In the two-locus heterogeneity model with X = Y + Z − Y Z, carry through the computations retaining the product term Y Z. In particular, let Km be the prevalence of the mth form of the disease, and let KmR be the recurrence risk for a relative of type R under the mth form. If K is the prevalence and KR is the recurrence risk to a relative of type R under either form of the disease, then show that K = K1 + K2 − K1K2 KKR = K1K1R + K1K2 − K1K1RK2 + K1K2 + K2K2R

− K1K2K2R − K1K1RK2 − K1K2K2R + K1K1RK2K2R.

Assuming that K1, K2, K1R, and K2R are relatively small, verify the approximation

λR − 1

=

K1 K

2

(λ1R − 1) + K2 K

2

(λ2R − 1)

+

K1K2 K2 [2K1 + 2K2 − K1K2 − 2K1R − 2K2R + K1RK2R]

≈

K1 K

2

(λ1R − 1) + K2 K

2

(λ2R − 1), where λmR = KmR/Km and λR = KR/K.