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 µ(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 d = −(B+ µ(t))x. 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 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. dx dt dn dt - -vßx-μ(t)n, where a) Let z = x/n, and show that z satisfies the initial value problem dz = dt -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). b) Find z(t) by solving equation (a).
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 µ(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 d = −(B+ µ(t))x. 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 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. dx dt dn dt - -vßx-μ(t)n, where a) Let z = x/n, and show that z satisfies the initial value problem dz = dt -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). b) Find z(t) by solving equation (a).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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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 µ(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 d = −(B+ µ(t))x. 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
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.
dx
dt
dn
dt
-
-vßx-μ(t)n, where
a) Let z = x/n, and show that z satisfies the initial value problem
dz
=
dt
-ßz(1 - vz). Observe that this initial value problem does not
depend on μ(t).
b) Find z(t) by solving equation (a).
Transcribed Image Text: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 dn = -v3x - μ(t)n, where
dn/dt is the death rate of the entire cohort, and the two terms
dt
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
dz
dź - Bz(1 - vz). Observe that this initial value problem does not
=
dt
depend on μ(t).
b) Find z(t) by solving equation (a).
z(t) =
= Choose one
c) Bernoulli estimated that v = B = 1/8. Using these values, determi
the proportion of 37-year-olds who have not had smallpox.
NOTE: Enter an exact answer.
Proportion:
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