By James Callahan, Kenneth Hoffman, David Cox, Donal O’Shea ,Harriet Pollatsek, Lester Senechal

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THE SPREAD OF DISEASE 15 Delta notation. This work has given us some insights about the way our model predicts future values of S, I, and R. The basic idea is very simple: determine how S, I, and R change. Because these changes play such an important role in what we do, it is worth having a simple way to refer to them. Here is the notation that we will use: ∆x stands for a change in the quantity x The symbol “∆” is the Greek capital letter delta; it corresponds to the Roman letter “D” and stands for difference.

10. Twain does not tell us how long the Lower Mississippi was in 1884 when he wrote the book, but he does say that 742 years later it will be only 1 34 miles long. How long must the river have been when he wrote the book? 11. Suppose t is the number of years since 1884. Write a formula that describes how much L has changed in t years. Your formula should complete the equation the change in L in t years = . . Copyright 1994, 2008 Five Colleges, Inc. DVI file created at 11:39, 1 February 2008 22 CHAPTER 1.

If S exceeds the threshold, I increases. Finally, I reaches its peak when S equals the threshold. The presence of a threshold value for S is purely a mathematical result. However, it has an interesting interpretation for the epidemic. As long as there are at least 7143 susceptibles, the infection will spread, in the sense that there will be more people falling ill than recovering each day. As new people fall ill, the number of susceptibles declines. Finally, when there are fewer than 7143 susceptibles, the pattern reverses: each day more people will recover than will fall ill.