32. A village of N + 1 people suffers an epidemic. Let X(t) be the number of ill people at time t, and suppose that X (0) = 1 and X is a birth process with rates λ = λi (N + 1 - i). Let T be the length of time required until every member of the population has succumbed to the illness. Show that N and deduce that E(T) = k=1 1 k(N + 1-k) E(T) = 2(log N + y) λ(N+1) +O(N-²) where y is Euler's constant. It is striking that E(T) decreases with N, for large N.

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