(The writer, having a PhD from Stanford University, is a former Dean at LUMS and Provost at Habib University, Karachi)

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