The mathematics of a pandemic

Kamuela E Yong, Ph.D.

March 22, 2020

Disclaimer: I am a mathematical epidemiologist who has published mathematical models of the spread of diseases, however, I do not claim to have any expertise in COVID-19. The information that is shared is mine and is not representative of the University of Hawaiʻi–West O'ahu.

In recent days, there has been a lot of news about COVID-19. With this news comes information and statistics. It can be difficult to understand what is happening with so much information being reported at once. As a mathematical epidemiologist, I would like to shed some light on the mathematics being reported. Please note, I do not have any expertise on COVID-19, I am a mathematical epidemiologist and I am simply explaining the mathematics of epidemiology and applying it towards the current data for COVID-19.

Some of the biggest questions being asked about the current COVID-19 are how deadly and how contagious is this virus. The virulence of COVID-19 is measured in what is called the Case Fatality Rate, CFR. The contagiousness of a disease is measured by what we called the Basic Reproductive Number, R0. Finally, we've been hearing a lot about flattening the curve, so we will discuss what it means to flatten the curve.

Case Fatality Rate, CFR

One method of assessing the severity of a disease is by examining what is the likelihood of dying if you are infected. In epidemiology, this is called the case fatality rate, or the case fatality ratio, CFR. The CFR measures the proportion of people who die from a disease compared to all the people who are diagnosed with the disease.

For example, if there are 100 people who have been infected with COVID-19, and 10 of people died from the disease, the case fatality rate would be

With many diseases, this is a fairly accurate method for determining the virulence of an outbreak. However, with COVID-19, there are many individuals who may be infected with the virus, but exhibit little to no symptoms. These people may not be tested for the virus and thus are not counted towards the total number of diagnosed individuals, making the CFR seem a lot deadlier than it really is.

If we return to our example where 10 people have died and 100 people have been diagnosed with COVID-19, let us also consider that there are an additional 100 people who have the disease, but with mild or no symptoms, thus have not been tested. So the true number of individuals infected with COVID-19 is actually 200 (100 individuals who have been diagnosed plus 100 individuals have not been diagnosed). A more accurate CFR is

This CFR shows the disease is a lot less deadly, however, without being able to properly diagnose individuals with mild or no symptoms, we are unable to get the true CFR.

It is important to note that the CFR can change based on individual and social responses such as how soon to receive treatment.

CFR during an outbreak

We must be careful when we examine the CFR during an outbreak because people may be at various stages in the development of the disease. For example, if we take the total number of deaths and the total number of infected people today, even if there are no new infections, some of the current people may die in a week, giving us a new total number of deaths, thus changing the CFR. Additionally, there are also people who may be in the early stages and not exhibiting symptoms, thus they too may not be counted towards the CFR.

Not all CFRs are the same

Global averages can be misleading. Some governments, such as Singapore and Hong Kong, have used aggressive measures to bring the outbreak under control, whereas others have been slow to implement such strategies. The global average hides these differences.

Additionally, the CFR varies by age. Data from the early stages of the outbreak in China show a variability in CFR by age, with people over the age of 80 being affected the most.

Source: NIH

Basic Reproductive Number, R0

The basic reproductive number, written R0, and pronounced “R naught,” is a term epidemiologists use to represent the contagiousness of a disease. R0 represents the number of secondary cases (new infections) from a single infective individual in a population that is completely susceptible to the disease. Therefore, if R0<1, then the disease will die down,  however, if R0>1, then the disease will begin to spread and an outbreak will occur.

The current estimate for COVID-19 is R0=2.2, although this may change (https://www.ncbi.nlm.nih.gov/books/NBK554776/). For simplicity, we will consider the case when R0=2. Beginning with one infective individual, the figure below illustrates how fast the disease can spread. In mathematics, we call this exponential growth.

Example of R0=2, meaning on average, each infected individual will infect two more individuals.(Image created by Yong with emoji)

Calculations of R0 are usually done by examining 1) the amount of time an individual is infectious, 2) the contact rate, and 3) the likelihood of infection per contact when a susceptible individual interacts with an infectious individual. Having more people in a room together can increase R0, while following measures to protect yourself, such as hand washing and avoiding crowded places, as recommended by health officials, can decrease R0.

Flattening the Curve

Implementing measures recommended by health officials, we can reduce R0 and slow the spread of the disease. Although the disease is slowed, it is not stopped, so some may wonder what is the point of disrupting our daily lives by canceling class and large gatherings such as concerts and sporting events. By limiting public gatherings and our movements, a practice called social distancing, we can reduce the number of cases at any given time, or “flatten the curve,” relieving the strain on our hospitals. Because there are a limited number of hospital beds, including ICU beds, if we see a large spike in the number of cases, our health care system may not be equipped to handle such volume, resulting in a shortage of available care. This is happening in Italy.

Image from Drew Harris illustrates flattening the curve.

To illustrate the importance of flattening the curve, we look at the 1918 Spanish flu. The cities of Philadelphia and St. Louis handled the outbreak in very different ways. Philadelphia continued to allow large gatherings and schools remained open for two weeks after the first case was reported, whereas St. Louis implemented measures only two days after the first case was reported. As a result, at the peak of the outbreak, for every 100,000 people in their city, Philadelphia had 257 deaths and St. Louis only had 31. Over the course of the outbreak, for every 100,000 people in their city, Philadelphia had 719 deaths and St. Louis had 347. Not only were there fewer overall deaths in St. Louis than in Philadelphia, but at the peak of the outbreak, Philadelphia had eight times as many deaths than the peak in St. Louis. This is why governments are implementing measures, so we don’t overburden the healthcare system with more patients than it can handle.

Actual data of death rate of Philadelphia and St. Louis from September 8–December 28, 1918. Image from Hatcher et al., 2007. https://doi.org/10.1073/pnas.0610941104