## Exponential Growth

The Coronavirus spread illustrates exponential growth and provides an opportunity for interesting discussion with students. For example, students could be asked to investigate and compare population isolation and focus on patients and subjects in mathematical terms, that is, how each solution affects the formula components in the exponential growth function.

The exponential growth function of the coronavirus is of the following form:

, where:

x (t) - the amount of infected / dead on a given day

x0 - the amount of infected / dead on the first day

b - the spread coefficient: how many people infect One person in one day

t - time in days

Suppose on the first day there was one patient, and the exponent base is 2 (one person infects two other persons in one day), then the equation will look like this:

Building a table in a spreadsheet can let us view the spread of the virus along a full month:

As can be seen from the graph, in thirty days the number of infected reaches over a billion people!

In practice, other factors need to be taken into account:

Those who are already infected cannot be infected again.

Worldwide spread prevention efforts slow down the rate of expansion.

The coronavirus is not well known, so many aspects of the spread rate are unknown.

The number of infections is not really known because the entire population cannot be examined.

Therefore, to make a more accurate calculation we will change the test data and metrics:

We will refer to the amount of deaths (which is a more exact figure) and not to the amount of contagious.

We will calculate the spread factor by looking at the amount of deaths worldwide in March 2020 with a linear regression.

Death data from the Coronavirus according to the worldometers.com website:

We first display the data on a graph:

The graph gradient is now much milder than the initial one..

## Linear Regression

Using a linear regression we try to find a linear relationship between an independent variable, x, and the dependent variable y, where the general form of the equation is of the form

y = a + b * x

(Source: __Wikipedia__)

The graph measurements (red dots) of y with respect to x, and the blue linear line is the line with which we try to explain the y-dependence of x, and so do future predictions of y values as x-dependent. The assumption underlying the linear regression is, of course, that in the model itself we expect a linear dependence of y on x.

In order for us to perform a linear regression on our model, which is recalled of an exponential rather than a linear form, we transform it to a logarithmic representation - which is a linear representation of the exponential form. That is, instead of displaying

We will display

Log (x0) + Log (b) * t

The resulting graph now looks as follows:

Displaying the logarithmic graph was done by calculating the base 10 logarithm of the cumulative amount of deaths and taking the logarithmic value column as the y axis of the graph.

Next, we will show the Trendline using the spreadsheet - in Google Sheets do this using the Trendline mark in Customize / Series:

The blue line is the Trendline. In order to find the line equation, we choose as the label (next to the Trendline mark) to display the equation (Use Equation) and get the line data we were looking for:

That is,

Log (X0) = 3.29

Log (b) = 0.0408

From the logarithm we consider the original data:

In words, the spread factor of the virus in terms of number of deaths per day is 1.1.

Let's go back to our initial graph and set these two values we found:

From this graph we can now estimate the amount of deaths in the near future.

Note: At some point, the graph data will change as a result of the fact that a large amount of the population is already infected (herd immunity), as well as the responses and actions of government systems in different countries greatly affect the rate of spread, so the ability to predict the spread of the virus is limited.

## Summary

We calculated the death spread factor of the Coronavirus as an exponential growth model.

The calculation was performed using a linear regression on the logarithmic graph of the global Coronavirus death data.

The quarantine strategies of the different countries try to influence both the initial rate of the deaths (X0) and the spread factor (b), which is the exponent base.

Question: In your opinion, which of the two factors - X0 and b - is affected and to what extent, by each of the following strategies:

limiting or preventing the entry of tourists to the country/state

Quarantine of citizens arriving from other countries

Social distancing

Quarantine of people who were close to patients

Isolating populations at risk

Wearing a mask

Washing hands

The Google Sheet used for the calculations and graphs can be viewed here: __https://docs.google.com/spreadsheets/d/1TPM6wiYGKD5GSoEmJgQlouxjwX-wUUwhy16ZTqa1gXQ/edit?usp=sharing__