Consider a 3-state system with one absorbing state, such as Healthy, Sick, and Dead. If the system satisfies the 1-step Markov conditions, the prevalence of the Healthy state will converge to a value that is independent of the initial distribution. This equilibrium prevalence and its variance are known under the assumption of time homogeneity, and provided reasonable estimates in the time non-homogeneous systems studied. Here, we derived the equilibrium prevalence for a system with more than three states. Under time homogeneity, the equilibrium prevalence distribution was shown to be an eigenvector of a partition of the matrix of transition probabilities. The eigenvector worked well for time non-homogeneous examples as well. We developed a test for whether the available sample was at equilibrium, and used it to explore whether there was selection bias in the baseline distribution of a large longitudinal cohort sample.
Diehr, Paula and Yanez, David, "Multi-state Life Tables, Equilibrium Prevalence, and Baseline Selection Bias" (June 2010). UW Biostatistics Working Paper Series. Working Paper 365.