# To Accumulate a Rate — Integrate!

Teaching High School Mathematics

# Covid-19 Calculus and Logistics

As the Covid – 19 virus is spreading across the whole world and keeping me(us) cooped up, I can’t help but keep an eye on the data. I’ve been following the WHO data for a few countries and the CDC for the USA for the past 30 days. Surely I am one of many. Every day I get on to see the damage. It feels very purposeful and gives me a focus. Honest reflection: I have taken a few days off here and there when the enormity of the situation seems to hit me. Then I’ll get back on and fill in the missing data. I felt compelled to use this data in my Calculus class in some way. Every day we have a new rate of growth for the spread of the virus and Calculus is all about rate of change.

The main challenge is trying to engage students with Calculus in some way remotely. Here is what I came up with this time… It’s not necessarily what I would do every time or at any point of the year, but it feels right to me for now.

The graph below shows data points for the total number of people confirmed to have Covid – 19 on a certain day. To be clear, each day includes every person that has ever contracted the virus regardless up to that day.
Yellow/orange is South Korea, Blue is China, Red is Italy, and Green is the United States of America. Desmos Data without curves. For each country I used day one to be when there were around 100 confirmed cases.

South Korea was the first country to stabilize so I was able to match an equation to their data first. You can see that it’s not perfect, but it seems pretty respectable.

I recognized that it looked like a logistic equation. For the past few years I have used the idea of the spread of a rumor to investigate logistic equations. The idea behind a logistic equation is that as a population grows it is able to grow faster until it starts to be restricted due to getting closer to the carrying capacity. For a rumor at our school the carrying capacity was the number of students in our school. It cannot grow as fast at the beginning when there are not many people who have heard the rumor, or have the virus. As more people know the rumor, or have the virus, it can now spread to more people more quickly. However, as you get closer to the total population knowing the rumor, or being infected, the rate it grows slows down because there are not many people left to tell or infect. With the virus, the carrying capacity will not be the entire population because not everyone will get the virus. So for each country we will have to choose a carrying capacity. Although I won’t talk about it here, it may be interesting to compare the carrying capacity that matches each country’s data with their population.

I keep hearing all this news about certain countries not releasing accurate results or even certain countries not testing enough people or testing at different rates each day… So, it is somewhat impressive to me that every country, state, county or city that I’ve looked at still roughly follows a logistic curve.

For South Korea my differential equation is: dP/dt = 0.31P (1- P / 8500).
The 8,500 is the Maximum population that will get the virus. South Korea has a much larger population, but they have been able to reduce the carrying capacity of people with the virus. The 0.31 has to do with the growth rate or spread rate of the virus.

When I solve this differential equation with an initial condition of  ** P(0)=93:
I get P(t) = 85000/(1+90e^-0.31t)

**(Not the real data, but it’s what allows me to get a nice equation to fit my data. I had to play around with this number.)**

I also used an online slope field generator to get a slope field and solution curve using Euler’s Method.

I used this slope field, both with the solution curve and without, to show my class how using a slope field and/or Euler’s Method can help to approximate the same solution we got by finding the actual particular solution to the initial value problem.

An activity to attempt:
1. Write a differential equation for each country listed.
2. Use Euler’s Method to find values for the particular solution
3. Find the particular solution by just substituting the Carrying Capacity and k into the appropriate spot in the South Korean equation that I listed above. You might have to play with the number in the spot that is currently 90 until it works.
4. Actually find the particular solution using separation of variables and partial fractions by hand. (advanced)

Some starting values I used:

China: Carrying Capacity = 81,000 and k = 0.23
Italy: Carrying Capacity = 140,000 and k = 0.17
USA: Carrying Capacity = 360,000 and k = 0.28

I made a video for my class to watch about all of the topics above using the Educreations app on my ipad. ( I’m not getting a kickback — But Educreations is a nice tool to make videos.)

I’d like to note that after South Korea started to flatten out, they then started to get more cases in a linear rate, as seen below. I’m not sure what change happened on that day or a short time before, but my guess is that was when they eased up on some restrictions. China also has this same result in their data. I thought this was a pretty cool opportunity for a piecewise function. The logistic listed above for 0<t<30 and a linear function:
y = 105t +5350 for t>30.

Back to the logistics:
For the USA I didn’t have enough data yet to determine a Carrying Capacity, so I just guessed a few numbers until my curve matched the data. Currently it is 360,000 but I’m sure I’ll have to change it as I get more data.

A cool activity would be for students to do the same work I’ve done above, but choose their own city, county, state, or country. Data can be found on the WHO situation report or from the CDC or research a more local source for cities or counties. If anyone is interested in doing this, here is how I went about it: I actually started with the logistic equation and did trial and error with different numbers for the carrying capacity, k value, and the number in front of e (That number helps determines the starting amount.)

Additionally, if you take the derivative of these logistic equations, you get the rate of change of new cases. It’s the curve from “flatten the curve.”