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 health. 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 dx/dt, the rate at which thenumber of susceptibles declines, is given by = -(ß+ µ(t))x. Thedtfirst 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 – µ(t)n, wheredn/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 problem= -Bz(1 – vz). Observe that this initial value problem does notdtdepend on µ(t).b) Find 2(t) by solving equation (a).z(t) = Choose onec) Bernoulli estimated that v = 3 = 1/8. Using these values, determithe 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:

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.EA: Extended Application Using Extrapolation To Predict Life Expectancy

Problem 5EA