Building a Long-Term Care State Model

Why Define the States Before the Rates?

A long-term care model needs to represent more than whether a person is alive or dead. For example, a hypothetical contract may pay a benefit while a covered person needs care, but not while that person is healthy. The model must therefore distinguish health status as well as death. Its states and permitted transitions describe what changes are possible; the intensities quantify how quickly those changes occur.

The central construction is a state list followed by a transition list. Only after both are clear should rates be placed into a matrix. This prevents a common modeling error: putting a plausible-looking rate in the wrong position or allowing a change of status that the assumptions exclude. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

These state descriptions are modeling assumptions, not universal descriptions of long-term care. “Care-needing” must be defined in a way that fits the hypothetical contract or population being modeled; for instance, it might mean meeting a specified care-eligibility criterion. The model also assumes that health and care-needing statuses do not overlap. If recovery is possible and relevant, it must be included as an allowed transition rather than silently treated as impossible.

From Permitted Transitions to the Intensity Matrix

Use the state order \(H,C,D\), with rows representing the state currently occupied and columns representing the destination state. Write \(q_{ij}(t)\) for the transition intensity from state \(i\) to a different state \(j\), at time \(t\). An intensity is a rate per unit time; the time unit must be stated. For example, an intensity expressed per year must be paired with time measured in years.

For each allowed transition, put its intensity in the corresponding off-diagonal position. Put zero in an off-diagonal position for a transition forbidden throughout the model. The diagonal entry in each row is the negative of the sum of the intensities of all permitted exits from that state. The resulting matrix \(Q(t)\) is called an intensity matrix or generator matrix.

The first row has two possible exits from health: entering care-needing status or dying. The second row has only one permitted exit, death. The third row has none. Consequently, the diagonal is negative in the two transient states and zero in the absorbing state. Every row sums to zero, which provides a quick arithmetic check on the construction.

Why the Diagonal Is the Negative Total Exit Intensity

The diagonal entries are not additional transitions. They are determined by the off-diagonal intensities. To see why, consider a short interval of length \(h\) in a state with outgoing intensities \(q_{ij}(t)\). Under the usual continuous-time intensity model, the probability of a particular permitted exit \(i\to j\) during that interval is \(q_{ij}(t)h+o(h)\), where \(o(h)\) denotes a term that becomes negligible compared with \(h\) as \(h\) tends to zero. The chance of remaining in state \(i\) must make the total probability one.

Adding the first-order probabilities of all permitted exits gives \(\bigl(\sum_{j\ne i}q_{ij}(t)\bigr)h+o(h)\). Thus the probability of remaining in \(i\) is \(1-\bigl(\sum_{j\ne i}q_{ij}(t)\bigr)h+o(h)\). The instantaneous rate of change of that stay probability at \(h=0\) is the negative sum of the outgoing intensities. This gives the diagonal rule.

This construction assumes finite, nonnegative transition intensities and the stated state structure. If a model allows a transition not included in the simple long-term care structure, the permitted-transition list and matrix must be revised before applying the diagonal rule.

Worked Matrix Constructions

Modeling Choices and Common Traps

The state labels alone do not determine the transition structure. In the simple no-recovery model, the allowed routes out of health are entering care-needing status and death, while the only allowed route out of care-needing status is death. A model that allows recovery has a different structure. Similarly, if the contract or population distinguishes different care levels, those levels may require separate states; combining them into one state is appropriate only if the distinction is unnecessary for the quantities being modeled.

When a rate depends on time, write the time argument explicitly. For example, \(q_{HC}(t)\) indicates that the intensity may change with calendar time \(t\). A matrix with constant entries describes a time-homogeneous intensity model; a matrix whose entries vary with \(t\) describes a time-inhomogeneous model. In either case, diagonal entries must use the rates at the same time as the off-diagonal entries.

A frequent exam trap is to interpret an observed zero as proof that a transition is forbidden. That conclusion is valid only when the model says the intensity is zero throughout the relevant interval. In a time-dependent model, an allowed rate can be zero at a particular time and positive at other times. The structural transition list, not a single matrix value, determines whether a route is forbidden.

• Check the state order. State the order once and use it consistently for both rows and columns.
• Check the direction. A rate for recovery belongs in row \(C\), column \(H\), not row \(H\), column \(C\).
• Check total exits. The magnitude of each diagonal entry is the sum of all outgoing rates from that state.
• Check units and time arguments. Rates per year require time in years; time-varying rates should be evaluated at the stated time.
• Check the assumptions. Recovery and other routes appear only if the model permits them; death is absorbing only under the stated model assumption.