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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 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 d -(3 + 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 = -√3x − μµ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dt == dn 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 = −ßz(1 – vz). Observe that this initial value problem does not dt 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 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 d -(3 + 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 = -√3x − μµ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dt == dn 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 = −ßz(1 – vz). Observe that this initial value problem does not dt depend on μ(t). b) Find z(t) by solving equation (a).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 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 d -(3 + 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 = -√3x − μµ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dt == dn 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 = −ßz(1 – vz). Observe that this initial value problem does not dt depend on μ(t). b) Find z(t) by solving equation (a).
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 d -(3 + 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 = -√3x − μµ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dt == dn 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 = −ßz(1 – vz). Observe that this initial value problem does not dt depend on μ(t). b) Find z(t) by solving equation (a).
a) Let z = x/ x/n, and show that z satisfies the initial value problem -ßz(1 - vz). Observe that this initial value problem does not dz = dt depend on μ(t). b) Find z(t) by solving equation (a). z(t): = Choose one ▼ c) Bernoulli estimated that v = ß = 1/8. Using these values, determi the proportion of 69-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
Transcribed Image Text:a) Let z = x/ x/n, and show that z satisfies the initial value problem -ßz(1 - vz). Observe that this initial value problem does not dz = dt depend on μ(t). b) Find z(t) by solving equation (a). z(t): = Choose one ▼ c) Bernoulli estimated that v = ß = 1/8. Using these values, determi the proportion of 69-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
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