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The Mathematic Pandemic

by Brian Zhang

We have all been learning math since elementary school. Word problems, systems of equations, and the quadratic formula are familiar to many students. But how can we apply the math that we learn to our daily lives? The COVID-19 pandemic is one scenario in which the math is not far beneath the surface. Epidemiologists utilize mathematical equations and models at every step of the way in mapping the spread of an outbreak and developing the vaccines to vanquish the disease. You may think that the mathematical tools used by scientific experts are far beyond the capabilities that you have now, but, in fact, many fundamental mathematical concepts of epidemiology are well within the grasp of any typical middle or high school student.

Any given infectious disease outbreak can be boiled down to its basic reproductive number and case fatality rate. The basic reproductive number, denoted by R0, is the average number of susceptible people that a single infected person will infect over the course of his/her infection. R0 can give an estimate as to how the disease will spread in a given population. If R0 < 1, then each infected person will infect less than 1 other person on average, so the outbreak will not be able to sustain itself and will soon die out. However, if R0 > 1, then each infected person will infect more than 1 other person on average, and the outbreak will see exponential growth in the number of cases. For example, suppose R0 = 3, and the length of time that individuals contract the disease from an infected person is one day – in other words, each day, a newly infected person spreads the disease to 3 other people. The number of new people infected on day n follows the geometric sequence 3n. Considering that a mild COVID-19 case lasts about 14 days, before the first infected person recovers, the total number of people being infected will be the geometric series 1 + 3 + 32 + … 314, which adds up to 7,174,452 in 14 days, more than 7 million people! The following figure illustrates how fast the number of infected cases grows over the course of 14 days.

Many factors affect the spread of COVID-19, including social distancing rules, human interaction behavior, population density, and vaccination rates. Therefore, the actual reproductive number fluctuates over time and depends on the geographic location. As a result, the number of infected cases changes over time and over different regions. The graphs below show the reproductive numbers in the United States and India during the period of January to May 2021. The high reproductive number in India in March and April explains the peak of infectious cases in that country.

In addition to the contagiousness of COVID-19, even more important information that people want to know is the virus’s deadliness. The chance of death caused by a disease is measured by the case fatality rate (CFR), calculated with the formula

CFR = number of deaths/ number of cases.

Obviously, the lower the CFR value is, the less dangerous the disease is. Similar to the basic reproductive number, the CFR is also affected by various factors, such as age, healthcare, and system accessibility, so it fluctuates over time as well. The average CFR in the United States from January 2020 (which marked the occurrence of the first confirmed COVID-19) to January 2021 was about 2.1%, suggesting that about 2 people would die among 100 cases. However, the CFR of the US was exceptionally high from April to May 2020, the early stage of the pandemic.  As shown in the graph below, the CFR peaked at close to 8%. 

Fortunately, when the CFR is high at a certain time point, the basic reproductive number is generally low, and vice versa. This is because if many patients quickly die from the disease, then they usually infect few other people, and if one person infects many other people, then is the disease could be considered less deadly since the infected people must have survived enough time for the disease to spread. 

Compared to many other infectious diseases, COVID-19 has been less deadly overall. The following table shows the CFR of COVID-19 and a number of other diseases.

While the basic reproductive number and case fatality rate offer a general idea of how contagious and deadly COVID-19 is, more complex mathematics is needed if we want to understand and even predict the progression of the spread of the disease. One of the most common models used is the SIR model, which separates the population into three categories: Susceptible, Infected, and Removed. People who have had no contact with the disease and are not immunized fall into the Susceptible category; those who have contracted the disease and are still able to infect others are Infected; and people who have recovered, died, or been immunized/vaccinated from the disease are called Removed. A set of differential equations is then established to describe how the numbers of the three populations are relevant to each other and how they change over time. For example, the Susceptible equation looks like this:

It means that the change in the number of Susceptible people S over the change in time t is the product of the average number of contacts a typical member of population connects with per time unit, β; the Susceptible fraction of the population, s; and the number of individuals already infected, I. The negative sign indicates that the Susceptible population shrinks when some of those within are infected. The model’s definition clearly conveys some of the implications of social distancing and quarantine: when people have little to no contact with each other, β is close to 0, so the rate of increase of S is going to be small. These measures essentially sever the connection between the Susceptible and Infected, the only mechanism by which the disease can spread. Therefore, strict social distancing and quarantine restrictions often prove to be successful in preventing the spread of a disease.

Finally, let’s take a look at how the effectiveness of vaccines is measured through mathematics. In a particular trial for testing a vaccine, the testing population is split into two groups: the vaccine group and the control group. The vaccine group receives the vaccine while the control group is given a placebo, a harmless solution. Over the course of the trial, volunteers are monitored for any COVID-19 cases.

After the monitoring period, the incidence rate (IR) of COVID-19 in each group is calculated by the formula

The incidence rates of the vaccine group and control group are denoted by IRvaccine and IRcontrol, respectively. To compare the results of the two groups, the relative risk (RR) of the vaccine is found using the equation

The final assessment of the vaccine is the vaccine efficacy (VE), which is simply

VE = 1- RR.

The vaccine efficacy measures how much the vaccine reduces the incidence of disease: the greater the VE, the more potent the vaccine. The  IRvaccine and IRcontrol of the Moderna and Pfizer vaccines are shown in the table below:

Now that you are better acquainted with the mathematics behind the spread of the COVID-19 pandemic and how vaccination trials are run, where else can math be found? The possibilities for applied mathematics are endless, appearing in fields from science and engineering to music and arts. The math we learn in the classroom truly empowers us to change the world and helps to pave the way for the future of mankind.

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