Question

[Markov Models] Consider the following situation:

There are three possible states: healthy, sick and dead.

Healthy people can get sick, and sick people can recover and become healthy.

Healthy people can die, and sick people can die. Dead people stay dead.  Each ‘cycle’ is one day long.

If you are healthy today, there is an 80% chance that you will be healthy tomorrow, and a 5% chance that you die. If you do not stay healthy or die, you become sick.

If you are sick today, there is a 60% chance that you will be sick tomorrow, and a 10% chance that you will die. If you do not stay sick or die, you recover (become healthy).

In ‘Cycle 0’, there are 1,000 healthy people, 800 sick people and 0 dead people.

a. (6 marks) Draw a Markov diagram for the situation above (hint: it’s the diagram with the circles and arrows).

b. (6 marks) Calculate the number of healthy people, sick people and dead people in Cycle 2. If necessary, round your answers to the nearest whole number. Show your work. (Hint: Cycle 2 is 2 cycles after Cycle 0. If Cycle 0 is today, then Cycle 1 is tomorrow, and Cycle 2 is the day after tomorrow.)

Healthy People in Cycle 2: ____________

Sick People in Cycle 2: ___________

Dead People in Cycle 2: ______

Explanation:

Answer 1

(6) Cycle 1 Number of healthy people who will die from cycle o - O•OSHO -000s x 1000 = so (i) fall sick! Number of healthy people who will from lyde o = 0.15 Ho = 0.15 x 1000 Number of Healthy people who will stay Healthy from cycleo - 0.8 x Ho 20.8 X 1000 - 800 -

You Number of sick people who will die from cycle 0 - 0.1 so - Qol x 800 - 80 - liv) Number healthy of sick people who will become from clecleo - 0.350 -0.3 x 800 = 240 - (W) Number of sick people who will stay sick from cycle o - 0.6so -0.6 x 800 = 480 (vi Hence, H - (ii) + (W) = 800+240 = 1040 Si= (ii) + (vi) = 150 + 480 = 630 D = + (iv) = so + 80 = 130 Cycle a No. of healthy people who will die from Cycle 1 - 0.05H = 0.05 x 1040 - (vii) = 52 No. of healthy people who will fall sick from cycle 1 – 0.15H, = 0.15X 1040 = 156 - (viii)

YOUVA Data No of healthy people who will stay healthy from cycle 1 = 0.8H, = 0.8 x 1040 - 832 (ix) No. of sick people who will die from / Cycle 1 = 0.is, = 0·1x630 = 63 No. of Sick people who will become healthy from cycle 1 - 0.3s, - 0-3 630 - 189 - (xis No. of from sick cycle people who will stay sick 1 – 0.65, = 0.6 x 630 = 378 - (xiis Hence, H₂ - (ix) + (xi - 832 + 189 - 1021 S2 = (viii) + (xii) = 156 +378 = 534 D2 = D, +(vii) + (x) = 130+52 +63 - 245