We shall consider a model for the spread of a disease in an isolated population, such as the students at a board- ing school. There are three variables: S = the number of susceptibles, the people who are not yet sick but who could become sick; I = the number of infected, the peo- ple who are currently sick; R = the number of recovered, or removed, the people who have been sick and can no longer infect others or be reinfected. 2.2. (a) Explain why the following differential equations are a reasonable model for the spread of the disease: ds =-aSI dt dI = aSI bI dt dR = bl dt Consider a school with 1000 students. Let's say that one student develops the flu, and that one day later two more students are infected. (a) Use the first equation above to estimate a on the basis of this information. (b) Let's say that b = 0.5. What is the real-world mean- ing of this? (c) Sketch the direction field for this system with S on the x-axis and I on the y-axis.