Some illnesses have a behavior that repeats over a period of time. One example of this behavior is the yearly cycle of the spread of the flu. Suppose the number of people sick from the flu after t months is modeled by

?

? M = is maximum number of sick people at any one time,?

? to = time, in months, where this maximum, M occurs,?

? r is a rate of spread and of recovery

a) Find the rate at which the number of sick people is changing. (Answer may include variables M , r. t0)

b) Suppose we use the parameters M = 25, 000, r = 1.5, and t0 = 1, to represent the spread of influenza in the United States. At what time t is the number of sick people increasing the fastest over the first month. That is, at what time t is the number of sick people increasing the fastest when 0 ? t ? 1?

c) Find the points of inflection of S on [0, 1].

S(t) M 1+r(t-to) 2