Use and Abuse of the Word “Exponential”

There has been much stress and panic over the COVID-19 problem. Included in that has been plentiful use of the word “exponential.” Sometimes that word is appropriate, often (by journalism majors who never took a math class) not. It occurs to me that a little remedial algebra might be useful, so here goes.

Things relate to one another in a variety of ways. One of these is “linear.” The essence of this relationship is that the output is proportional to the input. So, below on the left plot, you put in one “x” on the horizontal (bottom) axis and you get back one “y” on the vertical (left) axis.

Things relate to one another in a variety of ways. One of these is “linear.” The essence of this relationship is that the output is proportional to the input. So, below on the left plot, you put in one “x” on the horizontal (bottom) axis and you get back one “y” on the vertical (left) axis.

On the right plot you put in one “x” to get back two “y” but the relationship is still proportional (note the vertical axis is different while the horizontal axis remains the same). Whatever number you multiply “x” by gets you the same relative amount of “y.” Always. It can be y = 3x or y = .5x, it does not matter, the relationship is always represented by a line (hence: “linear”) and the output is proportional to the input. Such multiplication can make things larger but not “exponentially” larger and there is a difference.

The point is that “bigger” can be described as “proportionately bigger” or just simply “bigger” when the subject is in fact simple. There are quite a few linear relationships between physical objects. There are less of these when the objects are people because people are not simple.

Below are two power function curves, each involves a power relationship between x and y, a relationship which is non-linear. Notice that by increasing the exponent the curve ascends more rapidly and the values on the left (Output) axis rise faster.

Below is an interactive treatment. Change the exponent and you to see (a) what happens when linear switches to non-linear as you go from exponent = 1 to a larger exponent; (b) how rapidly the values on the left axis increase; and (c) how the curve eventually “explodes” or “points to the sky.” These results are because x is not being increased by a simple number but it is being multiplied by itself and sometimes many times over!!

There are various ways the word “exponential” may be properly used. Simple cases appear above. Here is another which is a bit more complex.

One sometimes hears that cases are doubling every __ days. The equation for this is

One sometimes hears that cases are doubling every __ days. The equation for this is

y = initial value

x

2

Again we have an exponent but the overall relationship is linear since 2 raised to any power is just a number, albeit larger. Often our interest here is in learning how long it will take, doubling in each cycle, for something to grow to a particular value. For instance, suppose you have 100 cases of a communicable disease that doubles every week (one cycle = 7 days). Your interest is in how long it will take to reach 10,000 cases. The equation is:

10,000 = 100

x

2

Where x is the number of cycles. If you solve for x (which involves the use of the exponential in a different context) you find that 100 = 10,000. Thus, just less than 7 weeks (46 days) from now at a constant rate of contagion the number of cases will have grown from 100 to 10,000

6.64386

2

Below is an interactive device that allows you to vary the inputs and display the resulting output. Lower results are more dramatic.

In what follows we will use the default values shown above of 900 initial cases, a cycle length in days of 8.3 and a target value of 45,000 cases where the number of days to reach the target is 46.844. This is a stylized illustration with arbitrary limits intended to demonstrate how the complexity of exponential relationships among many variables may be simplified to answer a specific question.

Below, in blue, is a plot of the equation which produces a result (the vertical axis showing the number of days to reach the target) from the two inputs, cycle length and initial cases. The plot is curved because the equation which produces it involves exponents. On top, in goldenrod, is a flat plot of the 45,000 case target, with a static value on the vertical axis of 46.844 days. The static plot “slices” the blue plot along a red line which represents all the combinations of initial cases and cycle length which may be combined to produce 46.844 days.

What, you may ask, does this do for us?

Below, in blue, is a plot of the equation which produces a result (the vertical axis showing the number of days to reach the target) from the two inputs, cycle length and initial cases. The plot is curved because the equation which produces it involves exponents. On top, in goldenrod, is a flat plot of the 45,000 case target, with a static value on the vertical axis of 46.844 days. The static plot “slices” the blue plot along a red line which represents all the combinations of initial cases and cycle length which may be combined to produce 46.844 days.

What, you may ask, does this do for us?

The answer to that question comes from peering down from above while placing gridlines as the next plot shows. Doing so tells us that the time to reach 45,000 cases will exceed 47 days only in some situations when the initial cases are less than 400 and the cycle length is more than 7.

The medical system is scaled based on a throughput of morbidity during times when sickness and health are “normal.” The economics of any system necessarily involve rationing which gets worse when demand increases. A calculation such as the one we just examined helps to allocate scarce resources among alternate ends, the central question of economics. On the “panic” end of that spectrum we find words such as “overwhelming the medical system” and the like. Mathematically this can be modeled by moving the red line in the plot revealing more or less cases above some flat plane of sustainability.

These are just a few very simple examples of the sort of thought that needs to go into using the word “exponential.” Since it has become such an instrument of hype it is unlikely that the media will abandon one of its favorite inflammatory terms. But a thoughtful and discriminating public can blunt that effect by better understanding of the meaning of technical terms.

These are just a few very simple examples of the sort of thought that needs to go into using the word “exponential.” Since it has become such an instrument of hype it is unlikely that the media will abandon one of its favorite inflammatory terms. But a thoughtful and discriminating public can blunt that effect by better understanding of the meaning of technical terms.