24. 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 the year t and who are therefore still susceptible. let ß be the rate at which susceptibles contract smallpox, and letv be the rate at which people who contract smallpox and die from the disease. finally, let µ(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
dz/dt = -[ß +µ(t)]x. (i)
the first term on the right side of eq. (i) is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. also
dn/dt=-vßx -µ(t)n, (ii)
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.
i know:
(a) the inital value problem is:
let z=x/n initial value problem
dz/dt=-ßz(1 - vz), z(0)=1
(b) general solution is z(1-vz)=ce^(-tb)
bernoulli estimated that v=ß=1/8.
questions:
(1) find the solution to the initial value problem
(2) determine the proportion of 20 year old's who have not had smallpox
note: on the basis of the model just described and the best mortality data available at the time, bernoulli calculated that if deaths due to smallpox could be eliminated (v=0),then approximately 3 years could be added to the average life expectancy (in 1760) of 26 years, 7 months. he therefore supported the inoculation program.