Occupancy Probabilities for Long-Term Care States

Why Occupancy Probabilities Matter

A long-term care model can describe more than whether a person has entered care. At a specified future time, the person may be healthy, receiving care, recovered, or dead. Occupancy probabilities give the probability of being in each of these states at that time. They provide a direct way to summarize a multi-state model over one or more periods.

This tutorial uses transition probability matrices: the entries are probabilities of moving from one state to another over a specified period. The calculations rely on the Markov assumption that the next period’s state probabilities depend on the current state and the transition probabilities for that period, rather than on the earlier path used to reach the current state. The matrices below are hypothetical. No interest rate is needed because the quantities being calculated are probabilities, not present values.

Moving an Occupancy Distribution Forward

Use a row vector for the current occupancy probabilities and multiply it on the left by the transition matrix. If the person is known to start in state \(H\), the initial vector is \((1,0,0,0)\). If the person starts in a mixture of states, the initial vector records the probability of starting in each one.

This formula follows from the law of total probability. For any destination state \(j\), the probability of being in \(j\) next period is the sum, over all possible current states \(i\), of the probability of being in \(i\) now multiplied by the probability of moving from \(i\) to \(j\). Thus \(\pi_{t+1,j}=\sum_i \pi_{t,i}p_{ij}\), which is exactly the \(j\)th entry of the row-vector product \(\boldsymbol{\pi}_tP\).

For a time-homogeneous model, the same matrix applies in every period. Repeating the one-period calculation gives \(\boldsymbol{\pi}_n=\boldsymbol{\pi}_0P^n\). If transition probabilities change by period, use the matrix for each period in chronological order: \(\boldsymbol{\pi}_n=\boldsymbol{\pi}_0P_1P_2\cdots P_n\). The matrices generally cannot be rearranged.

An absorbing death state is represented by a death row with probability 1 of remaining in death and probability 0 of moving elsewhere. An absorbing row is appropriate when death is permanent and the model includes no transitions out of death. The other states need not be absorbing: for example, a recovery state may allow relapse to care if the model specifies that transition.

Starting from a Distribution Across States

A policy cohort need not begin in one known state. For example, issue records or a model specification may assign probabilities to several possible initial states. The same matrix multiplication applies: each starting-state probability weights that state’s transition row. After propagation, probabilities of sets of states can be found by adding the relevant occupancy probabilities. The sum is appropriate when the states are mutually exclusive, as they are in a state model.

When Transition Probabilities Change by Period

A model may specify different transition matrices for successive periods, for example because transition probabilities depend on age or duration. The current occupancy vector must first be advanced using the first period’s matrix, then the resulting vector using the next period’s matrix. Each matrix’s rows and columns must use the same state ordering.

Checks and Common Interpretation Traps

An occupancy calculation is only as meaningful as the period and state definitions behind its matrix. Before multiplying, confirm that each matrix entry is a probability over the intended interval and that rows represent current states while columns represent destination states. Transition intensities are rates, not probabilities; an intensity matrix cannot be substituted directly for \(P\). Converting intensities into transition probabilities requires a separate model calculation.

• Check each row. A transition probability row must sum to 1. A row that does not usually signals an omitted destination or an arithmetic or transcription error.
• Check the occupancy vector. Its entries must be nonnegative and sum to 1, subject to rounding. The death entry should not decrease in a model with absorbing death.
• Keep state order fixed. If rows use \(H,C,R,D\), the initial vector and columns must use \(H,C,R,D\) too.
• Distinguish occupancy from entry. The probability of being in care at a given time is not necessarily the probability of ever having entered care. A person may have entered and then recovered, so occupancy at the endpoint does not record every prior transition.
• Combine events only after propagation. To find the probability of being in either care or recovery at the endpoint, add those endpoint probabilities; do not add transition probabilities from different starting states without weighting them by current occupancy.