Concept Architecture
Concept
Theoretically, Time-Homogeneous Model is a state-transition model in which transition probabilities remain constant throughout the model time horizon. The probability of moving between health states depends on the current state but not on calendar time or the number of elapsed model cycles. In health economics, time-homogeneous models are appropriate when disease progression, treatment effects and mortality risks can reasonably be assumed to remain stable over time.
Mathematically, a time-homogeneous model is characterised by a single transition probability matrix that is applied repeatedly during every model cycle. Because the transition matrix is invariant with time, future state distributions are obtained through repeated powers of the same matrix. This property simplifies both model implementation and analytical evaluation.
In practice, transition probabilities are estimated from clinical trials, observational studies or epidemiological data and assumed to remain unchanged throughout the simulation. Time-homogeneous models are commonly used for chronic diseases with approximately constant transition risks. When evidence indicates that risks change over time owing to ageing, disease duration or waning treatment effects, time-dependent models are generally preferred.
Purpose
Used to model disease progression and health outcomes under the assumption that transition probabilities remain constant throughout the model time horizon, simplifying long-term economic evaluation.
Mathematical Formulae
Primary Formula
State update:
????? = ?????
where:
- ??? = state distribution at cycle t
- ?? = constant transition probability matrix
Supporting Formulae
State distribution after n cycles:
??? = ?????�
Time-homogeneous transition probability:
P(X??? = j ? X? = i) = p??
where p?? is constant for all t.
Related Mathematical Methods
- Markov chains
- Matrix algebra
- State-transition modelling
- Cohort simulation
- Eigenvalue analysis
- Probabilistic sensitivity analysis
Example
A Markov model evaluating chronic glaucoma uses annual transition probabilities that are assumed to remain constant over a 20-year time horizon.
The transition matrix is:
?? =
?0.90 0.08 0.02?
?0.00 0.85 0.15?
?0.00 0.00 1.00?
This same matrix is applied during every annual cycle. Expected costs and quality-adjusted life years are calculated from the resulting cohort distributions over the entire model horizon.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(StateVector,TransitionMatrix) | Update the cohort using the same transition matrix each cycle |
| SUMPRODUCT | =SUMPRODUCT(StateVector,CostVector) | Calculate expected cycle costs |
| SUMPRODUCT | =SUMPRODUCT(StateVector,UtilityVector) | Calculate expected cycle QALYs |
| MUNIT | =MUNIT(NumberStates) | Create identity matrices for model development |
VBA (Optional)
Automate repeated application of a fixed transition probability matrix across all model cycles and summarise cumulative health and economic outcomes.
Sources
- Norris JR. Markov Chains. Cambridge University Press.
- Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322?338.
- Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Medical Decision Making. 2012;32(5):690?700.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- NICE. Health Technology Evaluation Manual.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press.
Related Concepts (4)
Library
Publications
1
An Introduction to Markov Modelling for Economic Evaluation — Briggs & Sculpher, Vol. 13, No. 4 ed., 1998 (PharmacoEconomics)
The foundational tutorial paper introducing Markov (state-transition) models for health economic evaluation, covering health states, cycle length, transition probabilities and the calculation of expected costs and outcomes. Widely cited as the standard entry point to Markov modelling.
Journal ArticleView source →
Frequently Asked Questions (7)
What is a time homogeneous model?
A Markov model in which transition probabilities remain constant throughout the entire modelled time horizon, regardless of how many cycles have passed.
Source: Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322-338. doi:10.1177/0272989X9301300409.
What constant-rate assumption defines a time-homogeneous model?
A time-homogeneous model is one in which the transition probabilities between states stay constant from cycle to cycle. Because the rates do not depend on how much time has passed, the model has the memoryless property associated with simple Markov chains. This assumption keeps the model easier to specify and to compute, since a single set of transition values applies throughout. It is reasonable when the underlying risks are broadly stable over the analytic horizon. Where risks clearly rise or fall with age or disease duration, a time-inhomogeneous formulation is more appropriate.
Source: Briggs, Claxton and Sculpher 2006
What is a time-homogeneous model?
A time-homogeneous model is a Markov model in which the transition probabilities remain constant throughout the entire time horizon, regardless of how many cycles have passed. The same probabilities govern movement between states at every cycle, so a patient in a given state faces the same transition risks whether early or late in the model. This is the simplest form of Markov model, appropriate when the risks of transition do not change over time, and it contrasts with a time-inhomogeneous model whose probabilities vary.
Source: Sonnenberg & Beck 1993
When is a time-homogeneous model appropriate?
A time-homogeneous model is appropriate when the transition probabilities can reasonably be treated as constant over the time horizon, so that the risks of moving between states do not change with time or age. This may hold over short horizons or where risks are stable. It offers simplicity, requiring one set of transition probabilities. Where risks genuinely change over time, such as mortality rising with age, the constant-probability assumption is inappropriate, and a time-inhomogeneous model is needed instead.
Source: Sonnenberg & Beck 1993
What are the advantages of a time-homogeneous model?
A time-homogeneous model is simple, since it uses a single, constant set of transition probabilities applied at every cycle, making it easy to build, parameterise, check, and communicate, and requiring less data than a model whose probabilities vary over time. Its behaviour is straightforward to analyse. These advantages make it a convenient choice where the assumption of constant transition risks is reasonable, providing a transparent and economical representation when risks do not change materially over the horizon.
Source: Sonnenberg & Beck 1993
What are the limitations of a time-homogeneous model?
The limitation of a time-homogeneous model is that its constant transition probabilities cannot represent risks that change over time, such as mortality increasing with age or event risk varying with time. Where such time dependency exists, assuming constant probabilities misrepresents the disease and can bias results, particularly over long horizons where age-related changes matter. In these cases a time-inhomogeneous model, with probabilities that vary by cycle, is required, so the time-homogeneous form suits only situations where risks are genuinely stable.
Source: Sonnenberg & Beck 1993
How does a time-homogeneous model differ from a time-inhomogeneous one?
A time-homogeneous model uses constant transition probabilities throughout the horizon, while a time-inhomogeneous model allows the probabilities to vary over time, for instance rising mortality with age. The homogeneous model is simpler but assumes stable risks; the inhomogeneous model is more flexible and realistic where risks change but requires more data specifying how probabilities vary. The choice depends on whether transition risks change materially over the horizon, with the inhomogeneous form used when they do.
Source: Sonnenberg & Beck 1993
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 7 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-MM-023
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