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Markov chains

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1) Create a Markov transition matrix that represents the following parameters.

Each week 4% of the total population becomes infected by a disease
Each week 10% of the infected population recovers from the disease
Each week 0.2% of the infected population dies from the disease

Write the corresponding Markov matrix in the space below.

Suppose the initial population consists of 100,000 healthy/susceptible people 0 people that are initially infected and 0 that are cured, and 0 that have died.

Assume the transition matrix above (from question 1). Assume each step in the Markov process represents 1 week. Round off any decimal numbers to the nearest whole number.

2a) How many people are infected according to this model, after 1 week?

b) How many people are cured/recovered after 2 weeks?

3) How many people have died after 1 year (52 weeks) according to this model?

4) On which week (approximately) is the percentage of infected people (as a fraction of those that are alive) the highest? What is this percentage?

5) What is the highest number of infected people during the pandemic as given by this model?

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Solution Summary

The solution contains a step-by-step explanation of the states and matrix. A calculator is used to assist with matrix calculations.

Solution Preview

1)
In this model, the categories are:
H = Healthy/susceptible
I = Infected
C = Cured/recovered
D = Died

The transitions are:
Since 4% of the Healthy population become Infected, we get
H --> I : 4%, or 0.04

Since 4% of Healthy people Infected, the other 96% stay Healthy, so we get
H --> H : 96%, or 0.96.

Since 10% of Infected become Cured/recovered, we get
I --> C : 10%, or 0.10

Since 0.2% of Infected become Died, we get
I --> D : 0.2%, or 0.002

The rest of the Infected remain Infected. 100% - 10% - 0.2% = ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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