COVID-19 and Exponential Change

At the time of this writing, the world has its full attention on the Coronavirus (COVID-19) pandemic. The outbreak has changed everything from travel, attendance at sporting events, school, and even how we work and greet one another. COVID-19 continues to wreak havoc on national economies, healthcare systems, and personal freedom.

People also have experienced a range of emotions during worldwide crises such as the one we all face now: fear, anxiety, and depression to name a few. And many of the negative emotions stem from a lack of knowledge and uncertainty. Yet information does exist which can inform and allay some of the despair and apprehension.

How to Reduce Transmission

As more information about halting the progression of COVID-19 becomes available, the more comfort people will feel. Two new phrases now in many people’s vocabularies fall into the category of understanding what each person can do: “social distancing” and “flattening the curve.”

Viruses have different means of infecting people. For example, COVID-19 spreads when an infected person coughs or sneezes which sends very small droplets into the air and on surfaces. If the droplets land in the mouth or nose of someone else, or if the uninfected person touches the surface where the droplet lies and rubs their eyes, the virus can take hold. 

Social distancing means making a concise effort to reduce contact with infected people thereby slowing the transmission of the virus. People practice social distancing, for instance, when they stay home instead of going to a concert or a bar where many people come into close contact with another. In essence, avoiding public places and not becoming infected or infecting others means the virus cannot spread to others.

Exponential Change

With social distancing, many people may have also heard of “flattening the curve.” The curve refers to the rising number of infected people. Flattening the curve, then, means reducing the increase of sick people that could come to overwhelm the healthcare system. How does that happen?

Many visuals in blogs, media, and professional journals use graphs to portray changes showing how the number of people contracting COVID-19 could shift dramatically. And understanding graphs and the story the data tells helps with the deluge of information out there.

The first graph shows exponential growth. Exponential growth refers to a quantity growing by a multiplicative constant. Let’s say in a laboratory a bacterium starts to divide into 2 every hour. The numbers go from 1 to 2, 2 to 4, 4 to 8, and so on. The growth factor or multiplicative constant comes to x2 or a doubling every hour. The graph shows how the change starts off slow but then really kicks in as time moves forward. 

The rapidly curving line means the absolute amount or quantity piles up faster and faster. In a mere 8 hours, the bacterium went from 1 to 128. And if we waited 16 more hours, at the end of the day the bacteria have grown to a staggering amount: 8,388,608! 

Exponential growth always has a deceptive and explosive phase. Many graphs in the media show how local communities, states, and countries have encountered exponential change. For example, the state of New Jersey had its first confirmed case of COVID-19 March 5, 2020 (Source: State of New Jersey Department of Health). However, across time more cases appear and the graph shows the telltale exponential curve. Each day the number of confirmed cases grows disturbingly fast until 29 days later 4,372 cases occurred in one day.

An exponential curve’s growth rate depends on various conditions and can fluctuate. In the first graph, the bacteria curve upward in a smooth fashion because of the constant growth rate of x2 or a doubling drove the change. But with the New Jersey data many factors such as uneven testing or lags in reporting affect the data. Still, the exponential curve ominously expresses itself giving rise to concerned consumers of information.

Linear and ratio (semilogarithmic) graphs

Many COVID-19 graphs available for public consumption exist to show how some count changes across time. Graphs displaying increases or decreases of an event or count go by the name time series. Recent time series counts have included the number of respirators delivered, tests successfully completed, and the number of deaths in a country. 

Observant viewers will notice the scale on the vertical axis for some graphs look different from others. Line graphs come in two versions, linear and ratio, or what some people also refer to as semilogarithmic. By changing the scale the resultant graph paints a very different picture of change.

Notice the scaling on the first exponential bacteria growth graph and compare it with the figure below. The second figure starts at 1 and goes to 10, 100, and then 1,000. Also, the distances from 1 to 10 looks exactly like the distance from 10 to 100 or 100 to 1,000. Going from 1 to 10 has a ratio or proportion of x10. An equal ratio of x10 occurs with 100 to 1,000. The beauty of ratio graphs comes with how they transform data.

Notice the orderly, straight line for bacterial growth. Because the bacteria grow at a constant rate of change at x2 or a doubling every hour, the line rises in an unbending course. Therefore, using ratio line graphs have an advantage of uncovering exponential functions - they will always appear as a straight line!

Because ratio graphs can detect the presence of an exponential, interested parties can forecast when things will really get out of control. Water use, forest loss, species extinction, world oil production, and the spread of diseases like COVID-19 all point to a future with alarming consequences. At some point, the relentless growth of the exponential overwhelms a system.

Take the example of the New Jersey chart converted to a ratio graph. Note the straight line and the startling spread of infected people. Because COVID-19 has differential effects, most people will not require hospitalizations. The fraction of those who do need admittance and specialized medical equipment to survive could face a dire situation. With more and more people becoming infected, the larger the number of people who will find themselves in need of a limited supply of hospital beds and life-saving devices like respirators.

The exponential growth also comes with a bright side. Several conditions exist that can slow down, halt, or move the changing trend into exponential decay. For instance, with the previous bacteria example, eventually resources will change. Bacteria thrive on nutrients, a source of water, and environmental preferences such as temperature or light. Change any of the previous conditions and bacteria cannot continue to grow exponentially.

Similarly, social distancing marks our current best effort to flatten the curve and push down the exponential. An infected person needs to seed the spread of the virus by sneezing or coughing. The virus lives on the surface for many hours or can persist for a limited period of time in the air. Then it can move to another person who breathes in the tainted air or touches their eye, nose, or mouth after contacting a contaminated surface. By staying home and avoiding the previously described situations, our society has the best chance to considerably reduce the spread.

Appreciating exponential change and how they look on linear and ratio graphs serves all consumers of information well. In a time of uncertainty, visual evidence provides clarity, awareness, and comfort. And people armed with knowledge can make decisions and take action that will save lives and help everyone get through the COVID-19 pandemic.