When the Current State Is Not the Whole Story
A multiple-state model often assigns each possible transition an intensity based on the current state and, in a time-inhomogeneous model, calendar time. But some transition rates depend instead on how long the individual has been in the current state. For example, a hypothetical disability contract might have a higher recovery intensity shortly after disability begins than after several years of continuous disability.
That distinction matters because two people can be in the same state at the same calendar time, yet have different durations in that state. If their transition intensities differ for that reason, a model whose state records only “disabled” cannot describe both rates using the same state-and-calendar-time intensity. The model needs additional information, such as duration since entry, or an expanded set of states that records duration categories.
In a state-only Markov model, knowing the current state and calendar time is sufficient to specify the rates of possible next transitions. This does not require the rates to be constant: \(q_{ij}(t)\) may vary with calendar time. It does require that the rate not also depend on how long the individual has occupied the state, unless that duration is included in the model’s state description.
Duration Time Is Not Calendar Time
Calendar time \(t\) measures time from a common origin, such as policy issue or the start of observation. Duration \(d\) measures time since a particular transition into a state. Individuals in the same state at calendar time \(t\) may have entered it at different times, so their durations can differ.
A calendar-time intensity can represent a change that affects everyone in the state at a given calendar date. A duration-dependent intensity represents a change associated with the length of time each individual has been in the state. Replacing duration with calendar time is generally not valid when entry dates differ.
Worked Example: Detecting Missing Duration Information
A hypothetical disability model assigns a recovery intensity of \(0.20\) per year during the first year of disability and \(0.05\) per year thereafter. Consider two people who are both disabled at calendar time \(t=5\): one entered disability at \(t=4.5\), and the other at \(t=2\). Compare their recovery probabilities over the next \(h=0.001\) years to first order.
Method. Use the defining short-interval property of an intensity: conditional on being in the relevant state, the probability of a specified transition during a short interval of length \(h\) is its intensity multiplied by \(h\), plus a term \(o(h)\), which is negligible relative to \(h\) as \(h\) tends to zero.
The first person has duration \(0.5\) years, so the recovery intensity is \(0.20\). The short-interval recovery probability is \(0.20(0.001)+o(0.001)=0.0002+o(0.001)\).
The second person has duration \(3\) years, so the recovery intensity is \(0.05\). The corresponding probability is \(0.05(0.001)+o(0.001)=0.00005+o(0.001)\).
The leading probabilities differ by a factor of \(0.0002/0.00005=4\), even though both people are in the disabled state at the same calendar time. A single recovery intensity \(q_{\text{disabled,recovered}}(5)\) cannot give both short-interval rates. The omitted information is duration in disability. The common mistake is to treat the two people as having the same rate merely because their current state and calendar time agree.
Why a Duration-Dependent Rate Changes the Model
The Markov property for the chosen state description requires that, once the current state and calendar time are known, past history does not supply further information needed to specify future transition rates. If the rate depends on duration in the state, the history of when the current state was entered is relevant. A state-only model therefore fails to encode the rate correctly.
This is a limitation of the chosen state description, not a claim that the underlying process cannot be modelled. One option is to enlarge the state to include duration, writing the augmented state as \((i,d)\). Another is to use a finite collection of duration categories. The second option can be convenient, but it must be interpreted carefully: categories may approximate continuous duration dependence, and a fixed duration threshold does not automatically produce an ordinary finite-state continuous-time Markov chain.
Deriving the No-Transition Probability
Consider a hypothetical process that starts in state \(i\), has no competing destination, and leaves \(i\) with duration-dependent intensity \(\mu(d)\) per year. Let \(S(d)\) be the probability of remaining in \(i\) without leaving through duration \(d\), given entry at duration zero. Over a short interval of length \(h\), the conditional probability of no exit is \(1-\mu(d)h+o(h)\). Thus \(S(d+h)=S(d)\{1-\mu(d)h+o(h)\}\). Subtract \(S(d)\), divide by \(h\), and let \(h\) tend to zero to obtain \(S'(d)=-\mu(d)S(d)\), with \(S(0)=1\). Solving this differential equation gives:
The integral accumulates the exit intensity over the time actually spent in the state. This expression applies to survival without exit when \(\mu(d)\) is the total exit intensity. With multiple possible destinations, the total exit intensity is the sum of their intensities; the probability of a particular destination also depends on its share of that total. The formula does not imply that a duration-dependent process has a finite-state, state-only intensity matrix.
Worked Example: Calculating Survival Across a Duration Change
A hypothetical long-term income benefit has one exit route from the disabled state: recovery. The recovery intensity is \(0.20\) per year for the first year of continuous disability and \(0.05\) per year after that. Assume the individual has just entered disability and the stated intensities apply exactly by duration. Find the probability of remaining disabled without recovery for two years, and the probability of recovery by then.
Method. Integrate the total exit intensity over each duration segment, then apply the no-exit formula. On the first year, the accumulated intensity is \(0.20(1)=0.20\). On the second year, it is \(0.05(1)=0.05\). The total accumulated intensity is \(0.20+0.05=0.25\).
Therefore the probability of no recovery during the two years is \(S(2)=e^{-0.25}=0.7788008\), or \(0.7788\) to four decimal places. Since recovery is the only exit, the probability of recovery by two years is \(1-S(2)=1-0.7788008=0.2211992\), or \(0.2212\) to four decimal places.
As a check, survival through the first year is \(e^{-0.20}=0.8187\), and conditional survival through the second year is \(e^{-0.05}=0.9512\). Their product is \(e^{-0.20}e^{-0.05}=e^{-0.25}=0.7788\), agreeing with the calculation. The common mistake is to use \(0.20\) for both years, which would give \(e^{-0.40}\) and ignore the stated duration change.
Using Duration Categories
A practical finite-state approach is to distinguish phases such as “recently entered” and “longer duration.” The intensity matrix can then assign different rates to the different phase states. This is a genuine state expansion: a person in the first phase and a person in the later phase are no longer represented by the same state.
For a finite-state continuous-time Markov model, transitions between phases have intensities, and the time to a phase transition is governed by those intensities. Therefore, if the intended rule is “move to the later phase exactly one year after entry,” replacing that rule with an exponential phase-transition intensity changes the model. Phase models can be useful approximations, but their timing assumptions should be stated rather than hidden.
Worked Example: Building a Two-Phase Approximation
A hypothetical model represents disability with two states: \(D_0\), an early-duration phase, and \(D_1\), a later-duration phase. From \(D_0\), recovery has intensity \(0.20\) per year and movement to \(D_1\) has intensity \(0.50\) per year. From \(D_1\), recovery has intensity \(0.05\) per year. Recovery state \(R\) is absorbing. Assume these intensities are constant while the process occupies each state. Find the intensity matrix and the first-order transition probabilities over \(h=0.01\) years from each disability phase.
Method. Put each outgoing intensity in its off-diagonal entry and choose each diagonal entry as the negative sum of the other entries in its row. Use the short-interval intensity approximation \(P(X(t+h)=j\mid X(t)=i)=q_{ij}h+o(h)\) for \(j\ne i\); the probability of remaining in the current state is one minus the sum of its outgoing transition probabilities, to first order.
From \(D_0\), the total exit intensity is \(0.20+0.50=0.70\), so its diagonal entry is \(-0.70\). From \(D_1\), the total exit intensity is \(0.05\), so its diagonal entry is \(-0.05\). Since \(R\) is absorbing, its row is zero. In the state order \(D_0,D_1,R\), the intensity matrix is
Each row sums to zero, as required for an intensity matrix. Starting in \(D_0\), the first-order probability of moving to \(D_1\) in \(0.01\) years is \(0.50(0.01)=0.005\), and the probability of recovery is \(0.20(0.01)=0.002\). The probability of remaining in \(D_0\) is \(1-0.005-0.002=0.993\), to first order. Starting in \(D_1\), the first-order recovery probability is \(0.05(0.01)=0.0005\), and the probability of remaining in \(D_1\) is \(1-0.0005=0.9995\). Recovery remains in \(R\) with probability one.
The calculation checks because the first-order probabilities in each starting row sum to one: \(0.005+0.002+0.993=1\), and \(0.0005+0.9995=1\). This model captures different recovery rates by phase and has a finite-state Markov specification. It is not an exact model of a switch occurring at a fixed duration of one year: the \(0.50\) intensity makes the time to move from \(D_0\) to \(D_1\) exponential, not deterministic. The common mistake is to interpret the phase transition as guaranteed at one year when its stated intensity does not say that.
Choosing a Suitable Representation
The right representation depends on what the model must preserve. If exact dependence on time since entry is essential, record duration as part of the state, or use a duration-based model that explicitly tracks the elapsed time. If a finite-state Markov framework is needed, duration phases may be used as an approximation or as part of a model with exponential phase movements. The assumptions should make clear which interpretation is intended.
Ask whether the rate changes with calendar time, duration in the current state, or both.
Compare individuals in the same state at the same calendar time but with different entry times. If their rates differ, the state-only specification omits duration information.
Track duration directly for an exact duration-dependent rate, or define duration phases and state whether their transitions are exact or approximate.
This distinction is useful in disability, long-term care, and other long-term product models: rates can change with time in a condition, not just with age or calendar date. A specification that confuses these time scales can assign the wrong transition rate to a person and consequently distort projected state probabilities or benefit cash flows.
Common Exam Trap
A frequent error is to see a rate stated as “after two years in the state” and place that change at calendar time \(t=2\). That is only correct for someone who entered the state at calendar time zero. For someone entering at \(t=3\), the two-year duration point occurs at calendar time \(t=5\). Another error is to use a fixed threshold to define a new phase and then treat the transition into that phase as if it had a constant exponential intensity. A fixed waiting time and an exponential waiting time are different assumptions.
Check Your Understanding
For each problem, state the relevant time scale and show how it affects the model.
- A hypothetical recovery intensity is \(0.12\) per year for the first two years in a state and \(0.04\) per year thereafter. At the same calendar time, two individuals have durations of 1 year and 3 years. What are their current recovery intensities? (Answer: \(0.12\) and \(0.04\) per year, respectively.)
- A single-exit state has duration-dependent intensity \(0.10\) per year for one year, followed by \(0.06\) per year for two years. Find the probability of remaining in the state without exit for all three years. Give the answer to four decimals. (Answer: \(e^{-(0.10+2(0.06))}=e^{-0.22}=0.8025\).)
- Two people are both in state \(C\) at calendar time \(t=6\). One entered at \(t=5\), and one entered at \(t=2\). The exit intensity is \(0.30\) per year for durations under two years and \(0.08\) per year thereafter. Can one state-only intensity \(q_{C,\text{exit}}(6)\) represent both rates? (Answer: No. Their durations are 1 and 4 years, so their rates are \(0.30\) and \(0.08\) per year.)
- A proposed two-phase model has intensity \(0.40\) per year to move from early to late duration, \(0.10\) per year for recovery from the early phase, and \(0.05\) per year for recovery from the late phase. What is the first-order probability of each outgoing transition over \(0.02\) years when starting in the early phase, and what is the probability of remaining there? (Answer: Early-to-late \(=0.40(0.02)=0.008\); recovery \(=0.10(0.02)=0.002\); remaining \(=1-0.008-0.002=0.990\), to first order.)
- A contract changes its recovery rate exactly three years after disability begins. Explain why a constant intensity from an early-duration state to a late-duration state does not represent this rule exactly, and name one representation that can. (Answer: A constant phase-transition intensity gives an exponential, random transition time rather than a fixed three-year waiting time. Tracking duration since entry and switching the recovery intensity at duration three years can represent the rule.)