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Table 1 Elements of the 2-state, 3-duration transition probability matrix

From: Recognizing duration effects in multistate population models

The 2-state, 3-duration transition probability matrix of Eq(1) is
\( \mathbf{A}=\left[\begin{array}{cccccc}{\uppi}_{10,10}& {\uppi}_{11,10}& {\uppi}_{12,10}& {\uppi}_{20,10}& {\uppi}_{21,10}& {\uppi}_{22,10}\\ {}{\uppi}_{10,11}& 0& 0& 0& 0& 0\\ {}0& {\uppi}_{11,12}& {\uppi}_{12,12}& 0& 0& 0\\ {}{\uppi}_{10,20}& {\uppi}_{11,20}& {\uppi}_{12,20}& {\uppi}_{20,20}& {\uppi}_{21,20}& {\uppi}_{22,20}\\ {}0& 0& 0& {\uppi}_{20,21}& 0& 0\\ {}0& 0& 0& 0& {\uppi}_{21,22}& {\uppi}_{22,22}\end{array}\right] \) (A.1)
To conveniently express the elements of A in terms of the underlying occurrence / exposure rates of interstate transfer, separate matrices, A13 and A46, are shown below for the first 3 and the second 3 columns of A. We then have
\( {\mathbf{A}}_{\mathbf{13}}=\left[\begin{array}{ccc}2\ {\mathrm{n}}^2{\mathrm{m}}_{10,20}{\mathrm{m}}_{20,10}/\left({\mathrm{D}}_0{\mathrm{E}}_0\right)& 2\ {\mathrm{n}}^2{\mathrm{m}}_{11,20}{\mathrm{m}}_{20,10}/\left({\mathrm{D}}_1{\mathrm{E}}_1\right)& 2\ {\mathrm{n}}^2{\mathrm{m}}_{12,20}{\mathrm{m}}_{20,10}/\left({\mathrm{D}}_2{\mathrm{E}}_2\right)\\ {}\left(2-\mathrm{n}\ {\mathrm{m}}_{10,20}\right)/\left({\mathrm{D}}_0\right)& 0& 0\\ {}0& \left(2-\mathrm{n}\ {\mathrm{m}}_{11,20}\right)/\left({\mathrm{D}}_1\right)& \left(2-\mathrm{n}\ {\mathrm{m}}_{12,20}\right)//\left({\mathrm{D}}_2\right)\\ {}2\ \mathrm{n}\ {\mathrm{m}}_{10,20}/\left({\mathrm{E}}_0\right)& 2\ \mathrm{n}\ {\mathrm{m}}_{11,20}/\left({\mathrm{E}}_1\right)& 2\ \mathrm{n}\ {\mathrm{m}}_{12,20}/\left({\mathrm{E}}_2\right)\\ {}0& 0& 0\\ {}0& 0& 0\end{array}\right] \) (A.2a)
where for j = 0, 1, and 2, Dj = 2 + n m1j,20 and Ej = 2 + n m1j,20 + n m20,10; and
\( {\mathbf{A}}_{\mathbf{46}}=\left[\begin{array}{ccc}2\ \mathrm{n}\ {\mathrm{m}}_{20,10}/\left({\mathrm{G}}_0\right)& 2\ \mathrm{n}\ {\mathrm{m}}_{21,10}/\left({\mathrm{G}}_1\right)& 2\ \mathrm{n}\ {\mathrm{m}}_{22,10}/\left({\mathrm{G}}_2\right)\\ {}0& 0& 0\\ {}0& 0& 0\\ {}2\ {\mathrm{n}}^2{\mathrm{m}}_{10,20}{\mathrm{m}}_{20,10}/\left({\mathrm{F}}_0{\mathrm{G}}_0\right)& 2{\mathrm{n}}^2{\mathrm{m}}_{10,20}{\mathrm{m}}_{21,10}/\left({\mathrm{F}}_1{\mathrm{G}}_1\right)& 2{\mathrm{n}}^2{\mathrm{m}}_{10,20}{\mathrm{m}}_{22,10}/\left({\mathrm{F}}_2{\mathrm{G}}_2\right)\\ {}\left(2-\mathrm{n}\ {\mathrm{m}}_{20,10}\right)/\left({\mathrm{F}}_0\right)& 0& 0\\ {}0& \left(2-\mathrm{n}\ {\mathrm{m}}_{21,10}\right)/\left({\mathrm{F}}_1\right)& \left(2-\mathrm{n}\ {\mathrm{m}}_{22,10}\right)/\left({\mathrm{F}}_2\right)\end{array}\right] \) (A.2b)
where for j = 0, 1, and 2, Fj = 2 + n m2j,10 and Gj = 2 + n m2j,10 + n m10,20.