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Table 2 Expressions for 6 rates and 12 probabilities in a 2-state, 3-duration semi-Markov model in terms of the other 6 probabilities.

From: Recognizing duration effects in multistate population models

In a model like that of Eq. (1), expressing the 6 underlying rates and 12 probabilities in terms of the probabilities πm0v0, πm1m2, πm2v0, πv0m0, πv1m0, and πv2v2, yields
mm0v0 = 2 πm0v0/[n (2 − πm0v0 – πv0m0)]
mm1v0 = 2 (1 − πm1m2 )/[n (1 + πm1m2)]
mm2v0 = 2 πm2v0 (2 – πm0v0 )/[n (2 − πm0v0 – πv0m0) (2 – πm2v0)]
mv0m0 = 2 πv0m0/[n (2 − πm0v0 – πv0m0)]
mv1m0 = 2 πv1m0 (2 – πv0m0)/[n (2 − πm0v0 – πv0m0 ) (2 – πv1m0)]
mv2m0 = 2 (1 – πv2v2 )/[n(1 + πv2v2)]
πm0m0 = πm0v0 πv0m0/(2 − πv0m0)]
πm0m1 = (2 − 2 πm0v0 − πv0m0)/(2 – πv0m0)
πm1m0 = πv0m0 (1 – [πm1m2 ]2)/[(4 – 2 πm0v0 – πv0m0 + πv0m0 πm1m2)]
πm1v0 = 2(2 − 2 πm1m2 − πm0v0 − πv0m0 + πv0m0 πm1m2 + πm0v0 πm1m2)/
[(4 – 2 πm0v0 – πv0m0 + πv0m0 πm1m2)]
πm2m0 = πv0m0 πm2v0 (2 – πm2v0)/[(4 – 2 πm0v0 – 2 πv0m0 + πm2v0 πv0m0)]
πm2m2 = (4 − 4 πm2v0 − 2 πm0v0 − 2 πv0m0 + 2 πm2v0 πm0v0 + πm2v0 πv0m0)/
[(4 – 2 πm0v0 – 2 πv0m0 + πv0m0 πm2v0)]
πv0v0 = πm0v0 πv0m0/(2 – πm0v0)]
πv0v1 = (2 − 2 πv0m0 − πm0v0)/(2 – πm0v0)
πv1v0 = πv1m0 πm0v0 (2 – πv1m0)/[(4 – 2 πm0v0 – 2 πv0m0 + πm0v0 πv1m0)]
πv1v2 = (4 − 4 πv1m0 − 2 πm0v0 − 2 πv0m0 + 2 πv1m0 πv0m0 + πv1m0 πm0v0)/
[(4 – 2 πm0v0 – 2 πv0m0 + πv1m0 πm0v0)]
πv2m0 = 2 (2 − 2 πv2v2 − πm0v0 − πv0m0 + πv2v2 πv0m0 + πv2v2 πm0v0)/
[(4 – 2 πv0m0 – πm0v0 + πv2v2 πm0v0)]
πv2v0 = πm0v0 (1 – [πv2v2 ]2)/[(4 – 2 πv0m0 – πm0v0 + πm0v0 πv2v2)]