A Semi-Markov Illness-Death Model

Structure of the Illness-Death Model

An illness-death model describes progression among three states: healthy, ill, and dead. Its main modelling choice is how to represent time: a person's chance of leaving a state may depend on how long they have already occupied it. For example, the risk of becoming ill may increase with time spent healthy, while the risk of death after illness may depend on the duration of illness.

The model below uses a reset clock. Time in the current state starts at zero upon entry to that state. Thus, the healthy-to-ill and healthy-to-dead hazards depend on time since entry into the healthy state; after a transition to illness, the ill-to-dead hazard depends on time since entry into illness.

Definition: A semi-Markov illness-death model has states \(0\) (healthy), \(1\) (ill), and \(2\) (dead), with allowed transitions \(0\to1\), \(0\to2\), and \(1\to2\). State \(2\) is absorbing. Conditional on entry into a state, the distribution of the next transition time and destination depends on the current state and its duration, but not on earlier history. The destination-specific hazards are \(q_{01}(u)\), \(q_{02}(u)\), and \(q_{12}(u)\), where \(u\) is time since entry into the state from which the transition occurs.

The two hazards out of state 0 represent competing ways to leave the healthy state. They must be considered together when calculating the chance of remaining healthy: a person who becomes ill can no longer make the direct healthy-to-dead transition. State 1 has only one allowed exit in this model, to death. Recovery, recurrence, and other transitions are excluded by specification, not ruled out as possibilities in every application.

For a transition hazard \(q_{ij}(u)\), the precise interpretation is conditional on still occupying state \(i\) at duration \(u\). Over a short interval of length \(du\), the probability of a transition to \(j\) is approximately \(q_{ij}(u)du\), with terms smaller than order \(du\) omitted. A hazard is a rate, not itself a probability.

Holding Times and Transition Densities

Let \(S_i(u)\) denote the probability of remaining in state \(i\) for at least duration \(u\), conditional on entry into that state. In state 0, either allowed transition ends the healthy-state sojourn. Therefore its total exit hazard is \(q_{01}(u)+q_{02}(u)\). In state 1, the total exit hazard is \(q_{12}(u)\).

S_0(u)=\exp\left\{-\int_0^u [q_{01}(v)+q_{02}(v)]\,dv\right\}, \qquad S_1(u)=\exp\left\{-\int_0^u q_{12}(v)\,dv\right\}.

These expressions require nonnegative, locally integrable hazards. They follow from the fact that the instantaneous rate of loss of state survival equals the current survival probability times the total exit hazard. The transition-specific density combines surviving in the current state until duration \(u\) with exiting to a particular destination at that duration.

f_{01}(u)=S_0(u)q_{01}(u),\qquad f_{02}(u)=S_0(u)q_{02}(u),\qquad f_{12}(u)=S_1(u)q_{12}(u).

For example, \(f_{01}(u)du\) is approximately the probability that the healthy sojourn ends in a transition to illness during \([u,u+du)\). The units of a hazard and a density are inverse time; \(S_i(u)\) is dimensionless. The destination-specific holding-time distributions are obtained by integrating these densities. If a state can be occupied indefinitely with positive probability, the corresponding exit densities need not integrate to one.

This formulation is the illness-death specialization of the semi-Markov transition-kernel and destination-specific holding-time ideas. It is essential to use the total exit hazard in \(S_0\): using only \(q_{01}\) would incorrectly treat direct death as though it could not pre-empt illness.

Worked Examples

Worked Example: Competing Exits Over a Short Interval

Suppose a hypothetical person is healthy and has already spent 2 years in state 0. Assume that over the next 0.1 year the hazards are constant at \(q_{01}=0.08\) per year and \(q_{02}=0.03\) per year. Calculate the probabilities of remaining healthy, becoming ill, and dying directly during that interval.

Under the stated piecewise-constant assumption, the total exit rate is \(0.08+0.03=0.11\) per year. The probability of no exit in 0.1 year is \(S=\exp[-(0.11)(0.1)]=\exp(-0.011)=0.9891\).

For constant competing hazards, the probability of an exit to each destination is its share of the total rate multiplied by the probability of any exit:

P(0\to1)=\frac{0.08}{0.11}\left(1-e^{-0.011}\right)=0.007956, \qquad P(0\to2)=\frac{0.03}{0.11}\left(1-e^{-0.011}\right)=0.002984.

The three probabilities sum to approximately \(0.9891+0.007956+0.002984=1.0000\), allowing for rounding. Thus, under these hypothetical rates, the chance of illness during the interval is about 0.796%, and the chance of direct death is about 0.298%. These are interval probabilities conditional on being healthy at its start, not hazard values. The calculation is exact for hazards constant over the interval; if they vary materially within it, their duration-specific functions must be used instead.

Worked Example: Duration-Dependent Hazards and Densities

Consider a hypothetical model in which, with \(u\) measured in years, \(q_{01}(u)=0.04u\) per year, \(q_{02}(u)=0.02\) per year, and \(q_{12}(u)=0.06u\) per year. The coefficients multiplying \(u\) have units of inverse years squared. Find the probability of remaining healthy for 3 years, the healthy-to-ill density at duration 3, and the probability of remaining ill for 2 years after entering illness.

The healthy-state cumulative exit hazard through duration 3 is \(\int_0^3(0.04v+0.02)\,dv=0.02(3^2)+0.02(3)=0.24\). Hence \(S_0(3)=e^{-0.24}=0.7866\). At duration 3, \(q_{01}(3)=0.12\) per year, so \(f_{01}(3)=0.7866(0.12)=0.09440\) per year. This density describes the rate of probability mass for a healthy-to-ill exit at that sojourn duration; it is not the probability of illness at exactly year 3.

After entry into illness, the cumulative exit hazard through duration 2 is \(\int_0^2 0.06v\,dv=0.12\). Therefore \(S_1(2)=e^{-0.12}=0.8869\): about 88.69% of comparable illness-state sojourns last at least 2 years under this model. The density of death at illness duration 2 is \(f_{12}(2)=0.8869(0.12)=0.1064\) per year.

Worked Example: Density of a Healthy-to-Ill-to-Dead Path

Use the hazards from the preceding example. For a hypothetical individual who first becomes ill after 3 years in the healthy state and then dies after 2 years in the ill state, calculate the joint density for those two sojourn durations and transition destinations.

The first transition has density \(f_{01}(3)=0.0943953\) per year, and, conditional on entry into illness, the second has density \(f_{12}(2)=0.1064304\) per year. The semi-Markov specification makes the latter distribution depend on the illness-state duration, not on the preceding 3 years spent healthy. Consequently, the joint density for these specified transitions and durations is the product:

f_{01}(3)f_{12}(2) =(0.0943953)(0.1064304) \approx 0.010047\ \text{per year}^2.

This is a joint density with respect to the two duration measurements, so its units are inverse years squared. It is not a probability for observing those exact times; a probability would require specifying intervals around the durations and integrating the density over them. The factorization also relies on the model assumption that the next sojourn law after entry into illness does not depend on the earlier healthy-state duration.

Why the Semi-Markov Specification Matters

A time-homogeneous Markov illness-death model uses transition rates that do not depend on how long a person has occupied the current state. Its holding times are memoryless. The semi-Markov model allows the exit rate to change with state duration, while retaining a concise rule for the next transition conditional on the current state and its entry time.

Feature	Time-homogeneous Markov illness-death model	Semi-Markov illness-death model here
Rate out of healthy state	Constant for each allowed destination	May vary with time since healthy-state entry
Rate out of ill state	Constant	May vary with time since illness-state entry
Clock at transition to illness	No duration clock is needed for constant rates	Illness duration is reset to zero
Flexibility	Simple, but imposes memoryless sojourn times	Allows non-exponential, destination-specific holding times

The extra flexibility is useful when, for example, risk changes with duration in illness or the competing healthy-state exits have different time patterns. But flexibility does not remove the need for a clearly defined time origin. This model uses duration since entry into the current state, not time since study enrollment for every transition.

The assumptions should be checked against the scientific setting and data structure. In particular:

Allowed transitions: the model excludes recovery and other transitions. If they occur and are relevant, the state diagram must include them.
Correct clock reset: the illness-to-death duration begins when illness starts. Using time since healthy-state entry instead would specify a different model.
Destination-specific hazards: both \(q_{01}\) and \(q_{02}\) contribute to loss of healthy-state survival. Omitting one distorts the holding-time distribution and the other transition density.
History restriction: conditional on the current state and its duration, the model assumes earlier path history adds no further information about the next transition. This is a substantive modelling assumption.
Observation process: when fitting the model to censored data, inference also needs an appropriate assumption about censoring. A common condition is that censoring is independent of the future event process conditional on the modeled information.

Useful diagnostics include plotting estimated or fitted hazards against state duration, comparing observed and model-implied sojourn distributions, and checking whether fitted hazards remain nonnegative and plausible over the durations supported by the data. In state 0, the two destination-specific hazards should be inspected separately as well as through their sum. A good fit to the total exit rate alone can conceal a poor allocation between illness and direct death. Sparse observations at long durations also make tail behavior and extrapolation particularly uncertain.

This model is not a general account of every form of history dependence. If earlier transitions, repeated state visits, or other features predict future risk even after conditioning on current state and duration, the semi-Markov restriction here is inadequate. Likewise, a fitted duration pattern does not by itself establish a causal effect of spending longer in a state.

Common Mistakes

Using a hazard as an interval probability. A rate of 0.08 per year is not an 8% probability over every interval. For a short interval of length \(\Delta\), the probability is approximately \(0.08\Delta\) only when the rate is nearly constant and competing exits are accounted for.
Ignoring the competing direct-death transition. Healthy-state survival uses \(q_{01}+q_{02}\), not \(q_{01}\) alone.
Failing to reset duration at illness onset. In this model, \(q_{12}\) takes illness duration as its argument. Substituting total time since initial healthy entry changes the model.
Calling a density a probability. Values such as \(f_{01}(3)\) have units of inverse time. Probabilities come from integrating density over a time interval.
Assuming every illness-death model is semi-Markov. A model with constant state-specific transition rates is a special memoryless case; duration-dependent hazards require the explicit duration specification given here.

Key takeaway: In the semi-Markov illness-death model, \(q_{01}(u)\) and \(q_{02}(u)\) jointly determine survival in the healthy state, while \(q_{12}(u)\) determines survival after illness begins. State duration resets at each transition, and each density is the relevant state-survival probability multiplied by its destination-specific hazard.

Check Your Understanding

Answer each question using the model definitions above. Short solutions follow the questions.

If \(q_{01}(u)=0.05\) and \(q_{02}(u)=0.03\) per year throughout a one-year interval, what is the probability of remaining healthy through that interval?
Why must \(q_{02}\) be included when calculating \(S_0(u)\)?
Suppose \(q_{12}(v)=0.04v\) per year, with \(v\) in years. Find the probability of remaining ill for 3 years after illness onset.
What are the units of \(f_{01}(u)\) and of \(f_{01}(s)f_{12}(v)\)?

Solutions. (1) The total rate is \(0.08\) per year, so \(S_0(1)=e^{-0.08}=0.9231\). (2) Direct death also ends the healthy sojourn, so \(S_0(u)\) uses the total exit hazard \(q_{01}(u)+q_{02}(u)\); omitting \(q_{02}\) overstates healthy-state survival. (3) The cumulative illness-state hazard is \(\int_0^3 0.04v\,dv=0.18\), giving \(S_1(3)=e^{-0.18}=0.8353\). (4) \(f_{01}\) has units of inverse time, and the product of two such densities for two durations has units of inverse time squared.