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COVID - 19 and the Chain Rule are Linked: Understanding R o through Calculus Eric P. Choate

• If R 0 < 1, on average an infected person re covers before infecting another person. New people still become infected, but recoveries outpace new infections so that transmission chains are easily broken, and the outbreak dies out before spreading rapidly. • If R 0 = 1, the outbreak neither grows nor de clines, and the number of infected people re mains a stable proportion of the population. This kind of disease is called endemic . • If R 0 > 1, the outbreak exhibits sustained growth, given a sufficient reservoir of people with no immunity to it. This is considered the definition of an epidemic .

Introduction

If any aspect of modeling an epidemic appears in the popular media, such as the first diagnosis of a deadly virus in the United States (Doucleff, 2014) or the 2011 film Contagion (Soderbergh, 2011) or news articles fact - checking Contagion (Kritz, 2020), it is the reproduction number R 0 . For an au dience that does not know calculus, R 0 has a practi cal definition and an intuitional effect: the average number of new infections that can be traced to a single infectious person. That is, if a college stu dent returns from spring break with a new disease with R 0 = 3, he infects his roommate, his chemistry lab partner, and his math professor when he makes up the test he missed leaving early for spring break. Each of these three in turn infect three more peo ple, for a total of nine new cases at this stage of the outbreak. These nine lead to 27 new cases in the third generation, and so the outbreak begins to grow exponentially.

Estimates of R 0 for COVID - 19 range from 2 to 4 (CDC, 2021).

In this article, we define R 0 in a different way as a parameter—and the only parameter—in a mathe matical model for an outbreak of a disease like COVID - 19, and then we connect the above effects of its value with causes in the model ’ s equations, with an eye toward how it could be useful to an

Given this loose definition, we see three different modes of outbreak behavior:

Virginia Mathematics Teacher vol. 47, no. 1

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