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Figure 1: A diagram of the two processes—infection and recovery—by which people move from one compartment to another.

• Recovered R (t): People who were infected but are no longer at risk of infecting others.

instructor of high school calculus. The model tracks the numbers of people who have not been infected, who have an active infection, and who have recovered and have immunity, and since it draws connections between the size of these groups and the rates of change of these groups, the model naturally takes the form of a system of differential equations. Students in a high school calculus class (and possibly their teachers) may not have studied differential equations, and so we will not attempt to solve the system. Instead, our focus here is on how the ideas of rates of change and proportionality can be used to create a model, and even though we may not know how to solve that model, we are still able to gain important qualitative information about how that unknown solution behaves by applying our knowledge of the concept of critical values. The definition of R 0 arises in the process of nondi mensionalizing the model, which involves an appli cation of the Chain Rule. A common framework for modeling a disease out break is a “ compartment model. ” Brief explana tions of this approach and other more complicated models may be found in (Abou - Ismail, 2020; Bertozzi, et al., 2020; & Hethcote, 2000). In the most common compartment model—the Suscepti ble - Infectious - Recovered (SIR) model—we let N be the fixed total population of a community. This could be a country, but it could also be a smaller community such as a rural town, a school, an as sisted living community, or a prison. At time t , the people of the community are split among three groups or “ compartments ”: The Susceptible - Infectious - Recovered Model • Susceptible S (t): People who do not have the disease and have no immunity to it. • Infectious I (t): People who have an active in fection that can spread to the susceptible.

Some modelers call R “ resistant ” because it is as sumed the previous infection provides immunity, and others called it “ removed ” because it also in cludes people who have died from the disease. Since these three groups account for the entire pop ulation, S (t) + I (t) + R (t) = N. More complicated models have more compartments and more con necting pathways. We build up the model as a system of differential equations for these unknown functions by connect ing the two aspects of the disease cycle we consid er—recovery and infection—with the rates of change of the compartments. A diagram of these two pathways is shown in Figure 1. People move from the infectious compartment I to the recovered compartment R when they are no longer able to infect others. Thus, the rate of change of R with respect to time is directly propor tional to the total number of infected at a given time. That is, the more infections there are, the more people who recover during a given time peri od. This gives us the equation: The proportionality constant γ is known as the re covery rate , and it is the reciprocal of the recovery time. That is, if it takes on average 10 days to stop being infectious, γ = 0.1 day - 1 . This term γ I appears twice in the model. In the righthand side of (1), it represents the number of people per day entering the recovered compart ment, but these people simultaneously leave the infectious compartment, and so the equation for the rate of change of I must include – γ I . However, that

Virginia Mathematics Teacher vol. 47, no. 1

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