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a) Let z = x/n, and show that z satisfies the initial value problem dz dt dź = −ßz(1 – vz). Observe that this initial value problem does not depend on μ(t). b) Find z(t) by solving equation (a). 1 z(t) = v+(1-v)eßt c) Bernoulli estimated that v = ß = 1/8. Using these values, detern

a) Let z = x/n, and show that z satisfies the initial value problem dz dt dź = −ßz(1 – vz). Observe that this initial value problem does not depend on μ(t). b) Find z(t) by solving equation (a). 1 z(t) = v+(1-v)eßt c) Bernoulli estimated that v = ß = 1/8. Using these values, detern

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 y 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 = −(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 d = -v3x − µ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dn 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 -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). dz = dt 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 y 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 = −(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 d = -v3x − µ(t)n, where dn/dt is the death rate of the entire cohort, and the two terms dx dn 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 -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). dz = dt b) Find z(t) by solving equation (a).
a) Let z = x/n, and show that z satisfies the initial value problem -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). dz = dt b) Find z(t) by solving equation (a). z(t) = 1 v+(1-v) eft c) Bernoulli estimated that v = ß = 1/8. Using these values, determi the proportion of 13-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
Transcribed Image Text:a) Let z = x/n, and show that z satisfies the initial value problem -ßz(1 - vz). Observe that this initial value problem does not depend on μ(t). dz = dt b) Find z(t) by solving equation (a). z(t) = 1 v+(1-v) eft c) Bernoulli estimated that v = ß = 1/8. Using these values, determi the proportion of 13-year-olds who have not had smallpox. NOTE: Enter an exact answer. Proportion:
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