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By Tan Wai-Yuan

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As an illustration, consider the genotype -£g at generation t. ) \q, |p}> Q = 1 — Pi respectively. Under self-fertilization with no selection, this gives the frequencies of the above ten genotypes at generation t + 1 as: AB AB 1 jP Ab Ab 9 aB aB ab ab AB Ab AB aB Ab ab aB ab AB ab aB Ab 1 2 1 2 1 2 1 -A1 -q -p -pq 1 -pq 1 -pq 1 -pq 1 2 -p 12 -q . The average fitness is T P 2 ( Z I + x2) + ^q2{xi + y2) + ^Q2(yi + x2) + J P 2 G / I + 2/2) + ^ ( z i + 1) + 2 ^ ( 1 + x2) + ^pq(l + 2/2) + ^ ( l + 2/i) 1 9 1 C5 9 + V+V =7Hence the frequencies of the ten genotypes at generation t + 1 under selection are given by: AB AB Ab Ab aB aB —p2(xi + x2) —q2(xi + y2) C5 C5 —q2(yi + x2) C5 Ab ab AB aB ab ab —2pq(l + x2) -2pq(l C5 C5 aB ab + y2) -2pq(l C5 + yx) 1 C5 J AB Ab °2(yi + 2/2) —2pq(xi C5 AB ab aB Ab 4 2 —P & C5 This gives the elements of the last row of P above.

Oo} and / n ( s ) the pgf of the above non-homogeneous Galton-Watson branching process. Then, it is obvious that fn(s) = fn1Hs) Ms) = fW[fi1}1(s)}, = fn2ltl[f£\s)}, if n < t l 5 if n > t x . 3. for »> x• The Structure and Decomposition of Markov Chains Consider a Markov chain {X(t), t e T } with parameter space T = ( 0 , 1 , . . , oo) and with state space S = { 0 , 1 , . . , }. In this section we will illustrate the basic structure and decomposition of these chains. For simplicity of illustration, we will restrict ourselves to homogeneous Markov chains, unless otherwise stated, although many results also hold in non-homogeneous chains.

J. Epidemiology 137 (1993) 1229-1240. I. M. Longini, W. S. Clark, G. A. Satten, R. H. Byers and J. M. Karon, Staged Markov models based on CD^+' T-lymphocytes for the natural history of HIV infection, in Models for Infectious Human Diseases:. Their Structure and Relation to Data, eds. V. Isham and G. Medley, pp. 439-459, Cambridge University Press (1996). G. Satten and I. M. Longini, Estimation of incidence of HIV infection using cross-sectional marker survey, Biometrics 50 (1994) 675-688. G. Satten and I.

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