September 19, 2021

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The Math of Ending the Pandemic: Exponential Growth and Decay

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Students in U.S. high schools can get free digital access to The New York Times until Sept. 1, 2021.

Featured article: “The Math That Explains the End of the Pandemic”

When the coronavirus came to the United States over a year ago, we developed a new mantra: “Flatten the curve.” The goal was to stem the runaway exponential growth in infections, in that way preventing hospitals from becoming overrun.

With the development and uptake of highly effective Covid-19 vaccines, we are now beginning to see the mathematical cousin of exponential growth — exponential decay — take over Covid case trends. According to experts, this is welcome news. If all goes well, current trends may foreshadow the end of the pandemic.

In this lesson, you will use the mathematical concepts of exponential growth and exponential decay to explain the spread and slowdown of the coronavirus. Then, you will use these models to explore different end-of-pandemic scenarios and the potential to reach herd immunity.

This tree diagram shows a potential coronavirus chain of transmission. Each dot represents an infected person. One infected person (the person at the top of the diagram) spreads the coronavirus to others, who then spread it to others and so on.

After looking closely at the graph above (or at this full-size image), answer these four questions. The questions are intended to build on one another, so try to answer them in order:

To understand exponential growth, let’s take a mathematical look at the tree diagram — using the table below.

The tree diagram can help us model the number of new infections over time. Specifically, let’s define each horizontal layer of dots as representing the number of new infections on a certain day.

The top layer, representing Day 0, depicts just one newly infected person (one dot). The next layer down, representing Day 1, depicts two newly infected people (two dots). The following layer, representing Day 2, depicts four newly infected people (four dots). Here’s a visual that shows how we are defining the days:

We can begin to fill in our chart with the number of new infections each day, based on the number of dots within these first few days:

Go ahead and fill in the rest of the table, based on the diagram. Note: You may notice a pattern, which can save you a lot of time from counting!

Answer the following questions, about the diagram and the table:

  • Fill in the blank with a number: In the diagram, each newly infected person spreads the disease to ___ new people the next day. Hint: How many lines come out from each dot?

  • Look at the table. What mathematical pattern do you notice?

  • Let’s connect our prior observations. Explain how the infection patterns we see in the tree diagram give rise to the mathematical pattern we see in the table.

Repeated multiplication creates what we call exponential patterns. If you’ve learned about exponents before, this name should make some sense. Exponents represent repeated multiplication. For example:

When we repeatedly multiply by a number greater than 1, we observe exponential growth. To get a sense of what exponential growth looks like, we’re going to visualize our table of values as a graph.

Treat the days as the x values and the number of new infections as the y values. Each horizontal row on the table represents a data value (an x-y coordinate pair). Graph these data values on the following x-y coordinate grid. After you graph the points, connect them using a curved line (not a straight line). Note: The first point is already graphed for you.

Respond to the following:

  • Describe what happens to the number of new infections over time.

  • Describe what happens to the rate of new infections over time.

  • Hospitals have limited capacity and bed space. Using this graph, discuss why it was so important in the early stages of the pandemic to follow social distancing guidelines and “flatten the curve.”

One way to test the quality of a mathematical model is to compare it to available data. Below is a graphic of the world’s coronavirus case count early in the pandemic (March 2020), using data from Our World in Data. In addition to the raw data (the black dots), we’ve fit an exponential growth mathematical model (the blue line).

  • Do you believe an exponential growth model is appropriate for modeling the initial spread of Covid-19? Justify using the graphics above.

  • The statistician George E. P. Box famously said, “All models are wrong, but some are useful.” Look at the model and the data. The model is not an exact fit to the data. Why do you think this is the case? Is the model still useful?

Read the featured article from the beginning through the following paragraph:

Every case of Covid-19 that is prevented cuts off transmission chains, which prevents many more cases down the line. That means the same precautions that reduce transmission enough to cause a big drop in case numbers when cases are high translate into a smaller decline when cases are low. And those changes add up over time. For example, reducing 1,000 cases by half each day would mean a reduction of 500 cases on Day 1 and 125 cases on Day 3 but only 31 cases on Day 5.

As people become vaccinated, they are less likely to catch and show symptoms of the virus. Think about the tree diagram from earlier. The vaccine effectively blocks severe infection pathways from person to person (it breaks the lines between dots). As more people are vaccinated, more pathways are blocked, and the spread of the virus begins to slow.

How much can the spread slow? The author poses an example in which the number of cases reduces by half each day. Mathematically, this can be written as:

Number of cases tomorrow = 0.5 * (Number of cases today)

Let’s create a table of values. Start out with the following:

To find the number of active cases on Day 1, we can follow the formula and multiply the Day 0 total by 0.5. This is shown below:

Go ahead and continue the pattern to fill in the rest of the table. You may get decimal answers for some days. At each day, round your answer to the nearest whole number, before proceeding to the next day.

Again, we see repeated multiplication. This means we have another exponential pattern. However, because we are multiplying by a number less than 1, we now have exponential decay. The article visualizes exponential decay using this graphic:

Answer the following questions:

  • In the table of values, how much did the number of cases fall from Day 0 to Day 1? How much did the number of cases fall from Day 5 to Day 6? Comment on any trend you notice.

  • Think about the trend you mentioned above. Does the graph show a similar trend? Explain.

  • The article says that “the worst of the pandemic may be over sooner than you think.” Assume people continue to get vaccinated and follow public health guidelines. Why does the exponential decay model indicate that the worst will be over “sooner” than we think? Why wouldn’t we have to wait awhile for the worst to pass?

As before, we can evaluate the quality of our model by comparing it to real data. Continue reading the article through the following paragraph:

This pattern has already emerged in the United States: It took only 22 days for daily cases to fall 100,000 from the Jan. 8 peak of around 250,000, but more than three times as long for daily cases to fall another 100,000.

Here is a graph from The New York Times’s Covid-19 database that illustrates this trend (focus on January 2021 and onward):

Answer the following questions:

  • The article says that case counts fell by 100,000 in 22 days. Afterward, it took three times as long for the cases to decline by another 100,000. Explain how this statement supports the exponential decay model.

Continue reading the article until you reach the following graphic:

This graph displays an important concept in statistics and the sciences: counterfactuals. Whenever you think to yourself, “I wonder what would happen if …” — you’re essentially constructing a counterfactual. A counterfactual is an alternate reality that would exist if you changed something about your world.

In the above case, the dashed line provides a counterfactual Covid-19 case scenario. In this counterfactual reality, we relax public health precautions “too soon” and break the exponential decay trend.

  • Imagine a counterfactual in which we started relaxing restrictions at an even earlier time, just as the cases began to trend downward. Would we see a larger or smaller gap between the solid line and the counterfactual? Why?

Option 1: Exponential decay in some countries. Exponential growth in others.

Use the Learning Network lesson on the recent surge of coronavirus cases in India. Connect the lesson to this visualization (from Our World in Data) of Covid-19 case trends in both India and the United States. Where do you see exponential decay? Where do you see exponential growth? What makes case trends differ between countries?

Option 2: Will we reach herd immunity?

Read the article “Reaching ‘Herd Immunity’ Is Unlikely in the U.S., Experts Now Believe.” Reflect on the following questions: What does herd immunity have to do with exponential growth and decay? What would need to happen to reach herd immunity? What factors affect our ability to reach herd immunity? Why are some experts still hopeful, even if we don’t quite reach herd immunity? You can also explore the topic visually with this “What’s Going On in This Graph?” activity.

Option 3: Explore the math of vaccine efficacy.

Use the Learning Network lesson on calculating vaccine efficacy with data from the main Pfizer trial. As you work through the lesson, you will use math, statistics and probability to get a practical sense of how the vaccine performed. (Spoiler: It did well.)

Option 4: Analyze vaccine hesitancy.

Explore the New York Times vaccination database to analyze vaccination rates in your community. Then, use this Learning Network lesson to learn about vaccine hesitancy and efforts to persuade vaccine skeptics.

This lesson was written by Dashiell Young-Saver, who is a high school statistics teacher and the founder of the site Skew The Script. Important contributions were made by Sharon Hessney, who writes the NYT Learning Network’s weekly feature “What’s Going On in This Graph?”

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