The four-parameter panel-observed multi-state likelihood requires the following additional assumptions.

Irreversibility: the chronic disease can progress but cannot improve, so $q_{21} = q_{32} = 0$ and state 3 is absorbing. Merely permitting transitions between adjacent states does not impose this; allowing regression would require additional transition intensities.
Piecewise-constant age: the current age is held at 50 and 55 over the respective observation intervals. Actual age changes continuously, and a five-year interval can be too long for this approximation when the age effect is substantial.
Log-linear age effects and common rates: the specified age functions apply to the patient, without unmodelled heterogeneity or other changing covariates.
Observation mechanism: disease states are recorded without misclassification, and examination times or dropout carry no additional information about the underlying process once the modelled history is given. A new visit only one year after a five-year gap might have been prompted by worsening symptoms, making the observation schedule informative.

If age is instead updated continuously, the model is time-inhomogeneous and the displayed homogeneous transition formula is not exact. For example, set

Λ (u, v) = \int_{u}^{v} q_{12} (A) d A, M (u, v) = \int_{u}^{v} q_{23} (A) d A .

(1)

Then the exact irreversible likelihood contribution is

L = e^{- Λ (50, 55)} [1 - e^{- Λ (55, 56)} - \int_{55}^{56} e^{- Λ (55, u)} q_{12} (u) e^{- M (u, 56)} d u] .

(2)

Thus the simple closed form uses an explicit frozen-age approximation; adjacency alone and continuously evolving age do not determine that form.

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 Incoming links: Chronic disease