Smallpox Controversy: to inoculate or not to inoculate (Reference: Boyce, DiPrima, 2005)

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.

Question 1. Who was Daniel Bernoulli?Why was he working on this project?

Bernoulli’s 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 =0$t=0$, and let ( )$n\left(t\right)$ be the number of these individuals surviving $t$ years later.Let ( )$x\left(t\right)$be the number of members of this cohort who have not had smallpox by year $t$ and who are therefore still susceptible. Let $\beta$ be the rate at which susceptibles contract smallpox, and let $\nu$ be the rate at which people who contract smallpox die from the disease. Finally, let ( )$\mu \left(t\right)$ be the death rate from all causes other than smallpox. Then $\frac{dx}{dt}$, the rate at which the number of susceptibles declines, is given by

=−[ + ( )] $\frac{dx}{dt}=-[\beta +\mu (t\left)\right]x$ (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 they die from all other causes. Also,

=− − ( ) $\frac{dn}{dt}=-\nu \beta x-\mu \left(t\right)n$, (2)

where $\frac{dn}{dt}$ 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.

Introduce a new variable ( )= ( ) ( ).$z\left(t\right)=\frac{x\left(t\right)}{n\left(t\right)}.$

Question 2.Formulatetheinitial value problem for ( )$z\left(t\right)$.

Question 3. Find the complete general solution to differential equation in Q.2 equation.

Question 4. Find the solution to the initial value problem.

Question 5. Draw the slope field( by hand, or with DField oruse MATLAB, for example, ode45) for this equation and draw some representative solutions. What does this picture tell youabout the solution to this initial value problem.

Question 6.Does the solutionin Q.5 agree with your analysis in Q. 4?

Question 5. Bernoulli estimated that = =1/8$\nu =\beta =1/8$.With that in mind, determine the proportion of 20-year-olds who have not had smallpox.

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, =0$\nu =0$, then the life expectancy among inoculated will increase by almost 10%.

Question 6. Check the internet for the information about what was the life expectancy in 1760 in the country where Bernoulli lived. How would the inoculation program change it?

As a result of this study,Bernoullisupported the inoculation program.