By Finkenstadt B. F.
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Extra info for A stochastic model for extinction and recurrence of epidemics estimation and inference for measles o
5) as pm+1 = k rk + k πm+1 sk [1 + = k sk pm (1−πm )(1−pm πm )sk −1 1−(1−pm πm )sk rk + (1−pm πm )sk 1−(1−pm πm )sk ] k ak sk pm (1−πm )(1−pm πm )sk −1 1−(1−pm πm )sk , where all sums extend over the ascertained families alone. 3. 3 oﬀer an opportunity to apply the EM algorithm. In this table the column labeled “Families n” refers to the number of families showing a particular conﬁguration of aﬀected and ascertained siblings. For these data the maximum 2. 2679 is consistent with the theoretical value of p = 1/4 for an autosomal recessive.
4) is dp dt = pq[s − (r + s)p] , ¯ w where q = 1 − p and w ¯ = 1 − rp2 − sq 2 . If we take this approximation seriously, then n n ≈ pn dt = 0 p0 ¯ w dp. pq[s − (r + s)p] Show that this leads to n ≈ 1 − pn 1 pn 1 − 1 ln − 1 ln + s p0 r 1 − p0 1 1 |s − (r + s)pn | + − 1 ln − r s |s − (r + s)p0 | when pn and p0 are both on the same side of the internal equilibrium point and neither r nor s is 0. Derive a similar approximation when s = 0 or r = 0. Why is necessary to postulate that pn and p0 be on the same side of the internal equilibrium point?
2083 . (Hints: Split the ﬁrst category into two so that there are ﬁve categories for the complete data. ) 7. In an inbred population, the inbreeding coeﬃcient f is the probability that two genes of a random person at some locus are both copies of the same ancestral gene. Assume that there are k codominant alleles and that pi is the frequency of allele Ai . Show that f pi + (1 − f )p2i 2. Counting Methods and the EM Algorithm 35 is the frequency of a homozygous genotype Ai /Ai and (1 − f )2pi pj is the frequency of a heterozygous genotype Ai /Aj .
A stochastic model for extinction and recurrence of epidemics estimation and inference for measles o by Finkenstadt B. F.