Question

3. [15 marks] A new tropical disease has emerged on an isolated island where the population is

1000. Observation shows that the disease has the interesting property that there needs to be

a minimum number of infected individuals before it can spread through the population. The

disease spreads according to the differential equation

dD/dt= 0.001(D - 100)(1000 - D)

where D(t) is the number of infected people on the island and t is measured in days.

(a) Find all the equilibrium points and determine their stability.

(b) A timely scientific breakthrough means that 5 people a day can be cured. Modify the equation

above to account for this.

(c) Use an Euler numerical scheme to find a numerical solution for the modified equation. Use

your scheme to solve the equation for initial conditions D(0) = 110 and D(0) = 150. Use

Dt = 0.5 and solve for t 2 [0, 10]. Plot your results on the same axis and comment.

Basic instructions on using Excel are given on page 2 of the Unit Notes.

(d) If the number of infected people on the island starts off at less than 100 then the disease

cannot spread. Use your numerical method to estimate how long it takes for the disease to

die out if D(0) = 99.

Answer 1