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Pandemics: How Are Viruses Spread? NCTM

Author: National Council of Teachers of Mathematics

“Pandemics: How Are Viruses Spread?” National Council of Teachers of Mathematics, Mar. 2020,

The spread of a virus is influenced by four factors. Manipulate those variables to create different outcomes. Results are modeled in a logistic curve, an S-shaped curve that describes population growth—of both viruses and people—as well as other phenomena in economics and science.



This math simulation models the spread of a virus through social contact.

The Spread of Virus Simulation has four variables you can control.

Population Size: Each small square represents a person

Virus variables include

Days ContagiousFor the coronavirus this varies between 2 and 14 days before significant signs appear. A selection of 6-10 seems reasonable to start.Chance of Contracting Virus per Contagious ContactThis is how likely an uninfected person is to catch the virus when contacted by a contagious person. This value is not known for coronavirus. Something between 10% and 4% may be reasonable.

Social variable is one we can control.

Number of Contacts per DayThis represents the number of contacts, other people, each person contacts per day. While in normal situations this could likely be in the 100s for students. In this setting between 10 and 2 seem reasonable.

NOTE: This is a simulation and it has been simplified to a small number of variables. These variables are applied uniformly across each person in the population. Real life is much more variable and complicated. Our goal is to help learners of all ages better understand how our actions impact the collective result and how mathematics can be used to better understand our world.

Each time you click Calculate Results or Recalculate Results, the simulation recalculates. For some cases the spread ends quickly. This is by chance given the variables. Click again for a new simulation. You may need to run multiple times to get a sense of the variability and impact of the settings. This simulation/model is not predictive but is intended to show how our actions can impact our community.


How Are Viruses Spread?

Recently, the COVID-19 respiratory disease (“coronavirus”) has sparked fear as the outbreak has been categorized as a pandemic. As the world of science trains a careful eye on the development of the virus, the world of mathematics, too, can play a role. For example, the applet below can help predict the spread of a virus over a population. Each person may pass on a germ or virus to others with whom they come in contact, but the disease will not be transmitted if the recipient has a resistance to the disease, has had a vaccination, or has already been infected. Consequently, there is a “population of opportunity” to whom the disease can spread.

The spread of a virus is influenced by various factors. Here are four factors:

  • the size of the population of opportunity;
  • the number of days contagious;
  • the number of people with whom an infected person comes in contact; and,
  • the probability of contracting the virus from contact with an infected person.

What is a Pandemic?

A pandemic is an epidemic (an outbreak of an infectious disease) that spreads across international borders, or at least across a large region. According to the World Health Organization (WHO), a pandemic can start when three conditions have been met:

  • a new virus subtype emerges;
  • it infects humans and causes serious illness; and,
  • it spreads easily and sustainably among humans.

Previous pandemics include the Swine Flu, cholera, and the Spanish Flu. Currently scientists are training a careful eye on the COVID-19 virus. By the second week in March 2020, there were 4226 confirmed cases of COVID-19 in humans and 75 deaths in the United States. According to WHO, there were 132,758 confirmed cases and 4955 deaths globally at this same time. The COVID-19 virus is a pandemic.

The History of Pandemics

An epidemic is an abnormally high occurrence of a disease in a particular population or geographic area. On the other hand, a pandemic is a global epidemic that crosses international boundaries. For example, the bubonic plague epidemic (the “Black Plague”) was confined to Europe in the 1300s, while the Spanish Flu pandemic of 1918 spread to all parts of the world.

The Antonine Plague (165-180)
This pandemic is believed to have been smallpox or measles that was brought to Europe by soldiers returning from the Near East. It may have killed as many as 5 million, and during a second outbreak in the middle of the third century, it was rumored that 5000 people a day were dying in Rome.

The first outbreak of cholera occurred in India in 1816, and it became a pandemic by spreading from Bengal, across India, and to China and the Caspian Sea. The second pandemic of cholera affected Europe and North America in the late 1820’s, and since then, there have been five other cholera pandemics. The seventh and most recent outbreak (1961-66) began in Indonesia and reached Bangladesh, India, and the Soviet Union.

The Spanish Flu Pandemic (1918)
The Spanish Flu of 1918 first appeared in Kansas. The first complaint was registered by the cook at the Fort Riley military base on March 11, 1918, followed by 107 additional patients before noon that same day. In the camp of 26,000 men, 1127 were infected, 46 fatally.

Men who carried the disease, believed now to have been transmitted from birds to humans, left Fort Riley to fight in World War I. They spread the flu at each stop, which led to a nearly-worldwide pandemic. The Spanish Flu claimed more lives than the fighting during World War I. Although estimates vary widely, more than 500,000 Americans died from the Spanish Flu; and worldwide, the death toll was between 10 and 100 million.

Asian Flu Pandemic (1957-58)
Unlike the virus that caused the 1918 pandemic, the 1957 Asian Flu pandemic virus was quickly identified, due to scientific advances. Vaccine production began shortly after the outbreak occurred in China, and a small supply of vaccines was available when the flu reached the U.S. Still, the virus spread quickly when school started in the fall, and nearly 70,000 Americans were killed; worldwide, the total reached 2 million.

During the 1957-1958 pandemic, a World Health Organization panel found that spread often followed public gatherings. They also observed that in many countries the pandemic broke out first in camps, army units and schools, suggesting that avoiding crowds may be important to controlling the spread.

How to Minimize the Threat of Pandemic

Even as early as 1918, officials knew that contact with infected persons caused the virus to spread. Consequently, they prevented large crowds from gathering in theaters, churches, and schools. The public notice to the left shows one town’s attempt to limit public gatherings.
Once an influenza pandemic begins, it is often unstoppable, because the virus spreads rapidly by coughing and sneezing. In fact, infected people can pass the virus to others before symptoms even appear.

Preventing people from coming in contact with one another is one way to prevent the spread of influenza. Providing vaccines to reduce the likelihood of contracting the disease, or using an antidote to treat the symptoms once they appear, are other means of control.

  • How would quarantine affect the spread? Which parameter (population, days contagious, number of contacts, or chance of contraction) does it change in the applet? Hold the other variables constant and change only this variable in the applet. What happens?
  • A medicine is developed that reduces how long infected people are contagious. How would this affect the spread? Which parameter does this medicine influence? Hold the other variables constant and change only this variable in the applet. What happens?
  • A medicine is developed for uninfected people to take that will decrease their likelihood of contracting the disease. How would this affect the spread? Which parameter does this medicine affect in the applet? Hold the other variables constant and change only this variable in the applet. What happens?

Try using the applet again, holding two variables constant.


  • Cartwright, Frederick F., and Michael Biddiss. Disease and History, 2nd revised edition. Darby, PA: Diane Publishing Company, 2000.
  • Nobilon International. Core Activities – Influenza Vaccines, Frequently Asked Questions, 2006,
  • World Health Organization. Influenza, 2003,

What is the Math?

The curve used to model this situation is called a logistic curve, an S-shaped curve that describes population growth-of both viruses and people-as well as other phenomena in economics and science.

Pandemics IMAGE PandemicCurve
Pandemics IMAGE Equation

Interesting Facts about the Logistic Curve

The point of inflection is the point where the curve changes from increasing faster to increasing slower. It also marks some symmetries for the curve.

  • Half of the people are accounted for below the point of inflection, and half are accounted for above the point of inflection.
  • Half of the time is accounted for to the left of the point of inflection, and half of the time is accounted for to the right of the point of inflection.

Once the point of inflection is known, it is possible to estimate:

  • How long a pandemic will last. Since half the time occurs to either side of the point of inflection, the point of inflection is the midpoint of the curve. Therefore, if a pandemic has lasted x days in getting to the point of inflection, it will likely continue for another x days.
  • The total number of people that will be infected. Because half of the people are infected on either side of the point of inflection, it gives the midpoint. Therefore, the number of people who will be infected after the point of inflection is roughly equal to the number infected up to the point of inflection.

Before the point of inflection is known, it is not easy to make accurate predictions.


Objectives and Standards

NCTM Standards and Expectations

  • Algebra
  • Discrete Math
  • Functions and Trigonometry
  • Algebra
  • Pre-K to 2nd
  • 3rd to 5th
  • 6th to 8th
  • 9th to 12th

DMU Timestamp: May 11, 2020 21:16

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