Please solve & show steps... Daniel Bernoulli's work in 1760 had the goal of appraising theeffectiveness of a controversial inoculation program against smallpox,which at that time was a major threat to public hecalth. His modelapplies equally well to any other disease that, once contracted andsurvived, confers a lifetime immunity. Consider the cohort ofindividuals born in a given year (t = 0), and let n(t) be the number ofthese individuals surviving t years later. Let x(t) be the number ofmembers of this cohort who have not had smallpox by year t and whoare therefore still susceptible. Let 3 be the rate at which susceptiblescontract smallpox, and let v be the rate at which people who contractsmallpox die from the disease. Finally, let u(t) be the death rate fromall causes other than smallpox. Then dr/dt, the rate at which thenumber of susceptibles declines, is given by = -(B+ µ(t))x. The%3Ddifirst term on the right-hand side of this equation is the rate at whichsusceptibles contract smallpox, and the second term is the rate atwhich they die from all other causes. Also = -vßx – u(t)n, wheredtdn/dt is the death rate of the entire cohort, and the two termson the right-hand side are the death rates due to smallpox and to allother causes, respectively.a) Let z = x/n, and show that z satisfies the initial value problemdz= -Bz(1 – vz). Observe that this initial value problem does notdepend on u(t).b) Find z(t) by solving equation (a).z(t) = |Choose onec) Bernoulli estimated that v = ß = 1/8. Using these values, determinethe proportion of 50-year-olds who have not had smallpox.NOTE: Enter an exact answer.Proportion:

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Description: Please solve & show steps... Daniel Bernoulli's work in 1760 had the goal of appraising theeffectiveness of a controversial inoculation program against smallpox,which at that time was a major threat to public hecalth. His modelapplies equally well to

n(t)= number of individuals surviving t years later x(t)= number of members of cohort who have not…

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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 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 wl are therefore still susceptible. Let ß be the rate at which susceptibles contract smallpox, and let v be the rate at which people who contra smallpox die from the disease. Finally, let 4(t) be the death rate fro all causes other than smallpox. Then dx/dt, the rate at which the de number of susceptibles declines, is given by = -(B + µ(t))x. The

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 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 wl are therefore still susceptible. Let ß be the rate at which susceptibles contract smallpox, and let v be the rate at which people who contra smallpox die from the disease. Finally, let 4(t) be the death rate fro all causes other than smallpox. Then dx/dt, the rate at which the de number of susceptibles declines, is given by = -(B + µ(t))x. The

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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 hecalth. 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 dr/dt, the rate at which the
number of susceptibles declines, is given by = -(B+ µ(t))x. The
%3D
di
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 – u(t)n, where
dt
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
dz
= -Bz(1 – vz). Observe that this initial value problem does not
depend on u(t).
b) Find z(t) by solving equation (a).
z(t) = |Choose one
c) Bernoulli estimated that v = ß = 1/8. Using these values, determine
the proportion of 50-year-olds who have not had smallpox.
NOTE: Enter an exact answer.
Proportion:

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