(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.
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:
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Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised.
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Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised.
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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
(b) 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],
v=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.)
(c) Use the result of part (b) 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?
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