Exponential Population Growth Equation: N = N_{o}e^{rt}

In the above population growth equation N is the final population after a certain length of time (t); N_{o} is the starting population; e is a constant (base of natural logarithms = 2.71828); r = growth rate (decimal value). In the above equation when rt = .695 the original starting population (N_{o}) will double. Therefore a simple equation (rt = .695) can be used to solve for r and t. The growth rate (r) can be determined by simply dividing .695 by t (r = .695 /t). In this case t is the doubling time. In the above equation t is the length of time to determine the size of final population (N). Note: Number of decimal places & rounding off can really affect the results when plugging decimal values into the equation. In addition, when projecting future population numbers with COVID-19, growth rates and doubling times can change significantly depending on strict or relaxed mandates from the government regarding mask-wearing, social distancing, etc.

Another example of exponential growth: 500 deaths grow to more than one million after 11 doubling times. 500 x 2^{11} = 1,024,000. After 10 more doubling times it would be more than one billion deaths. 500 x 2^{21} = 1,048,576,000.

The exponential COVID-19 infection increased from 100,000 to 160,000 between Friday 27 Mar and Monday 30 Mar 2020 (3:00 P.M.). This increase in infections during the same time period using the exponential population growth equation was about 172,000. In disease cases, rates of infection and doubling times are complex and are constantly changing. That is why results from Worldometer are not exactly the same as my exponential growth equation calculator (Table 2.) On Friday 3 April 2020 the number of cases in U.S. increased to 270,000.

Friday 27 March 2020 U.S. 100,000 Cases Doubling Time: 3 Days r = 0.232

If You Want To Try Plugging COVID-19 Data Into Population Growth Equation:

Doubling Time
In Days

2

3

4

5

6

7

8

9

10

11

12

Growth Rate (r)
In Days

0.348

0.232

0.174

0.139

0.116

0.099

0.087

0.077

0.069

0.063

0.058

Table 1. Eleven Doubling Times In Days & Corresponding Growth Rates

1. From news on TV or Internet find current doubling time (in days) and number of people infected with COVID-19 for given area (state or federal).

2. Look up doubling time on above Table 1 to find corresponding growth rate (r). Or just divide 0.695 by the doubling time in days (106 according to one website 21 Sept 2020).

3. Place decimal value of growth rate (r) in following Table 2 along with current number infected (cases) with COVID-19 (N_{o}) and projected number of days (t).

4. Press Calculate to get Answer: Number of people infected with COVID-19 (N) at projected number of days and at current doubling time. This is the size of infected population (N). Note: The doubling times are constantly changing, so this is only an estimate. It does not come out exactly the same as Worldometer.

As of 21 Sept 2020 the Worldometer projection for COVID-19 deaths in the US on 1st day of Jan 2021 is 378,321. My Wayne's Word Javascript exponential growth calculator (Table 2.) is 385,414. See following data used in Wayne's Word calculator. This is a deadly pandemic virus that I'm afraid many Americans (including President Trump) have underestimated, or they just don't understand exponential growth. Without strict mandates for mask-wearing & social distancing (mandate easing) the Worldometer projection for daily deaths in the US by 1 Jan 2021 is 8,000! This alarming number is based on little or no mandates and wholesale reopening of businesses/schools.

Data Used In Wayne's Word Exponential Growth Calculator:

N_{o} = 200,000

r = .00656

t = 100 days

N = 385,414

number in initial population (N_{o})

growth rate (r) (decimal value)

length of time (t) (same units as r)

number in final population (N)

enter value

enter value

enter value

answer

Table 2. Final population size with given annual growth rate and time. Be sure to enter the growth rate as a decimal (for example 6% = .06).

number in initial population (N_{o})

growth rate (r) (decimal value)

length of time (t) (same units as r)

number in final population (N)

1 Wolffia

0.56

125 days

2.5154386709191598e +30 or 2.515 x 10^{30} *

8000

0.56

4 days

75147

8000

0.56

20 days

585,043,535

*This would be a ball of wolffia plants roughly equivalent to the plant Earth!

Table 3. Population sizes with different growth rates and time periods.

The Exponential Growth Of Wolffia microscopica From India!

In the population growth equation (N = N_{o}e^{rt}) the growth rate for Wolffia microscopica may be calculated from its doubling time of 30 hours = 1.25 days. In the above equation when rt = .695 the original starting population (N_{o}) will double. Therefore a simple equation (rt = .695) can be used to solve for r and t. The growth rate (r) can be determined by simply dividing .695 by t (r = .695 /t). Since the doubling time (t) for Wolffia microscopica is 1.25 days, the growth rate (r) is .695/1.25 x 100 = 56 percent. Note: When projecting the population growth to longer time intervals, use a value for t that corresponds to the length of time (i.e. number of days). For example, try plugging in the following numbers into the above population growth equation where t is 16 days: N_{o} = 1, r = 0.56 and t = 16. Note: When using a calculator, the value for r should always be expressed as a decimal rather than a percent. The total number of wolffia plants after 16 days is 7,785. This exponential growth is shown in the following graph where population size (Y-axis) is compared with time in days (X-axis). Exponential growth produces a characteristic J-shaped curve because the population keeps on doubling until it gradually curves upward into a very steep incline. If the graph were plotted logarithmically rather than exponentially, it would assume a straight line extending upward from left to right.

Sixteen days of exponential growth in Wolffia microscopica.

Wolffia plants have the fastest population growth rate of any member of the kingdom Plantae. Under optimal conditions, a single plant of the Indian species Wolffia microscopica may reproduce vegetatively by budding every 30 hours. One minute plant could mathematically give rise to one nonillion plants or 1 x 10^{30} (one followed by 30 zeros) in about four months, with a spherical volume roughly equivalent to the size of the planet earth! Note: This is purely a mathematical projection and in reality could never happen!

The following illustration shows a comparison of the size of one minute wolffia plant, roughly intermediate between a water molecule and the planet Earth!

If a water molecule is represented by 10^{0}, then a wolffia plant is about 10^{20} power larger than the water molecule. The earth is about 10^{20} power larger than a wolffia plant, or 10^{40} power larger than the water molecule.