Maribrevas
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- Apr 10, 2020
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In 1760 Daniel Bernoulli developed a model for the spread of smallpox, which at the time was a major threat to public health. His goal was to determine whether or not a controversial inoculation program would be effective. His model applies equally well to any disease where survival after contraction of the disease results in lifelong immunity. Consider the cohort of individuals born in a given year (ie: at time t = 0). Let n(t) be the total number of these individuals who are still alive t years later. Let x(t) be the number of members of the cohort who have not contracted smallpox by year t, and are therefore still susceptible. Let β be the rate at which susceptibles contract smallpox, and let ν the chance that people who contract smallpox die from the disease. Finally, let µ(t) be the death rate from all causes other than smallpox. Then the population is governed by the following differential equations:
dx/dt = − (β + µ(t)) x
dn/dt = −νβx − µ(t)n
(a) Explain why these equations govern the population described above.
(b) Let z = x/n and show that z satisfies the initial value problem dz/dt = −βz (1 − νz), z(0) = 1.
(c) Describe what z represents in terms of the population being modeled.
(d) Solve this new equation for z(t). (Hint: The equation now resembles the differential equation for the logisitic model. You may cite the formula for the solution to the logistic model from the lecture notes.)
(e) Bernoulli estimated that ν ≈ 1/8 yr^(−1) and β ≈ 1/8 . Using these values, sketch or plot z(t), describe your plot, and determine the proportion of 20-year-olds who have not had smallpox. Note: Based on the model described above and using the best mortality data available at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated (if you could somehow make ν = 0), then approximately 3 years could be added to the average life expectancy in 1760: 26 years and 7 months. Therefore Bernoulli supported the proposed inoculation program.
dx/dt = − (β + µ(t)) x
dn/dt = −νβx − µ(t)n
(a) Explain why these equations govern the population described above.
(b) Let z = x/n and show that z satisfies the initial value problem dz/dt = −βz (1 − νz), z(0) = 1.
(c) Describe what z represents in terms of the population being modeled.
(d) Solve this new equation for z(t). (Hint: The equation now resembles the differential equation for the logisitic model. You may cite the formula for the solution to the logistic model from the lecture notes.)
(e) Bernoulli estimated that ν ≈ 1/8 yr^(−1) and β ≈ 1/8 . Using these values, sketch or plot z(t), describe your plot, and determine the proportion of 20-year-olds who have not had smallpox. Note: Based on the model described above and using the best mortality data available at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated (if you could somehow make ν = 0), then approximately 3 years could be added to the average life expectancy in 1760: 26 years and 7 months. Therefore Bernoulli supported the proposed inoculation program.