Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
where P =n+p+h is the total population, and a and b are some real numbers (which you do not need to bother calculating.) (e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of people in the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of the population. What are these percentages?

where P =n+p+h is the total population, and a and b are some real numbers (which you do not need to bother calculating.) (e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of people in the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of the population. What are these percentages?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please just answer d) and e)

The last time I posted it and due to the rules only a),b),c) were answered.

Much apprciated!

In this question, the real numbers in your answers should be given with three significant digits of accuracy after the decimal point. The public health authorities of a small town have divided the population into three categories: covid-negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed that in each successive week: • Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised. • Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised. • Among those who are hospitalised, 60% recover and become negative, 30% are released from hospital but remain positive, and 10% remain hospitalised. (a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h represent the number of people in the population who are negative, positive, and hospitalised respectively, write down a matrix M for which [n', p',h'] = M[n; p; h], where [n',p', h'] represents the column vector of negative, positive, and hospitalised members of the population in the following week. (b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that d1 > A2 > A3. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen command in Pari-GP.) (C) Write down eigenvectors V1, V2, V3 attached to A1, A, and A3 respectively. (d) Show that v1 can be normalised so the the sum of its coordinates is equal to 1, and, with this normalisation, show that, for any triple v = [n; p; h], = Pv1 + av2 + bv3
Transcribed Image Text:In this question, the real numbers in your answers should be given with three significant digits of accuracy after the decimal point. The public health authorities of a small town have divided the population into three categories: covid-negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed that in each successive week: • Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised. • Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised. • Among those who are hospitalised, 60% recover and become negative, 30% are released from hospital but remain positive, and 10% remain hospitalised. (a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h represent the number of people in the population who are negative, positive, and hospitalised respectively, write down a matrix M for which [n', p',h'] = M[n; p; h], where [n',p', h'] represents the column vector of negative, positive, and hospitalised members of the population in the following week. (b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that d1 > A2 > A3. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen command in Pari-GP.) (C) Write down eigenvectors V1, V2, V3 attached to A1, A, and A3 respectively. (d) Show that v1 can be normalised so the the sum of its coordinates is equal to 1, and, with this normalisation, show that, for any triple v = [n; p; h], = Pv1 + av2 + bv3
where P = n+p+h is the total population, and a and b are some real numbers (which you do not need to bother calculating.) (e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of people in the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of the population. What are these percentages?
Transcribed Image Text:where P = n+p+h is the total population, and a and b are some real numbers (which you do not need to bother calculating.) (e) Use the result of part (d) to show that, no matter what the initial state of the population, the proportion of people in the population who are negative, positive, and hospitalised will, over time, stabilise to fixed percentages of the population. What are these percentages?
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS