## Introduction

As we talk generally about mathematics and statistics, we often hear the phrase, “X is **increasing exponentially**“. We look at the graph that comes with such a comment and, yes, the curve shows the data are increasing slowly, slowly, more rapidly, much more rapidly … and we conclude, that’s **exponential growth**. Is it though? And what does exponential growth actually look like?

## COVID-19: exponential growth?

Newspapers and the media in general do talk constantly about exponential growth in the context of the COVID-19 pandemic. This article discusses this aspect of the pandemic by demonstrating what exponential looks like and how we can use **Goal Seek** in Excel to model it.

## COVID_19 Charts

Here are a few charts based on Covid-19 for the UK:

We do not directly see anything exponential there, do we? Well, let’s look at August and September 2020, then:

There is a definite increasing slope to that curve, isn’t there. Is it increasing exponentially, though? Let’s see when we add the exponential trend line, equation and R^{2} value:

The R^{2} value is very high, at 0.8862 but, since it is not equal to 1, it is not perfect! How about the cumulative number of Covid-19 cases in the UK, then? Take a look:

As with the raw data, not conclusive. Try again:

Looks promising in terms of exponentiality but when I tried to add the exponential trend line to that graph, it was greyed out: not possible to evaluate it. Not exponential, then.

## Another Approach

Let’s take another approach by using an exponential template and fitting our data to that. I found a google sheets file on the web that discusses this topic and took their lead in the following screenshot:

**You can download my spreadsheet to follow along in more detail with what follows. You will see I have made a number of changes to the original file** and there is a link to the original file in my Excel version.

The use the **R** and **CFR** metrics as the foundation of this model. R is the **reinfection rate**: a value of 1 says that, on average, we will infect one other person if we have the disease. The R of 1.98 in the screenshot is for the USA and it tells us that everyone who catches covid-19 will, on average, infect almost two other people.

The CFR metric is the **Case Fatality Rate**, shown as a percentage and it tells us that if 100 people die when 1,000 people were infected, the CFR is 10%. As you can see from the screenshot, the USA CFR was 2.88% at the end of September 2020.

## Formulas in the Table

The formulas in the table are as follows:

B5=A5+B2

B6=B5+(B5-A5)*B$2 … copy that down to B26 in this case, to 29th January 2021

C5=B5*B$3 … copy that down to C26 in this case

D15= 11th September 2020

D16=D15+10 … the table is modelling changes to the case and death data every 10 days from 11th September 2020 to 29th January 2021

In the range F3:M8, I have added the actual total cumulative case and deaths data for several countries that I have then added to the table by using Goal Seek. **Watch the video that accompanies this page to see how I did that.**

What we see is that each country has its own R and CFR values. For example, while the UK has a relatively low R value, its CFR is veery nearly 10%, way above the others. Of course, the USA has a massive number of deaths but their CFR is one third of that of the UK.

## The Graph

As you change the data in cells B26 and C26, you will see how the model illustrates the data for each country **EXPONENTIALLY**:

## Conclusions

We can see that the term Exponential means something specific and while the covd-19 rates of increase are often high and very high, that does not mean they are increasing exponentially.

Download the Excel file and the video that I have created for this page and check your further understanding of this page

Spreadsheet:

Video file:

Duncan Williamson

3rd October 2020