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 8 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 4(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 dr/dt = -[B + u(t)]r. (1) The first term on the right of Eq. (1) is the rate at which susceptibles contract smallpox, and the second term is the rate at which the die from all other causes. Also, dn/dt = -vßx- H(t)n, (2) where dn/dt is the death rate of the entire cohort, and the two terms on the right side are 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 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 8 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 4(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 dr/dt = -[B + u(t)]r. (1) The first term on the right of Eq. (1) is the rate at which susceptibles contract smallpox, and the second term is the rate at which the die from all other causes. Also, dn/dt = -vßx- H(t)n, (2) where dn/dt is the death rate of the entire cohort, and the two terms on the right side are 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
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 dr/dt, the rate at which the number of susceptibles declines, is given by
dr/dt = -[B + µ(t)] r.
(1)
The first term on the right of Eq. (1) is the rate at which susceptibles contract smallpox, and
the second term is the rate at which the die from all other causes. Also,
dn/dt = -vßr – p(t)n,
(2)
where dn/dt is the death rate of the entire cohort, and the two terms on the right side are the
death rates due to smallpox and to all other causes, respectively.
(a) Let z = 1/n, and show that z satisfies the initial value problem
dz/dt = -Bz(1 – vz), z(0) = 1.
(3)
Observe that the initial value problem (3) does not depend on p(t).

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