For the Bernoulli model of a community with endemic smallpox (i.e. the model in the lecture notes) find the life expectancy at birth in terms of the parameters , and . (For this question assume to be a constant.) Bernoullis intention was to determine whether life expectancy was improved by the use of variolation, that is, deliberate infection of susceptibles with a mild strain of smallpox to confer immunity. Calculate the life expectancy at birth of a cohort if variolation is practised at a young age (assume at t = 0) and assuming that the practice is fatal in a fraction of cases and that the time period between infection by smallpox and either death or recovery is short compared with the time scale of the model. Show that life expectancy of the cohort is increased by variolation if

What is the critical value of , above which variolation decreases life expectancy, if = = 1/ 8 and = 1/ 25 ? (In his original paper, Bernoulli used a value of = 1 200 , which was criticised by DAlembert as being far too low.)

Bernoullis model follows the proportion surviving of a cohort of N0 individuals born in a particular year. Let the total population surviving to time t be N(t). This population is divided into two groups. The first group of size S(t), called susceptibles, are people who have not contracted the disease. The remaining group of size N(t) S(t) have contracted the disease and survived, and are immune to further infection. N(t) and S(t) are assumed

to be sufficiently large (at least in the thousands) that they can realistically be thought of as continuous functions of time. Let the background death rate of the cohort from causes other than smallpox be (t), that is, the backgound probability of dying in the time interval t to t + dt, conditional on being alive at time t, is (t)dt. We further assume a fixed rate of infection of smallpox 8, that is, the probability of a susceptible person being infected with smallpox in the intervalt to t+dt is Bdt. We further assume that the infection is fatal in a fraction a of cases, where 0