Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 5 (b)

MATH152 April 2015

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Question B 5 (b)

The percentage of people with the disease, March Madness, is recorded every week. Note that it is possible to recover from March Madness one week and catch it again the following week. Records indicate that the disease can be modelled by a random walk and that if 50% of the population is infected with March Madness one week, then 60% of the population will be infected the next week. Records also indicate that if 100% of the population is infected one week, then 90% of the population will be infected the next week. It is known that 10% of the population has March Madness this week. (b) What percentage of the population will have March Madness two weeks from now?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Set up this problem like in part (a), and use your answer from that part (i.e., use the matrix).

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Represent the proportion of people who are infected as column vectors like so: ${\begin{pmatrix}{\text{ proportion of people who are not infected }}\\{\text{ proportion of people who are infected }}\end{pmatrix}}$ Because ten percent of the people are infected this week , the vector is ${\begin{pmatrix}0.9\\0.1\end{pmatrix}}$ To see the percentage of people infected in two weeks, we calculate $M^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}$ By part (a), $M={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}$ So $M^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}^{2}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}{\begin{pmatrix}0.7&0.1\\0.3&0.9\end{pmatrix}}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.52&0.16\\0.48&0.84\end{pmatrix}}{\begin{pmatrix}0.9\\0.1\end{pmatrix}}={\begin{pmatrix}0.484\\0.516\end{pmatrix}}$ Hence, the proportion of the population that will be infected in two weeks is $\color {blue}51.6{\text{ percent}}$