Doing the (Coronavirus) Math: Exponentials, Bell Curves and Flattening

A simple lesson in the math behind coronavirus COVID-19 predictions increases the appreciation for self-isolation and social distancing.

At first, the spread of coronavirus COVID-19 trickled in slowly. But that trickle turned into an exponentially increasing torrent very quickly.

From past epidemics and pandemics, we know that at some point the rise of infected people will begin to taper off – thanks to an increase in the number of recovered people and anything that limits the spread of the disease. The resulting infectious curve takes on a bell shaped (normalized Gaussian) curve. The steepness of the bell curve is determined by how quickly a population becomes infected. If that rate can be spread out over time, then the curve becomes flattened out. This is a desirable outcome as it gives the resource constrained health officials, hospitals and the like more time to deal with the sick.

There are only a few ways to slow down the spread of a virus to flatten its bell-shaped curve:

  • Quarantine and Isolation – Quarantines restrict the movement of well people who may have been exposed to a disease to see if they get sick. Isolation is used to separate ill people from those who are healthy. The goal of both approaches is to prevent people from coming in contact with one another. That is why large crowds have been banned during the spread of COVID-19. Social distancing also helps.
  • Vaccines or Antidotes – Vaccines are used to reduce the chance of contracting the disease. Antidotes provide a way to treat people who already have the disease.
  • Predictive Models – Predictive models help health professionals and government agencies decide when and where to apply resources to stop or lessen the devastating economic and health effects of communicable diseases like COVID-19. But it’s important for the general population to appreciate the basic math that makes these predictions possible.

Exponentials: The Rising Curve

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Population growth of viruses and many other natural phenomena are modeled using a logistic or S-shaped curve. Such curves represent an exponential function that is used in math to model growth. In simpler terms, exponential growth means that the number of cases of infected people doubles for a given time period. With the COVID-10 outbreak, that doubling occurs on average every two or three days. This doubling also pertains to the rate of the number of fatalities.

No virus can grow at an exponential rate forever. Virus growth is bounded by available resources, such as uninfected hosts, transmission mediaum, nutrients, water, etc. Still, the initial exponential growth of viruses do increase at an alarming rate.

For COVID-19, we know the number of infected people doubles every 2 to 3 days. To be conservative, let’s choose a 3-day doubling period. If there were 1000 confirmed COVID-19 cases today and the number was 500 cases three days ago, then the infection rate doubles every three days. As long as the deaths are doubling at a constant rate, the growth is exponential. This means that the original 500 cases will grow to more than 1 million cases after 11 doubling times or 3 days x 11 = 33 days (roughly one month).

Fortunately, this type of infectious growth rate doesn’t continue infinitum. As the growth rate peaks and begins to fall, the curve changes from an exponential one to a normal distribution or bell curve.

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