Exponential 2

Bunkered down in self-quarantine as the world reacts to SARS-CoV-2, aka coronavirus - which causes the disease COVID-19 - I’ve been shifting between productive work at home, news fatigue, distractions on Facebook and Youtube, and sometimes fits of anxiety. My worries are mostly about the health and safety of family, friends, and co-workers, but also the future of humanity, rethinking social constructs, the collapse of end-stage capitalism - you know, lightweight stuff. It’s the anxiety that often keeps me up late at night pouring over articles about the virus and statistics about infection growth, deaths, and recoveries. Not great for quieting the mind when it’s well past time for sleep.

It’s looking at the numbers that gets me thinking again and again about exponential growth. In my head I try to calculate the powers of 2 using base2 prefixes because I’ve spent a many years working with computer storage and a fair amount of that involves converting from base10 to base2 and back again. But I keep getting lost around pebi and then having to start over: 1, 2, 4, 8, ... 256ti, 512ti, 1pi, 2pi - wait what square am I on? It's far easier to use a calculator. For more on decimal and binary prefixes check out this Wikipedia page about Binary prefixes - I’d say it’s a fun discussion for another time, but some people might argue about the “fun” part.

Grains on a Chess Board

An old story goes something like this:

According to Wikipedia the first mention of this story dates to 1256AD (Wheat and Chessboard Problem), and there are many variations - different grains (rice vs wheat), maybe a sultan or emperor instead of a king, perhaps a traveling sage or con man instead of the inventor. Some versions include the king exacting revenge by requiring the inventor count each grain before collecting the prize - which is impossible because the time required to count all of the grains - at say 1 grain per second - is millions of years.

I prefer the version about the origins of an Indian rice pudding called Paal-Payasam (from the story of Ambalappuzha Sri Krishna Temple) in which the god Krishna disguises himself as a wandering sage who plays against a king who loves chess. When the enraged king realizes he cannot pay the reward, Krishna reveals himself and says the king and his people can pay it over time by serving it in the temple every day to pilgrims. It’s a lot nicer than killing someone.

Here is an illustration of a chess board and the number of grains in each square which makes it a little easier to visualize the rapid growth of these numbers:

Another perspective is to consider the length of rice grains placed end to end. A standard long grain of rice is between 6.61mm and 7.5mm according to the International Rice Research Institute. Let’s use 7mm as the average length for each grain. Here are the length of grains placed end to end and some distances in nature for comparison:

We can also visualize exponential growth in the graph below:

As you can see it is hard to follow what is happening from 20 to around 254 because the numbers are so small, and then they grow rapidly to over 9.2 quintillion (263). And ironically the numbers above aren’t completely accurate - it turns out spreadsheets such as Google Sheets and Microsoft Excel lose precision after 15 digits.

Infection Rates

Right, enough with agricultural products… What does this have to do with coronavirus?

The rate of infection varies for different viruses. Coronavirus infection rates with no intervention have been observed to double approximately every three days. In other words a single person will infect people nearby and three days later the number of infected people will be two, and after three more days four people, and three more days eight people, and so on. This is an exponential growth curve based on the powers of two and in the first 30 days it looks like this with over 1,000 people affected:

The big problem is that unless something changes in the environment, the rate of doubling continues such that over 1 million people are infected after 60 days:

Wait - that looks like the same curve - and yes it’s the same exponential growth curve, only three orders of magnitude (1,000 times) larger. Actually to be more precise, it’s 1024 times larger - that is the nature of exponential growth for 2n - the value grows 3 orders of magnitude for every 10 powers:

If the rate of growth does not change, within 90 days over 1 billion people will be infected:

Well, it isn’t that simple - the real world is more complex with different continents, dense urban and sparse rural areas, and despite modern forms of transit there are ways the virus can slow and speed it's transmission that will not match the smooth mathematical graph each day.

Thankfully several countries have already put significant measures into place that are reducing the rate of infection. For example, China has slowed the virus to just a few cases per day through drastic social quarantine in multiple regions - compared to thousands of new cases per day just a few weeks ago. Japan seems to have responded early and significantly, perhaps due to experience with SARS a few years ago which had a major impact on the country, and to date the rate of growth in Japan has been much slower than others.

But most countries have not reacted as quickly or as dramatically which means potentially very large spread in some large populations. As an example, the United States is the third largest country on Earth by population with ~330 million people, yet infection rates are growing quickly while testing remains very limited and many states and cities have yet to implement significant physical isolation.

And as a wise friend of mine pointed out, infection rates are not strictly exponential - rather they are sigmoid, or “S-shaped” because eventually a population runs out of people who are not infected and thus the rate of growth tapers off:

Also, people do not stay infected forever - they eventually recover or unfortunately sometimes they die. An overly simplified example of the eventual slow down and then decline in number of cases is often portrayed as a normal distribution curve:

There are many factors such as population centers and interaction patterns, gender and age impacts on viral infection, and others which make the real world graphs much more complex and rarely smooth like the examples above. This is where epidemiologists and data scientists have provided much better insights about infection growth, factors, and potential outcomes. Imperial College in the United Kingdom has a 16-Mar-2020 paper on non-pharmaceutical interventions for treatment of COVID-19 that includes a lot of excellent information: Imperial College - COVID-19 NPI Modeling

Okay, okay - so what’s the big deal?

Surely many people have heard by now about the importance of “flattening the curve” which is vital to avoid overwhelming our health systems. Several articles have now covered the literal life and death implications of limiting the number of people infected because there simply aren’t enough hospital beds, ventilators, or trained medical staff to handle the number of critical cases if infection rates are not quickly and aggressively slowed.

Thankfully there does not appear to be a lot of people who remain sick from COVID-19 for a very long time. However, most municipalities and countries are likely run into massive health, social, and economic problems long before the tapering off happens - unless we take significant actions to reduce the rate of new infections.

According to the World Health Organization’s interim guidance dated 13-Mar-2020, approximately 14% of COVID-19 cases require hospitalization and oxygen support, and 5% of cases require admittance to Intensive Care Unit (ICU). One of the big problems most countries will face if the number of infected people grows very large is a lack of ICU beds and ventilators to treat the critically ill.

Let’s consider Italy for a moment:

But we know the entire population of Italy won’t get infected all at the same time. Several models have discussed saturation (eg. simultaneous infection) at 50% or 60% of a given population. Let’s be even more conservative - say 10% of the population is infected at the same time:

Instead let’s consider when Italy would run out of ICU beds - assuming all of them are vacant and available for use (which is typically not the case). At roughly day 52 the number of critical cases requiring ICU beds exceeds the available number of ICU beds in the entire country.

Consider the United States of America which has more ICU beds per capita than Italy:

The examples above are grossly oversimplified, but hopefully it becomes clearer why the situation is potentially so dangerous. And as discussed earlier, there are multiple resources that need to be carefully managed in addition to ICU beds - also ventilators, urgent care staff, hospital supplies, and more.

Running Out of Beds

The big problem actually isn’t running out of beds, it is the rapid decline in patient outcomes when there are no beds or ventilators available. People requiring critical care may not get the help they need and die as a result. Medical staff are pushed beyond their limits, their health can suffer and they may be more susceptible to infection despite protective equipment, and mistakes could be made amidst the chaos dealing with hospital overload.

There are two main solutions for preventing an urgent crisis in the healthcare system:

1. Flattening the Curve - at it’s simplest, reduce the rate of infection to keep the number of people requiring hospitalization, ventilation and ICU care below capacity of the healthcare system

2. Add Capacity - this is often harder, although China has already demonstrated the massive resources they can bring to bear in building a new hospital to treat COVID-19 cases in roughly 2 weeks

Unfortunately many countries do not have the ability to build new hospitals in a short period of time, but there are other options. For example, build temporary facilities in tents, adapt existing buildings or military ships. These may seem far fetched, but in a time of crisis it could be the difference between thousand and tens or hundreds of thousands of people dying.

Flattening the Curve

Here in the Republic of Ireland there are approximately 260 ICU beds for a population of roughly 4.8 million people. If we flatten the curve of critical cases we may be able to avoid overwhelming hospitals and have far better outcomes for a majority of people - patients and medical workers combined.

Washing hands thoroughly is very important, but flattening the curve is best done by isolating people so the rate of infection is far lower than normal spread of the virus. China and South Korea are good examples where dramatic isolation led to a rapid reduction in infection rates. Unfortunately in both cases there were still large spikes and stresses on their respective healthcare systems, but it could have been much, much worse.

So stay home, work from home, checking on neighbors, but avoid in person social gatherings. If you do have to go out, limit your trips to essential supplies and perhaps exercise, but avoid coming in contact with others.

General recommendations are to maintain at least 2 meters (about 6 feet) distance from others, continue to wash hands diligently with soap for at least 20 seconds, and avoid touching un-sanitized surfaces.

Hopefully most communities will have better outcomes by avoiding major issues in their healthcare systems through flattening the curve.

We will get through this

At the same time that we physically isolate ourselves to slow the rate of infection, it is equally or possibly more important to keep in touch with family, friends, and our communities - either at a safe distance or via phone and social media.

Staying connected is an important part of our mental and emotional health. We human beings are social creatures. We need community - it helps build and sustain compassion and understanding, provides context and reminders of the importance of the whole in addition to the individual. And it is not uniquely human - numerous studies have shown many species crave and thrive companionship and working together as a collective.

In the meantime consider what else we might do with the power of 2. During these challenging times, try reaching out to 2 people you know every day on Facebook, Messenger, Instagram, WhatsApp, or whichever your favorite social media or video chat app is. Ask how they are doing, if they need anything, or just say hi. And if each of those people reach out to 2 more people they know, and so on, and so on...

It grows exponentially and that is incredibly powerful!