MATH152 April 2015
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Question B 5 (a)
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.
- (a) What is the probability transition matrix for this system?
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To find the matrix, it is enough to recall that matrix multiplication is a linear transformation and that the columns of a matrix are the images of the standard basis vectors under that matrix.
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Represent the proportion of people who are infected as column vectors like so:
And let be the desired transition matrix. The data given in this question becomes,
The first column of is and the second column of is . So, it is enough to find . We have that
So as is a linear transformation,