 The
plot illustrates:
 Linear growth:
red
curve: y = a*x.
 Cubic growth:
blue
curve: y = a* x**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 numberit'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 efolding constant
t_e = 1/k is
multiplicative inverse
(AKA reciprocal)
of k.
The efolding 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 accountsa wonderful
discovery
in banking.
 Actually a key
biotic population
in the Covid19 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:
 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
Covid19
with NO social precautions
(see Wikipedia:
Basic reproduction number R_0;
How Long Is COVID19 Contagious? 2020 Nov04).
 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 COVID19 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 = (β1/τ)A = [(βτ1)/τ]A ,
which has the solution
(which is what is true for all time for a particular
initial condition)
A = A_0*e**[(R_01)/τ]t ,
where
A_0 is the infectious population
at time zero,
the efolding time
t_e = τ/(R_01),
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:
 R_0 > 1 giving
exponential increase.
 R_0 < 1 giving
exponential decrease.
 R_0 = 1 giving
constant infectious population.
This steadystate case can hold approximately,
but holding nearly exactly must be very rare.
 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.
 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 postinfection 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.
 Note that
basic reproduction number R_0
for the original
Covid19
with no social precautions is estimated to be in the range 3.35.7 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 COVID19 Contagious? 2020 Nov04).
That gives β = 6/10 = 0.6 persons infected per day and
efolding time
t_e = τ/(R_01) = 2 days.
So in 1 efolding time = 2 days, the
infectious population
increases by a factor of
e = 2.71828 18284 59045 23536 ...
and in
5 efolding times = 10 days,
the infectious population
increases by a factor of 5e ≅ 150
(CAC32).
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
infectedexponential 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).
 For more information on the mathematical modeling of
infectious disease, see
Wikipedia:
Mathematical modelling of infectious disease.
 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 semilog plot
where the yaxis
is logarithmic
infection rate
and the xaxis
is time.
On such semilog 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 semilog plots
of infection rate.
The common assumption seems that the general public
CANNOT understand
semilog plotseven though
a onesentence description suffices for anyone to understand them.
 Image 2 Caption:
Fortunately, the Our World in Data has
semilog plots.
For the Covid19 pandemic, see
Our World in Data: Coronavirus (COVID19) Cases
(via the displayed placeholder image
alien_click_to_see_image.html)
and click on the log option.