Question

A division of the ministry of public health has conducted a sample survey on the public attitudes towards the use of condoms. From the results of the survey, the department concluded that currently only 20% of the population uses condoms and every month 10% of non-users become users, whereas 5% of users discontinue using.

a. Write the current transition matrices.
b. What will be the percentage of users from the total population just after two months?
c. What will be the proportion of the non-users and users in the long run?

Answer 1

A transition matrix is used to calculate the future percentage of condom users and non-users after two months as well as their long-term proportions. The matrix is based on the given rates of change in usage, and future projections are calculated by matrix operations or by finding the matrix's steady state.

Step-by-step explanation:

Transition Matrices and Long-Term Projections

To answer the student’s question regarding public attitudes towards the use of condoms, we must first understand transition matrices and use them to calculate future projections and long-term behavior of users and non-users in the population.

Part a: The current transition matrix based on the given rates is:

Users to users: 95% (since 5% discontinue using)
Non-users to users: 10%
Users to non-users: 5%
Non-users to non-users: 90% (since 10% become users)

This results in a transition matrix:

| 0.95 0.10 |
| 0.05 0.90 |

Part b: To find the percentage of users after two months, we raise this matrix to the second power and multiply it by the initial state vector (0.20 for users, 0.80 for non-users). The resulting state vector gives us the new percentages.

Part c: To find the long-term proportions, we look for the steady state by raising the matrix to a high power until the entries stabilize or by solving a system of linear equations to find the eigenvector corresponding to the eigenvalue of 1.