Exponential growth

The spread of covid-19 and epidemics in general is all about exponential growth. Although, I deal with exponential growth in my work daily, I don’t have an immediate intuitive grasp for what it means in life until I actually think about it. To me I write down functions with the form e^{rt} and treat them as some abstract quantity. In fact, I often just look at differential equations of the form

\frac{dx}{dt} = r x

for which x = A e^{rt} is a solution, where A is any constant number, because the derivative of an exponential function is an exponential function. One confusing aspect of discussing exponential growth with lay people is that the above equation is called a linear differential equation and often mathematicians or physicists will simply say the dynamics are linear or even the growth is linear, although what we really mean is that the describing equation is a linear differential equation, which has exponential growth or exponential decay if r is negative.

If we let t represent time (say days) then r is a growth rate. What x = e^{rt} actually means is that every day x, e.g. number of people with covid-19, is increasing by a factor of e, which is about 2.718.  Now, while this is the most convenient number for mathematical manipulation, it is definitely not the most convenient number for intuition. A common alternative is to use 2 as the base of the exponent, rather than e. This then means that tomorrow you will have double what you have today. I think 2 doesn’t really convey how fast exponential growth is because you say well I start with 1 today and then tomorrow I have 2 and the day after 4 and like the famous Simpson’s episode when Homer is making sure that Bart is a name safe from mockery, goes through the alphabet and says “Aart, Bart, Cart, Dart, Eart, okay Bart is fine,”  you will have to go to day 10 before you get to a thousand fold increase (1024 to be precise), which still doesn’t seem too bad. In fact 30 days later is 2 to the power of 30 or 2^{30}, which is only 10 million or so. But on day 31 you have 20 million and day 32 you have 40 million. It’s growing really fast.

Now, I think exponential growth really hits home if you use 10 as a base because now you simply add a zero for every day: 1, 10, 100, 1000, 10000, 100000. Imagine if your stock was increasing by a factor of 10 every day. Starting from a dollar you would have a million dollars by day 6 and a billion by day 9, be richer than Bill Gates by day 11, and have more money than the entire world by the third week. Now, the only difference between exponential growth with base 2 versus base 10 is the rate of growth. This is where logarithms are very useful  because they are the inverse of an exponential. So, if x = 2^3 then \log_2 x = 3 (i.e. the log of x in base 2 is 3). Log just gives you the power of an exponential (in that base). So the only difference between using base 2 versus base 10 is the rate of growth. For example 10^x = 2^{(log_2 10) x}, where \log_2 (10) = 3.3.  So an exponential process that is doubling every day is increasing by a factor of ten every 3.3 days.

The thing about exponential growth is that most of the action is in the last few days. This is probably the hardest part to comprehend.  So the reason they were putting hospital beds in the convention center in New York weeks ago is because if the number of covid-19 cases is doubling every 5 days then even if you are 100 fold under capacity today, you will be over capacity in 7 doublings or 35 days and the next week you will have twice as many as you can handle.

Flattening the curve means slowing the growth rate. If you can slow the rate of growth by a half then you have 70 days before you get to 7 doublings and max out capacity. If you can make the rate of growth negative then the number of cases will decrease and the epidemic will flame out. There are two ways you can make the growth rate go negative. The first is to use external measures like social distancing and the second is to reach nonlinear saturation, which I’ll discuss in a later post. This is also why social distancing measures seem so extreme and unnecessary because you need to apply it before there is a problem and if it works then those beds in the convention center never get used. It’s a no win situation, if it works then it will seem like an over reaction and if it doesn’t then hospitals will be overwhelmed. Given that 7 billion lives and livelihoods are at stake, it is not a decision to be taken lightly.

 

2 thoughts on “Exponential growth

  1. One of my relatives once asked me to figure out the rate of return on her investments or bank acount–she had taken calculus but couldn’t remember the formula or recipe.I had taken calculus as well but i had to rederive the exoponential by hand (the professor i took the calculus course from was likely a grad student with a foreign accent i couldn’t understand–he mostly rewrote the textbook on a chalkboard–i do not know if they still use these in colleges.) i still can’t really derive the tsallis distribution.

    One probably needs at least a logistic or ‘tanh’ s-curve . (I think one needs a ‘nonlinear fokker-planck equation’ or its discrete analog which is even more complex.)

    https://arxiv.org/abs/1409.7588 this is the kind of thing i read since you can’t go outside (tho i can a bit or go for a hike)–i just noticed its ‘local’. my bank account is more going on a linear or exponential decrease.
    un/fortunately its raining now. (one of my favorite songs is ‘i can’t stand the rain’ by rare essence of dc).

    Like

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