Designing the Long-Term Care Transition Graph

Why the Transition Graph Comes First

A long-term care model is easier to interpret when its states and permitted movements are made explicit before rates or benefit values are assigned. The graph records which changes of state the model allows; it does not, by itself, say how likely those changes are or what the contract pays in each state. Those are separate pieces of the model.

For this tutorial, consider four broad states: healthy, care-needing, recovery, and death. They are placeholders, not universal clinical categories. In a particular contract, the definitions must be consistent with the policy’s eligibility rules and with the information the model needs to track. For example, “care-needing” might mean that the insured meets a specified benefit trigger, rather than simply having a particular diagnosis.

Definition: A transition graph is a directed graph whose nodes represent the states of a model and whose directed edges represent the state changes permitted by that model. An edge \(i\to j\) represents a possible direct move from state \(i\) to state \(j\). It does not represent a move that must pass through an intermediate state.

Graph design starts with the state definitions, the contract, and the stated model assumptions. It is not determined by intuition alone. If the model says recovery is a distinct state between care-needing and healthy, a direct care-needing-to-healthy edge is excluded unless the model also explicitly allows a direct transition that bypasses recovery.

State Definitions and Edge Decisions

Suppose a hypothetical policy model uses these definitions:

Healthy (H): the insured is alive and does not currently meet the model’s long-term care eligibility definition.
Care-needing (C): the insured is alive and currently meets that definition.
Recovery (R): the insured has left the care-needing state but is represented separately while recovering or being monitored before return to the healthy state.
Death (D): the insured is dead.

The labels alone do not determine all the edges. A model must also specify what kinds of changes are allowed. For instance, it might permit a relapse from recovery to care-needing, or it might assume that relapse is excluded. It might allow death from every living state, or make a simplifying assumption that death can occur only from some states. Either choice changes the graph and should be stated.

A directed edge is about a direct transition. If \(C\to R\) and \(R\to H\) are permitted, then \(C\) can reach \(H\) by a two-transition path. That does not imply that \(C\to H\) is itself an edge. Keeping paths separate from direct edges prevents accidental inclusion of transitions that the state definitions or assumptions rule out.

Modeling rule: Include an edge when the stated contract and model assumptions permit that direct state change. Exclude it when the contract or assumptions rule it out. A structural edge records permission; an intensity describes the modeled transition rate and may depend on time or other conditions.

Death is commonly modeled as an absorbing state: once the process enters death, it remains there, so there are no edges from \(D\) to a living state. Remaining in a state during a period is not usually drawn as a self-loop in a continuous-time transition graph; it is represented by the process not making a transition during that period.

A Baseline Graph and Its Audit

Worked Example: Drawing a Graph with an Intermediate Recovery State

A hypothetical policy model uses \(H,C,R,D\) as defined above. Its assumptions are: a healthy insured may become care-needing or die; a care-needing insured may recover or die; a recovering insured may return to healthy, relapse into care-needing, or die; and death is absorbing. A direct move from care-needing to healthy is excluded because recovery must be represented as a distinct intervening state. Construct the graph, its adjacency matrix, and the number of directed edges.

Method. List permitted direct destinations for each origin state, then encode each allowed edge as 1 and each excluded edge as 0. Use the state order \(H,C,R,D\) for both rows and columns.

From \(H\), the permitted destinations are \(C\) and \(D\), giving row \((0,1,0,1)\). From \(C\), they are \(R\) and \(D\), giving \((0,0,1,1)\). From \(R\), they are \(H,C,D\), giving \((1,1,0,1)\). From \(D\), there are no permitted destinations, giving \((0,0,0,0)\).

\begin{array}{c|cccc} & H & C & R & D\\ \hline H & 0 & 1 & 0 & 1\\ C & 0 & 0 & 1 & 1\\ R & 1 & 1 & 0 & 1\\ D & 0 & 0 & 0 & 0 \end{array}

The edge count is the sum of the entries: \(2+2+3+0=7\). The direct edge \(C\to H\) is absent, even though the path \(C\to R\to H\) is available. The row for \(D\) is all zeros because death is absorbing.

Check. The listed edges are \(H\to C\), \(H\to D\), \(C\to R\), \(C\to D\), \(R\to H\), \(R\to C\), and \(R\to D\): seven in total. A common mistake is to add \(C\to H\) merely because a person can eventually return to healthy. That would count an indirect path as a direct transition and contradict the stated intermediate-state assumption.

The adjacency matrix is a compact audit tool, not a substitute for explaining the assumptions. In particular, a zero can mean that a direct transition is structurally excluded. It should not be confused with a transition that is permitted but has an intensity of zero at a particular time.

Contract Rules and Health Assumptions Are Different

A contract helps determine which state distinctions matter for benefits. For example, if benefits are payable only while eligibility conditions are met, the model must be able to distinguish eligible care-needing periods from noneligible periods. The contract’s payment rule does not automatically determine whether recovery is a separate state or whether relapse is possible. Those are modeling choices about the process, which must be made explicit.

A useful edge-by-edge test is to ask three questions:

Are the origin and destination distinct states?
Confirm that the proposed move changes the condition represented by the model, rather than merely describing a change that is irrelevant to its state definitions.

Does the model permit this direct change?
Check the contract-related rules and health assumptions. If the change must pass through an explicitly represented intermediate state, do not add a bypass edge unless it is separately allowed.

Is the destination consistent with absorbing states and exclusions?
For example, if death is absorbing, no edge can lead from death back to a living state.

These checks also expose ambiguity. If it is unclear whether “recovery” means no longer eligible or fully returned to the healthy condition, the state definitions need clarification before the graph can be finalized. A graph that is internally consistent but uses unclear states will not reliably represent the contract or the health process.

Structural Permission Versus Transition Intensity

In a continuous-time model, an intensity describes the instantaneous rate of a transition under the model’s conditions. It is not the same thing as an edge definition. A model may specify a permitted edge and then assign an intensity that varies with time. To infer that a transition has positive probability over an interval, a rate cannot merely be positive at one isolated instant: it must contribute positive integrated rate while the process is in the origin state.

The distinction follows from how transition probability accumulates. In a simple competing-risks setting starting in state \(i\), let \(q_{ij}(t)\) be the intensity of transition \(i\to j\), and let \(q_i(t)\) be the sum of intensities of all exits from \(i\). Under the usual nonnegative, locally integrable intensity assumptions, the probability that the first exit by time \(T\) is to \(j\) is

\int_0^T \exp\!\left(-\int_0^t q_i(s)\,ds\right)q_{ij}(t)\,dt.

The exponential factor is the probability of still being in \(i\) immediately before time \(t\); multiplying by \(q_{ij}(t)\,dt\) gives the probability of exiting to \(j\) in that small time interval. Integrating adds the mutually exclusive possibilities over time. If \(q_{ij}\) is positive only at a single instant and is zero elsewhere, its integral over time is zero, so it contributes no transition probability under this model. By contrast, an intensity positive over an interval of positive length can contribute positive probability, provided the process can still be in \(i\).

Important distinction: A graph may record a structurally permitted edge even when its modeled intensity is zero over a particular period. Conversely, a rate value that is positive at just one isolated instant does not establish positive transition probability. State whether a graph represents structural permissions or only transitions with positive probability under a specified intensity model.

Worked Example: Auditing a Time-Varying Edge

In a hypothetical model, an insured starts in \(H\). Over the first half-year, the intensities are constant at \(q_{HC}=0.2\) per year and \(q_{HD}=0.1\) per year. After the first half-year, both are zero except that \(q_{HC}(t)=0.4\) per year at the single instant \(t=1\). For the first-exit calculation, assume these are the only exits from \(H\). Determine whether \(H\to C\) belongs in the graph and calculate the probability that the first transition is \(H\to C\) by time \(0.5\).

Method. Treat \(H\to C\) as structurally permitted because its intensity is positive throughout the first half-year. Then use the competing-exit formula on that interval. Do not infer positive probability from the isolated value at \(t=1\).

During the first half-year, the total exit intensity is \(q_H=0.2+0.1=0.3\) per year. The probability that the first exit is to \(C\) by \(T=0.5\) is

\begin{aligned} P(\text{first exit to }C\text{ by }0.5) &=\int_0^{0.5} e^{-0.3t}(0.2)\,dt\\ &=\frac{0.2}{0.3}\left(1-e^{-0.3(0.5)}\right)\\ &=\frac{2}{3}\left(1-e^{-0.15}\right)\\ &=\frac{2}{3}(1-0.86070798)\\ &=0.09286135. \end{aligned}

Rounded to four decimals, the probability is \(0.0929\). The edge \(H\to C\) is included because the model permits it over a time interval with positive length, and it has positive first-exit probability by 0.5. The isolated rate value at \(t=1\) contributes no additional probability: a single point has zero duration in the integral.

Check. The probability of any exit by 0.5 is \(1-e^{-0.15}=0.13929202\), and the probability of an exit to \(C\), \(0.09286135\), is smaller. The common error is to say the rate at \(t=1\) alone creates a possible transition with positive probability. The correction is to assess integrated rate over time during which the process can be in the origin state.

When an Assumption Changes the Graph

Worked Example: Excluding Relapse Under a Contract Model

A second hypothetical model uses the same four states but assumes: healthy insureds may enter care-needing or die; care-needing insureds may recover or die; recovering insureds may return to healthy or die; relapse from recovery to care-needing is excluded; and death is absorbing. Construct the adjacency matrix, count its edges, and determine whether \(C\) can eventually reach \(H\).

Method. Apply the stated direct-transition rules separately to each origin state. In the state order \(H,C,R,D\), the rows are \(H:(0,1,0,1)\), \(C:(0,0,1,1)\), \(R:(1,0,0,1)\), and \(D:(0,0,0,0)\). Thus

\begin{array}{c|cccc} & H & C & R & D\\ \hline H & 0 & 1 & 0 & 1\\ C & 0 & 0 & 1 & 1\\ R & 1 & 0 & 0 & 1\\ D & 0 & 0 & 0 & 0 \end{array}

The edge count is \(2+2+2+0=6\). The path \(C\to R\to H\) exists, so \(H\) is eventually reachable from \(C\), even though the direct edge \(C\to H\) is absent. The edge \(R\to C\) is also absent, because relapse is explicitly excluded.

Check. The six edges are \(H\to C\), \(H\to D\), \(C\to R\), \(C\to D\), \(R\to H\), and \(R\to D\). A common error is to include \(R\to C\) simply because relapse is clinically conceivable. The graph represents this model’s stated assumptions, not every conceivable health outcome.

Comparing this matrix with the baseline graph shows the practical effect of one assumption: excluding relapse removes exactly one edge. It does not remove the path from care-needing to healthy, nor does it change the absorbing status of death. When a question asks which transitions are allowed, answer from its specified assumptions rather than silently substituting a different model.

Common Graph-Design Traps

Confusing reachability with a direct edge. A path \(C\to R\to H\) does not imply a direct edge \(C\to H\).
Adding every clinically imaginable transition. Include only transitions permitted by the state definitions and assumptions of the model being used.
Treating benefit eligibility as the whole health model. Contract rules help identify relevant states, but do not by themselves settle recovery, relapse, or intermediate-state assumptions.
Interpreting a single positive intensity value as positive probability. For a time-varying intensity, the rate must contribute over a set of positive duration while the process can occupy the origin state.
Drawing self-loops for remaining in a state. In a continuous-time transition graph, remaining in a state is not ordinarily a separate transition.
Allowing an exit from absorbing death. If death is defined as absorbing, every edge from death to a living state is excluded.

Key takeaway: Define each state precisely, write down the assumptions that govern direct changes, and then draw the directed edges. Keep structural permission, actual intensity, and positive transition probability distinct.

Check Your Understanding

For each question, identify the direct edges or graph property requested from the stated hypothetical assumptions.

A model has \(H,C,R,D\). From \(H\), transitions to \(C,D\) are allowed; from \(C\), to \(R,D\); from \(R\), to \(H,D\); and \(D\) is absorbing. Give the edge count and state whether \(C\to H\) is a direct edge. (Answer: 6 edges; no, only the path \(C\to R\to H\) is allowed.)
A model permits \(H\to C\), \(H\to D\), \(C\to D\), \(R\to H\), and \(R\to D\), with no other transitions. Is \(C\) able to reach \(R\), and how many edges are in the graph? (Answer: No; 5 edges.)
In a model starting in \(H\), \(q_{HC}(t)=0.3\) per year on \(0\le t\le 0.2\), and is zero otherwise except for \(q_{HC}(t)=2\) per year at \(t=0.8\). There are no competing exits. Is \(H\to C\) permitted with positive transition probability by time \(0.2\), and what is that probability? (Answer: Yes; \(1-e^{-0.3(0.2)}=1-e^{-0.06}\approx0.0582\).)
A graph has \(H\to C,D\), \(C\to R,D\), \(R\to H,C,D\), and absorbing \(D\). A new assumption excludes relapse from \(R\) to \(C\). Give the revised edge count and identify the removed edge. (Answer: 6 edges; \(R\to C\) is removed.)
A model specifies a positive intensity for \(R\to C\) only at the single instant \(t=2\), and zero at all other times. Under the usual locally integrable intensity model, does that value alone create positive probability of a transition over a time interval containing \(t=2\)? Explain briefly. (Answer: No; its integrated intensity over the interval is zero, so the isolated value alone creates no positive transition probability.)

What You'll Learn