Page 148 - 20dynamics of cancer

THEORY II 133

9 (a) 97.5–100
95.0–97.5
7
Acceleration 5 00.0–90.0
92.5–95.0
90.0–92.5
3
1
–1
0 (b)
Fraction affected –2
–1

–3

–4
9
Acceleration 5
7

3

1 (c)
30 40 50 60 70 80
Age

Figure 7.9 Explanation of the drop in aggregate acceleration caused by popu-
lation heterogeneity. Each panel shows patterns for different segments of the
population stratified by transition value, u. The legend in (a) shows that each of
the first four strata comprise 2.5% of the population, that is, the top 2.5% of u
values, the next 2.5%, and so on. The fifth stratum includes the rest of the pop-
ulation, with individuals that have u values that fall between the 0th and 90th
percentiles. All panels have parameters that match the curve labeled s = 0.6
in Figure 7.8. (a) The contribution of each stratum to the aggregate LLA of the
population. I calculated each curve from Eq. (7.8), with denominators integrated
over all values of u, and numerators integrated over values of u within each stra-
tum and divided by the total probability contained in the stratum. Total LLA
equals the sum of the curves. (b) Fraction of individuals within each stratum
who suffer cancer by age 80, calculated as 1 − b in Eq. (7.7), integrated over u
values within each stratum, and divided by the total probability contained in
the stratum. (c) LLA calculated within each stratum by integrating numerators
and denominators in Eq. (7.8) over values of u within each stratum.