Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity. Consider the cohort of individuals born in a given year (t = 0), and let n(t) be the number of these individuals surviving t years later. Let r(t) be the number of members of this cohort who have not had smallpox by year t and who are therefore still susceptible. Let 3 be the rate at which susceptibles contract smallpox, and let v be the rate at which people who contract smallpox die from the disease. Finally, let µ(t) be the death rate from all causes other than smallpox. Then dæ/dt, the rate at which the number of susceptibles declines, is given by = -(B+ µ(t))æ. The first term on the right-hand side of this equation is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also = -vßx – µ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms on the right-hand side are the death rates due to smallpox and to all other causes, respectively. a) Let z = x/n, and show that z satisfies the initial value problem dE = -Bz(1 – vz). Observe that this initial value problem does not depend on µ(t). b) Find 2(t) by solving equation (a). z(t) = Choose one c) Bernoulli estimated that v = 3 = 1/8. Using these values, determi the proportion of 67-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity. Consider the cohort of individuals born in a given year (t = 0), and let n(t) be the number of these individuals surviving t years later. Let r(t) be the number of members of this cohort who have not had smallpox by year t and who are therefore still susceptible. Let 3 be the rate at which susceptibles contract smallpox, and let v be the rate at which people who contract smallpox die from the disease. Finally, let µ(t) be the death rate from all causes other than smallpox. Then dæ/dt, the rate at which the number of susceptibles declines, is given by = -(B+ µ(t))æ. The first term on the right-hand side of this equation is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also = -vßx – µ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms on the right-hand side are the death rates due to smallpox and to all other causes, respectively. a) Let z = x/n, and show that z satisfies the initial value problem dE = -Bz(1 – vz). Observe that this initial value problem does not depend on µ(t). b) Find 2(t) by solving equation (a). z(t) = Choose one c) Bernoulli estimated that v = 3 = 1/8. Using these values, determi the proportion of 67-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 33EQ
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Transcribed Image Text:Daniel Bernoulli's work in 1760 had the goal of appraising the
effectiveness of a controversial inoculation program against smallpox,
which at that time was a major threat to public health. His model
applies equally well to any other disease that, once contracted and
survived, confers a lifetime immunity. Consider the cohort of
individuals born in a given year (t = 0), and let n(t) be the number of
these individuals surviving t years later. Let x(t) be the number of
members of this cohort who have not had smallpox by year t and who
are therefore still susceptible. Let 3 be the rate at which susceptibles
contract smallpox, and let v be the rate at which people who contract
smallpox die from the disease. Finally, let u(t) be the death rate from
all causes other than smallpox. Then dx/dt, the rate at which the
number of susceptibles declines, is given by = -(ß+ µ(t))x. The
dt
first term on the right-hand side of this equation is the rate at which
susceptibles contract smallpox, and the second term is the rate at
which they die from all other causes. Also = -vßx – µ(t)n, where
dn/dt is the death rate of the entire cohort, and the two terms
on the right-hand side are the death rates due to smallpox and to all
other causes, respectively.
a) Let z = x/n, and show that z satisfies the initial value problem
= -Bz(1 – vz). Observe that this initial value problem does not
dt
depend on µ(t).
b) Find 2(t) by solving equation (a).
z(t) = Choose one
c) Bernoulli estimated that v = 3 = 1/8. Using these values, determi
the proportion of 67-year-olds who have not had smallpox.
NOTE: Enter an exact answer.
Proportion:
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