Image 1 Caption: Exponential growth is when the rate of increase of a quantity is proportional to the amount of the quantity.

Features:

The plot illustrates:
1. Linear growth: red curve: y = a*x.
2. Cubic growth: blue curve: y = a* x**3.
3. Exponential growth: green curve:
  y = a*e**x ,
  where e = 2.71828 18284 59045 23536 ... is a mathematical constant just called e: it's an irrational number---it's mad, bad, and dangerous to know. Actually, irrational number are real numbers that CANNOT be written as the ratio of two integers: i.e., they are NOT rational numbers. Their trailing numerical digits go on forever without terminating in 0 and without becoming a repeating decimal sequence.
  The function e**x is called exponential function.
Exponential growth occurs whenever the rate of change is proportional to the amount: i.e., in calculus notation
dA/dt=kA ,
where t is an independent variable (e.g., time), A is amount, dA/dt is rate of change of A (i.e., the derivative of A with respect to t), and k is the rate constant.
The mathematical solution of this 1st order linear differential equation is
A = A_0*e**(kt) = A_0*e**(t/t_e) ,

where A_0 is the amount of A at t = 0 and the e-folding constant t_e = 1/k is multiplicative inverse (AKA reciprocal) of k. The e-folding constant is the change in t needed to change A by a factor of e.
Exponential growth is actually very common.
Biotic populations without some controling factor (i.e., death) exhibit exponential growth.
There was a time long ago when bank accounts exhibited exponential growth. In the early 1990s, banks discovered they did NOT have to pay interest on bank accounts---a wonderful discovery in banking.
Actually a key biotic population in the Covid-19 pandemic is the infectious population. In a simple model, the infectious population will increase exponentially or decrease (inverse) exponentially. Of course, public heath organizations want to avoid the former and have the latter.
We can do a simple analysis to prove the exponential increase/exponential decrease:
1. Say β is the mean rate at which an infectious person infects: e.g., 0.6 persons infected per day which is of order the rate for Covid-19 with NO social precautions (see Wikipedia: Basic reproduction number R_0; How Long Is COVID-19 Contagious? 2020 Nov04).
2. The change in infectious population due to new infections is
```
  (dA/dt)_new = βA , 
```
  where A is the current infectious population and we have assumed that an infected person becomes an infectious person with NO time delay. Now we hypothesize the probability of continuing to be infectious has exponential decrease according to P=e**(-t/τ). This formula leads to τ being the mean length of the infectious period (e.g., 10 days: see How Long Is COVID-19 Contagious? 2020 Nov04) and also leads to
```
  (dA/dt)_ending = -A/τ 
```
  being the rate of decrease of infectious population due to the ending of infection. The total change in infectious population then obeys the 1st order linear differential equation (which is what is true at every instant in time)
```
  (dA/dt) = (dA/dt)_new + (dA/dt)_ending = (β-1/τ)A = [(βτ-1)/τ]A , 
```
  which has the solution (which is what is true for all time for a particular initial condition)
```
  A = A_0*e**[(R_0-1)/τ]t  , 
```
  where A_0 is the infectious population at time zero, the e-folding time t_e = τ/(R_0-1), and R_0 = βτ is the famous basic reproduction number R_0, the mean number of persons an infectious person infects during the mean infectious period.
  We see there are 3 cases:
  1. R_0 > 1 giving exponential increase.
  2. R_0 < 1 giving exponential decrease.
  3. R_0 = 1 giving constant infectious population. This steady-state case can hold approximately, but holding nearly exactly must be very rare.
3. Now social precautions and medical factors probably CANNOT change τ much. However, obviously β and R_0 can be changed by social precautions. Isolation of all infectious persons reduces them to zero and the infectious population exponential decreases as rapidly as τ allows.
4. The above analysis is very simplified, but it captures some of the main effects. One effect it does NOT capture is that β and R_0 effectively decrease as there are fewer people to infect. This will happen as the infectious population becomes a significant part of the total population or, if there is post-infection immunity, as the infectious population plus the immune population becomes a significant part of the total population. So there is a saturation and the infection rate slows and the infected/immune population approaches the total population. If a large enough part of the total popultion becomes immune either naturally through infection or artificially through vaccination, then herd immunity is achieved and β and R_0 become very small and infectious population exponential decreases ideally to effectively zero as rapidly as τ allows.
5. Note that basic reproduction number R_0 for the original Covid-19 with no social precautions is estimated to be in the range 3.3--5.7 which is pretty high (see Wikipedia: Basic reproduction number R_0). Say we take R = 6 as a round number and 10 days for the infectious period (see How Long Is COVID-19 Contagious? 2020 Nov04). That gives β = 6/10 = 0.6 persons infected per day and e-folding time t_e = τ/(R_0-1) = 2 days. So in 1 e-folding time = 2 days, the infectious population increases by a factor of e = 2.71828 18284 59045 23536 ... and in 5 e-folding times = 10 days, the infectious population increases by a factor of 5e ≅ 150 (CAC-32). If A_0 = 10,000, by day 10 there are 1.5*10**6 people infected and by day 20 there are 2.25*10**8 = 225 million people infected---exponential growth. Of course, saturation is likely to start before 225 million people are infected in a country with the population size of the United States (2020 population estimate 329,484,123: see Wikipedia: Demographics of the United States).
6. For more information on the mathematical modeling of infectious disease, see Wikipedia: Mathematical modelling of infectious disease.
7. Since epidemics and pandemics naturally tend to increase/decrease exponentially, the natural and most useful way to display their time dependent behavior is on a semi-log plot where the y-axis is logarithmic infection rate and the x-axis is time. On such semi-log plots exponential increase/decrease regions are regions of straight lines with positive/negative slope. You are losing/winning if the slope is positive/negative.
  Alas, it's almost impossible to find websites with semi-log plots of infection rate. The common assumption seems that the general public CANNOT understand semi-log plots---even though a one-sentence description suffices for anyone to understand them.
8. Image 2 Caption: Fortunately, Our World in Data has semi-log plots. For the Covid-19 pandemic, see Our World in Data: Coronavirus (COVID-19) Cases (via the displayed placeholder image alien_click_to_see_image.html) and click on the log option if the good folks at Our World in Data ever restore the log option.

Images

Credit/Permission: User:Lunkwill, 2005 / Public domain.
Image link: Wikipedia: File:Exponential.png.
Credit/Permission: Our World in Data, Current year / No permission: click on placeholder image alien_click_to_see_image.html.
Image link: Our World in Data: Coronavirus (COVID-19) Cases.

Local file: local link: exponential_function_plot.html.
File: Mathematics file: exponential_function_plot.html.

y = a*e**x ,

dA/dt=kA ,

A = A_0*e**(kt) = A_0*e**(t/t_e) ,