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