Exponential Outbreaks: The Mathematics of Epidemics

Note: We have published two additional lessons about the Ebola virus in West Africa and the United States.

Overview | What mathematical principles describe the spread of disease? How can we mathematically model an epidemic?

In this lesson, students explore the fundamental mathematical concepts underlying the spread of contagious diseases. Using a simple exponential model, students compare and contrast the effects of different transmission rates on a population and develop an understanding of the nature and characteristics of exponential growth. Students can then compare their projections with actual Ebola data from West Africa, to create context for analyzing the strengths and limitations of this simplified model.

Materials | Computers with Internet access; graphing utilities

Warm-Up: Have students get up to speed on the Ebola outbreak in West Africa by reading The New York Times’s Ebola Facts page. Students can quickly learn about where Ebola comes from, what the symptoms are, and how contagious the virus is.

After they have a sense of what Ebola is, have students focus on some of the quantitative aspects of the outbreak: How many new cases are being reported? How many people have been infected in West Africa? How many people could become infected?

Ask students to think about why it is important to understand how a disease spreads, and how that knowledge can be used to help treat victims and respond to an epidemic.

Related: In “As Ebola Spreads, So Have Several Fallacies,” Carl Zimmer compares Ebola with a far more common virus: the flu.

Unlike Ebola, the influenza virus is truly airborne. And if recent history is any guide, it will kill thousands in the coming months.

Flu viruses and Ebola viruses take different routes to the same biological goal: to get into new hosts and replicate. Scientists have learned a great deal about the devious ways in which they manage to do it.

Yet misconceptions about how they travel continue to circulate, including the persistent notion that Ebola, like influenza, is airborne. The uncertainty only grows when possible new cases are identified

Read the entire article with your class, then answer the questions below.

Questions | For discussion and reading comprehension:

1. How can the flu transmitted?
2. How can Ebola be transmitted?
3. What makes a virus an “airborne” virus?
4. What kinds of symptoms do flu and Ebola victims share?
5. What kinds of symptoms do flu victims suffer than Ebola victims generally do not?

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Activity

In this activity, students will explore the behavior of exponential functions by using them to model the spread of disease. It is important to note that, while meaningful in this context, the exponential growth model offers a very simplified view of a very complex phenomenon. Students are asked to consider some of the limitations of this model at the end of the initial activity.

Begin by asking students to consider how a rumor might spread among a population. Suppose on Day 1 a single person tells someone else a rumor, and suppose that on every subsequent day, each person who knows the rumor tells exactly one other person the rumor. Have students ponder, discuss and answer questions like: “How many days until 50 people have heard the rumor? 100 people? The whole school? The whole country?”

In the situation with the rumor, the number of people who have heard the rumor doubles every day; this is because, each day, every person who knows the rumor tells it to a new person. In other words, there is a 100 percent transmission rate: 100 percent of those who know the rumor spread it to someone else. A transmission rate this high means that the number of people who know the rumor will grow very quickly. In fact, in this simplified exponential model, one person could spread the rumor to the entire population of the United States in less than a month!

Mathematically, the spread of disease can be modeled in a manner similar to the spread of a rumor. Although a number of simplifying assumptions must be made, the simple exponential model captures the basic impact of transmission rates on the dispersion of a disease among a population. Students can explore the consequences of transmission rate using multiplication, algebra, graphing utilities and elementary statistics.

One way to explore the exponential model is through simple multiplication. In the rumor example, the number of people who know the rumor doubles every day, and so repeated multiplication by 2 is used to calculate how many people know the rumor. For example, after four days, the total number of people who know the rumor has doubled four times, and so is 1 x 2 x 2 x 2 x 2 = 16.

If the transmission rate were, say, 50 percent instead of 100 percent, then repeated multiplication by 1.5 would be used. This would mean that, each day, 50% of those infected would infect someone else. In general, if the rate of transmission as a percent is r %, then the number used in repeated multiplication would be 1 + r/100. For example, if the transmission rate is 6 percent, use 1 + 6/100 = 1.06; if r = 50%, use 1 + 50/100 = 1.5.

This repeated multiplication can be expressed using exponential functions. In the initial rumor example, the function would be y = 2^x, or two raised to the power of x. Here, y represents the number of people infected and x represents the number of days that have passed since day zero. If the transmission rate is 50 percent, the function would be y = (1.5)^x, and in general for a transmission rate of r %, the function would be y = (1+ r/100)^x. Students with more knowledge of exponentials can use the familiar function y = Pe^(rt).

Have students use calculators and graphing utilities to explore the consequences of different transmission rates. For various values of r, like 50, 10, 1, 0.5, 0.1, 0.01 percents and so on, have students investigate questions like “How long until 100 people are infected? 1,000 people? 10,000 people?” Have students graph a variety of exponential models with different transmission rates using calculators or graphing utilities like Desmos, the free online graphing calculator, and ask them to compare and contrast how the populations are affected based on the graphs.

One important characteristic of exponential growth is doubling time. For a fixed rate of growth, the amount of time it takes for the infected population to double in size will be constant. That is, the amount of time it takes for the number of cases to increase from 100 to 200 is the same as the amount of time it takes for the number of cases to go from 1,000 to 2,000. Have students find the doubling times for each of the above rates. Doubling time is a useful tool in quickly estimating growth in a population.

Have students present their graphs and calculations, and discuss their findings. Have them compare their graphs with the graphs of actual data on infections from West African countries to see if their models seem reasonable, and ask students to ponder what impact lowering the transmission rate can have on a population.

Facilitate a conversation with students about the strengths and weakness of this particular model. For example, one strength of this model is its relative simplicity: It’s easy to work with, while still capturing an essential characteristic of the spread of disease, namely that each infected person each day has some chance of infecting other people.

However, there are many limitations to this model. Typically, a person can only transmit a disease for a fixed period of time, but in this simple model, one person continues to infect others indefinitely. Also, in this simple model the rate of transmission never changes; in actuality, rates of transmission may go down as knowledge of the epidemic spreads or as governments take measures to stem an outbreak, as this interactive graph demonstrates.

Going Further

Compare Projections: Students can compare their projections with actual data from The Times or other sources. A simple comparison could involve data points and graphs, while more sophisticated math students could use graphing utilities to run regressions on the real world data to estimate the actual transmission rates of Ebola in various locations.

Use the Logistic Growth Model: The logistic growth model is a more complicated, but more realistic, model for the spread of disease. Have students investigate how this model differs from the simple exponential growth model. Students with stronger mathematical backgrounds can use computing technologies to run logistic regressions on the actual data.

Compare Contagious Diseases: Have students compare Ebola with other known viruses and diseases in this chart that maps a disease’s deadliness against its contagiousness, as measure by its “reproduction number.” Students can further investigate how a disease’s reproduction number is determined and applied in studying outbreaks.

Gamify Epidemic Prevention: Students can play this fun graph theory-based game that illustrates how disease can spread through a network. And they can read about how scientists are using other networks, like cellphones and social networks, to analyze and study the spread of Ebola.

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