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Why? This does mean that the disease is "endemic" in the population. 6. , when c < 1 t—»oo What do the numerical simulations do in the cases you ran when c < 1 ? 5. Conjectures After you try various simulations, make a conjecture about lim s[t] = a formula in terms of c t—>oc in case c > 1. This formula should be easy to guess if you have done simulations with values such as c = 2,3,4 and compute your observed limit in terms of c. Once you have made clear conjectures, you should use the equations for change to show that things change in the direction of your conjecture.

54 Chapter 5 Differential Equations from Increment Geometry The three curves in this chapter illustrate an important idea of calculus: they are each described by a differential equation, which in turn comes from studying properties of a tiny increment of the curve without knowing the curve itself in advance. The curves have a physical description - they are quite concrete - but the tractrix and catenary provide important examples in differential geometry and the calculus of variations. In other words, these curves are important in mathematics.

6 for polio. ) When s[0] < 1/c, when does the i[t] "peak"? 5. The Second Peak of an S-I-R Epidemic When is the disease spreading fastest in terms of the growth of new cases? What does a si measure? If we want to maximize the function f[t] = a s[t] i[t] for t > 0, we need to find zeros of its derivative ds . di d(s x • i) , J = — -2 + s- — at dt dt = (—asi)i -f s(asi ( = (asi) - \ ( • — bi) b \s — i Show that this is zero when s — i = 1/c. When is f[t] increasing? When is f[t] decreasing? What is the slope table of its graph?

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Analyse numerique by Temam R

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