Consider the spreading of a highly communicable disease on an isolated island with population size N. A portion of the population travels abroad and returns to the island infected with the disease. You would like to predict the number of people X who will have been infected at some time t. Consider the following model, where k > 0 is constant:

dX/dt = k X (N - X)

(a) List two major assumptions about the disease or the island implicit in the preceding model. How reasonable are the assumptions?

(b) Graph X versus " t " if the initial number of infections is X1 < N/2 . Graph X versus " t " if the initial number of infections is X2 > N/2

(c) Solve the model given earlier for X as a function of " t ".

(d) From part (d), find the limit of X as " t" approaches infinity.

(e) Consider an island with a population of 5000. At various times during the epidemic, the number of people infected was recorded as follows:

T (days)

2

6

10

X (people infected)

1887

4087

4853

ln (X/(N