Introduction
The phrase “exponential growth” has found its way into our vocabulary. People use the phrase as a synonym for rapid growth. This is unfortunate since exponential growth is a particular form of growth. In the short term linear growth might actually be faster than exponential growth, but not in the long run.
Here we will explore exponential growth as the simplest model for Epidemiological growth. It fails miserably, in the long run, but might work well in the short run. However the failure shows the way to improve the model.
Exponential growth never lasts forever.
Let start with the simplest epidemiological model: Exponential Growth. When someone talks about doubling time, they are talking about exponential growth. Doubling time is the length of time it takes something to double in size. During exponential growth this time is fixed. If the number of birds in the back yard doubles in 1 week, it will double again in the next week, again in the next week, …, until you are awash in birds. In linear growth the same number of birds would be added every week; for instance 10 the first week, 10 the second week, 10 the third week,…, until you are awash in birds.
All growth is ultimately limited. Therefore, growth ultimately stops, something limits it. What is unique to exponential growth is how fast this happens.
Let’s consider an exponential growth of an infection in a population with a doubling time T. The infection grows as the following
| Doubling times | Number infected | Bigger than |
| 0 | 1 | |
| 1 | 2 | |
| 2 | 2×2=4 | |
| 3 | 4×2=8 | |
| 4 | 8×2=16 | |
| 5 | 16×2=32 | |
| 6 | 32×2=64 | |
| 7 | 64×2=128 | |
| 8 | 128×2=256 | |
| 9 | 512 | |
| 10 | 1024 | |
| 11 | 2048 | |
| 12 | 4096 | |
| 13 | 8192 | |
| 14 | 16,384 | |
| 15 | 32,768 | |
| 16 | 65,536 | About population of Cayman Islands |
| 17 | 131,072 | |
| 18 | 262,144 | |
| 19 | 524,288 | |
| 20 | 1,048,576 | About Population of Rhode Island |
| 21 | 2,097,152 | |
| 22 | 4,194,304 | |
| 23 | 8,388,608 | About Population of New York City |
| 24 | 16,777,216 | > Population of Wuhan China |
| 25 | 33,554,432 | |
| 26 | 67,108,864 | >Population of Italy |
| 27 | 134,217,728 | |
| 28 | 268,435,456 | |
| 29 | 536,870,912 | >Population of United States |
| 30 | 1,073,741,824 | |
| 31 | 2,147,483,648 | >Population of China or India |
| 32 | 4,294,967,296 | |
| 33 | 8,589,934,592 | >Global Population |
So 33 doubling times and the number of infected exceeds the total number of people on the Globe. At this point there is no one left to infect! So exponential growth must stop.
For example, let’s assume a single individual gets infected in the Cayman Islands. Let’s also assume that the doubling time is 1 week. 16 weeks later the entire population of the Cayman Islands is infected. 17 weeks after than the entire population of the world is infected. Thus, the time to infect the first 65 thousand is the same time required to infect the next 4.3 billion (approximately). While, the next doubling exceeds the number of people on the planet.
What about the Current situation? (3/24/2020)
Now lets apply what we have learned to the current situation. The number confirmed cases of COVID-19 in the United States from 3/15/2020 to 3/24/2020 is shown in the graph below. Also shown is and an exponential fit, along with the mathematical equation. The data was recorded from the John Hopkins site.
During this time the US followed an exponential curve with an exponent of 0.3129. As a consequence, every day the number of cases in the US increased by 38%. It turns out that the equivalent doubling time is given by ln(2)/0.3129. Here ln() is the natural logarithm and ln(2) is approximately 0.693. As a result, the US cases were doubling every 2.2 days.
As I edit this John Hopkins reports there are 46,485 known cases in the United States. I know this is low, but for the argument that follows it doesn’t matter much. Go to the table above. The closest entry is between the 15th and 16th doubling. That means if we continued at this rate for 18 doublings (18*2.2 days = 40 days) the epidemic would end because everyone in the world would be infected. THIS IS NOT A PREDICTION! What this is telling us is that the current pattern will change sometime within the next 40 days. Given the nature of exponential growth, sooner is better than later.
Evaluating Exponential models
To be useful, models need to be solved either with pen and paper, or on a computer. Follow this link to look at the exponential model as an ordinary differential equation, and a simple python program to evaluate it. https://davidalarrabee.com/?page_id=845
What’s next (Part II)?
In conclusion, although exponential growth can happen in the early phases, it cannot last. Therefore, exponential growth is a short term situation. Perhaps the first issue we can explore with modeling is an answer to the question “when does exponential growth stop?”
This link will take you to the next level of modeling, where the total number of people who can be infected is limited to the population. https://davidalarrabee.com/?page_id=717