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 by some time t . Consider the following model, where k > 0 is constant:

a. List two major assumptions implicit in the preceding model. How reasonable are your assumptions?

b. Solve the model given earlier for X as a function of t.

c. From part (b), find the limit of X as t approaches infinity.
d. Consider an island with a population of 5000. At various times during the epidemic the number of people infected was recorded as follows:

Do the collected data support the given model?

e. Use the results in part (f) to estimate the constants in the model and predict the number of people who will be infected by t = 12 days.