Math & Beyond

Exponential Growth in Maths and its relation with Covid

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July 30, 2020

Reading Time: 4 minutes

Introduction

Do you know that if you take a fine paper(thickness 0.01mm), and fold it 45 times, you can actually reach the moon?

Yes, that is possible due to the fact that the thickness of the paper shows an exponential growth as you fold it!

Exponential growth is important to discuss because so many people underestimate the power of it. It is much more potent than linear growth and that’s why it is discussed so much in the context of covid.

Recently, you may have come across a lot of news headlines like “The coronavirus is growing exponentially”, “The curve of the coronavirus cases is rising sharply” etc.

What do these phrases really mean? Why is there so much buzz surrounding this?

This article would give the reader a good understanding of Exponential function, Exponential growth with examples, and its significance in the context of covid.

You can check this video by Ted-Ed which explains the scale of exponential growth using the paper folding analogy.

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What are the Powers and exponents?

An expression that represents repeated multiplication of the same factor is called a power.

In Mathematics, Exponent is a quantity that represents the power to be raised for a given number or an expression, usually expressed as a raised number or variable as a superscript of the number or an expression.

Exponent Description            

Special symbol, the up-arrow "↑" or “^” (in the keyboard) also can be used to represent an exponent.

v↑k = vk

s^b = S

The exponent of a number is the number of times that number should be multiplied by itself. 

For example: 7^4 = 7 × 7 × 7 × 7 = 2401. The exponent "4" says the number 7 to be multiplied four times by 7 itself.

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What are the exponential functions?

To the list of mathematical functions namely linear functions, quadratic functions, rational functions, and radical functions, in addition, comes Exponential Functions. 

 Description of exponential functions

Exponential functions are further classified into Exponential Growth and Exponential Decay and this blog is confined to discuss Exponential Growth.

Bar graph representing exponential growth


Exponential Growth

Exponential growth is a mathematical term that represents a quantity that increases without limit based on an exponential function. The important concept is that the rate of change continues to increase over time.

For example, the nuclear fission through which we generate electricity in nuclear power plants involves a chain reaction in which atoms keep dividing exponentially.

 Nuclear chain reaction with exponential growth            

Exponential growth occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.


Exponential Growth Formula

The original exponential formula is y = ab^x. However in Exponential growth, ‘b’ value (growth factor) has been replaced by (1 + r) and hence growth "rate" (r) is determined as b = 1 + r.

Exponential Growth formula

where,

a = initial value (the amount before measuring growth)

r = growth (most often represented as a percentage and expressed as a decimal)

x = number of time intervals that have passed

 

Example: The population of Vatican City in 2016 was estimated to be 55,000 people with an annual rate of increase of 2.4%.

  1. What is the growth factor for Vatican City?

           After one year the population would be 55,000 + 0.024(55000).

          By factoring, we have 55000(1 + 0.024) or 55000(1.024).

          The growth factor is 1.024. (Growth factor should be greater than 1)

  1.  Write an equation to model future growth.

             y = abx = a(1.014)x = 55000(1.024)x

Most naturally occurring Exponential growth phenomena grow continuously. For example, bacteria will continue to grow over a 24 hours period, producing new bacteria which will also grow. The bacteria do not wait until the end of the 24 hours, and then all reproduce at once.

The exponential ‘e’ is used when modeling continuous growth that occurs naturally such as populations, bacteria, etc.

You can think of 'e' as a universal constant representing how fast you could possibly grow using a continuous process.

Also, the beauty of 'e' is that not only is it used to represent continuous growth, but it can also represent growth measured periodically across time (such as the growth in the previous example).

Continuous Exponential Growth

The following formula is used to illustrate continuous growth. If a quantity grows continuously by a fixed percent, the pattern can be depicted by this function.

exC

A = final value (amount after growth)

Ao = initial value (amount before measuring growth)

e = exponential e = 2.71828183...

k = continuous growth rate (also called constant of proportionality)

(if k > 0, the amount is increasing (growing); 

t = time that has passed


Example: A strain of bacteria growing on your desktop doubles every 5 minutes. Assuming that you start with only one bacterium, how many bacteria could be present at the end of 96 minutes? (Hint: bacteria continuously grow)

neweqq

Initially, there is only one strain of bacteria. (Ao=1). For finding k, use the fact that after 5 minutes there will be two bacterias.


Now, form the equation using this k value, and solve the problem using the time of 96 minutes.

expAA


Significance of exponential growth in Covid

The coronavirus cases are growing exponentially in many countries. You may have heard that countries are trying to ‘flatten the curve”. It is because the curve of daily cases of coronavirus is rising sharply.

Flattening the curve would mean a decrease in the rate of growth of coronavirus cases and hence less burden on the healthcare system.

If you want to take a feel of what exponential growth really means, read the short story in the below image.
 

Shocking facts about exponential growth and story about chess

 

Exponential growth inversely proportional to Exposure Reduction

The exponential growth of coronavirus cases is inversely proportional to social exposure reduction. 

It means that the more you avoid going out and reduce exposing yourself, the lesser is the rate of growth of coronavirus cases. 

It also means that the chances of an infected person spreading viruses to other people are high when the social distancing norms are not followed properly. 
 

Growth of coronavirus with respect to exposure reduction

Credits- Kit Yates, Senior Lecturer of Biology, University of Bath, England

 

Graph depicting simulated exponential growth

This is the graph which is simulating the exponential growth of covid-19. The cases increase sharply as the curve moves upwards.

Notice that the cases were increasing very slowly in the initial days, but after day 42, the curve takes off. That is what happened in most countries especially like USA, India, and Brazil. 
 

simulated exponential growth of coronavirus

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Conclusion

Exponential growth is a pattern of data that shows sharper increases over time that is versatile across various sectors like medical, finance, marketing, economy, banking, insurance, information technology etc. Surf the net to find more about Exponential growth.

If you have any interesting applications of exponential functions on your mind, you can share them with us on Twitter.


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Frequently Asked Questions (FAQs)

1. What is Exponential growth?

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.

For example, if at a particular location, the number of coronavirus cases doubles every day starting with two cases on the first day, the cases would be 4 on the second day, 16 on the third day, and so on. The cases are growing to the power of 2 each day in this case (i.e., exponentially).

2. What is the difference between power and exponent?

An expression that represents repeated multiplication of the same base factor is called a power while the number which is raised to that base factor is the exponent.

For example, 3^2 is called the power where the base is 3 and the exponent is 2.  


External References


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