Hold the Hurrahs

By Anjum Altaf

A number of congratulatory articles have lauded the containment of the Covid 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 appreciation of the nature of exponential growth. Take a look at the number of Covid-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, 192. Ignoring the initial turbulence, the number of deaths are 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, 196,608. By 23 June, just two months from now, the number of deaths could be in the neighbourhood 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 is 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 deaths 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 number 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, 5,000 infected persons had arrived in the USA, 2,500 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 for ever 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 has a PhD from Stanford University. This opinion appeared in The Sindh Courier on April 27, 2020 and is reproduced here with the author’s permission.

Back to Main Page

4 Responses to “Hold the Hurrahs”

  1. Faizaan Qayyum Says:

    It’s hard to disagree with the crux of your message. What may happen over the next few months is indeed terrifying & the quality of our leadership really isn’t helping.

    I’m just thinking out loud here without empirical backing & wondering what you think: exponential growth is limited by the total population (that you’ve already pointed out). Should it not be limited by social interactions as well? We don’t go out and meet everybody everyday. Most people would have a certain (fixed?) routine, interacting with a fixed set of people, facilities, spaces, and networks. What is the probability that they would come into contact with completely new people at one (or more) points in their everyday routine?

    This would bring every infected individual (transmitter) to a micro level saturation point of sorts, where they’ve already interacted with everybody they could interact with over a certain duration. This is similar to the limit of total population that you mentioned, but at an individual scale.

    So then hypothetically everyone in those networks of everyday interactions would be infected, but it would take a long time for the virus to reach people outside of those networks. If this holds the proportion of population at risk of infection at any given time would effectively be much smaller than the total population, and the rate of growth could remain slow for much longer.

    This would change in cities with functional transit systems. If you take the subway everyday in NYC you are stuck together with hundreds of unknown individuals, & the odds of running into the same people in the same train everyday aren’t too high. Each infected individual would transmit the virus to many more people and possibly never reach a micro level saturation point.

    Rules like lockdown and social distancing work by further limiting these interactions. In countries like Pakistan it would mean upper class households would receive the initial brunt of infections (from people traveling from abroad). As they start to stay at home and isolate themselves, the latter part of spread would mostly hit people who can’t afford it. And because we don’t have functional transit & the handful of available facilities aren’t running, a large part of the population may remain protected from exposure for much longer.

    This could explain why the trajectory of infections has been so different in CA than, for example, in NY. This may also help explain the difference in vulnerability within these states.

    Alternatively the inflection point that you mention would make all this irrelevant after a few weeks and we’d be underwater before we know it.

    What do you think?

    • Anjum Altaf Says:

      Faizaan: You are extrapolating from your’s (perhaps) and my (certainly) life styles, which is very flattering. But the vast majority isn’t at all like that. People go out to eat more than once a week, they go to drink in the evenings, to gyms or swimming pools in the afternoons, to dances, games, or beaches on the weekends. They commute using public transport. Etc., etc. And virus transmission is not limited to face-to-face interaction. You can pick it up from all sorts of surfaces and in public spaces (campuses, cafeterias, coffee shops, restrooms, classrooms) where you have no idea who has touched what.

      Lockdowns and social distancing are simply intended to cut down the rate of transmission by reducing all the types of social interactions mentioned above. The following explains that very well: https://www.bbc.com/news/health-52473523

      The infections in NYC and LA are different for a host of reasons — density, reliance on public transit, proportion of poor Afro-Americans in the population, proportion of uninsured in the population, the fact that NYC is a much bigger hub for air travel from Europe, etc.

      I don’t believe you have the right picture of the spread of the virus in Pakistan. The bulk of it came in with middle and lower-middle class religious pilgrims — from Iran, the Tableeghi congregation, people coming back from Umra. A lot of these people went back to their homes and were hugged by hundreds of well-wishers. Many of the poor can’t sustain social distancing or hand-washing so transmission was most likely fairly rapid. We actually don’t know yet how lethal the infection is because we don’t have prevalence data.

      The reason the reported numbers are low to date is not because there is much less public transit, etc. It is more likely related to the very different scale of the initial seeding as I have highlighted in the article, a fact that has not received the attention it should. Also, note that the numbers continue to double every ten days or so. If this continues, and there is no indication at the moment that it won’t, the numbers won’t stay small very long. That is the thing about exponential growth that just fails to sink in — by the time you get to square 64 on the chessboard, you are talking trillions.

  2. Anjum Altaf Says:

    I had argued in the article based on common sense that the very different speeds at which the virus is spreading in various locations may be due primarily to nothing more than the initial conditions, what i called the seeding. This still remains a neglected explanation in favour of many other optimistic ones like weather, exposure to BCG, etc.

    Now, enough data has accumulated for the numbers to be crunched and the seeding hypothesis is coming out as a lot more credible.

    Recall that Wuhan sealed the city but NYC did not. Wuhan had over 80,000 confirmed cases; the city with the second highest number of cases in China had about 1,500. We also allowed the virus to disperse all over from Taftan and Raiwind.

  3. Anjum Altaf Says:

    The numbers problem could be the same for India.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: