## O

{ tq(t)/St° Q(y)dy if t < to f (t\a(to) =0) = q(t)(l - f)/ f 0 (1 - 1 )q(y)dy, t < to.

t00t0

Consequently, if to ^tt lim qf(t\to) = lim qb(t\to) = Q(t)/m t°—t°—

4 Early Detection Disease Model

In this section, the results of the chronic disease model will be adapted to the early detection model. The states for this model are So, Sp and Sc having the natural history So ^ Sp ^ Sc. Define the probability of the transition So ^ Sp during (t, t + dr) by w(t)dr where t refers to age. The point incidence refers to the transition Sp ^ Sc. The relation between w(z) and I(z) is

Jo where q(t) is the pdf of the sojourn time in the pre-clinical state. Assume that at time t0 the disease is detected early and the age at which the disease is detected is z. Then the expression for the forward and backward recurrence time distribution are

Equation (23) enables the calculation of the pdf's of the forward and backward recurrence time distribution. Note that aside from definitions the only change is the substitution of w(t) for I(r) in (4) and (6).

Similarly, the distribution of the time in the pre-clinical state T = Tb + Tf q(t) f0z w(t)dr/P(z) if t> z f (t\z) = { (24)

Jt Jo

Jt Jo