Concept Architecture
Concept
Theoretically, Transient State is a health state in a Markov chain or state-transition model that, once left, may never be revisited with certainty. Individuals occupying a transient state have a non-zero probability of transitioning to other states, and over time the probability of remaining indefinitely in the transient state approaches zero. In health economics, transient states typically represent temporary clinical conditions such as disease stages, treatment phases or recovery states preceding progression to other health states or an absorbing state.
Mathematically, a transient state is defined within Markov chain theory by the probability of eventual return to that state. A state is transient if the probability of ever returning after departure is less than one. In finite Markov chains containing absorbing states, transient-state occupancy declines over successive cycles as individuals transition permanently into absorbing states or other long-term states.
In practice, transient states are specified during model construction to represent health conditions from which patients may progress, recover or die. Transition probabilities are estimated from clinical or observational data, and repeated state updates determine the changing occupancy of transient states throughout the model horizon. State occupancies are combined with costs and utilities to estimate cumulative health and economic outcomes.
Purpose
Used to represent temporary health states through which individuals may pass before progressing to other states, enabling realistic modelling of disease progression and healthcare interventions.
Mathematical Formulae
Primary Formula
A state i is transient if:
P(return to i ? X? = i) < 1
Supporting Formulae
State update:
????? = ?????
Canonical decomposition of the transition matrix for an absorbing Markov chain:
?? = ??? ???
?????? ???
where:
- ?? = transitions among transient states
- ?? = transitions from transient to absorbing states
- ?? = identity matrix for absorbing states
Fundamental matrix:
?? = (?? ? ??)??
where ?? gives the expected number of visits to transient states before absorption.
Related Mathematical Methods
- Markov chains
- Absorbing Markov chains
- Matrix algebra
- State-transition modelling
- Cohort simulation
- Fundamental matrix analysis
Example
A Markov model for chronic kidney disease contains the states Mild Disease, Moderate Disease, End-Stage Renal Disease and Death.
The first three disease states are transient because patients may leave them through disease progression or death. Death is an absorbing state because patients remain in that state permanently. During each annual cycle, the model updates the occupancy of the transient states using the transition probability matrix and accumulates associated healthcare costs and quality-adjusted life years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(StateVector,TransitionMatrix) | Update occupancy of transient states after each cycle |
| SUMPRODUCT | =SUMPRODUCT(StateVector,CostVector) | Calculate expected costs associated with transient states |
| SUMPRODUCT | =SUMPRODUCT(StateVector,UtilityVector) | Calculate expected QALYs accumulated in transient states |
| MINVERSE | =MINVERSE(IdentityMatrix-QMatrix) | Calculate the fundamental matrix for absorbing Markov chains |
VBA (Optional)
Automate repeated calculation of transient-state occupancies and expected time spent in each state before absorption.
Sources
- Kemeny JG, Snell JL. Finite Markov Chains. Springer.
- Norris JR. Markov Chains. Cambridge University Press.
- Grinstead CM, Snell JL. Introduction to Probability. American Mathematical Society.
- Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322?338.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- 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.
Related Concepts (3)
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 (6)
What is a transient state?
A health state within a Markov model that a patient can leave and potentially return to, unlike an absorbing state such as death.
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 is an example of a transient state?
A transient state is one a patient can occupy for a while and then leave, and possibly re-enter later. States such as being in remission, experiencing a relapse, or living with a manageable complication are transient, because a patient can move out of them into other states and sometimes back again. They contrast with an absorbing state such as death, which is never left. Most states in a chronic disease model are transient, since the condition keeps evolving. Briggs and colleagues (2006) describe them.
Source: Briggs et al. 2006
What characterises a transient state?
A transient state is characterised by having transitions leading out of it, so the probability of leaving over the horizon is positive and patients do not remain permanently. Patients can move from a transient state to other states, and in some models return to it, reflecting conditions that can improve, worsen, or recur. This contrasts with an absorbing state, which patients cannot leave. Transient states thus represent the conditions of the disease course that patients experience temporarily as they move through the model.
Source: Sonnenberg & Beck 1993
How does a transient state differ from an absorbing state?
A transient state differs from an absorbing state in that patients can leave a transient state, moving to other states, whereas they cannot leave an absorbing state once entered. Transient states represent temporary conditions patients pass through, such as stages of disease that can change, while absorbing states represent permanent end points, most commonly death. Over a long horizon, patients drain from transient states into absorbing ones, so the distinction is between states that are passed through and states that retain patients permanently.
Source: Sonnenberg & Beck 1993
Can patients return to a transient state?
Whether patients can return to a transient state depends on the model's transition structure. In some models, transitions allow patients to move back into a state they have left, representing conditions that can recur or improve, such as remission and relapse. In others, the structure only allows forward movement through progressively worse states, so a state, though transient, is not re-entered. Transient states are defined by being leavable; whether they are also re-enterable is a feature of the particular transition structure.
Source: Sonnenberg & Beck 1993
Why are transient states important in a Markov model?
Transient states are important because they represent the conditions patients experience as the disease progresses, the stages, symptoms, or treatment phases through which patients move, so they carry the costs and health effects of those conditions. Most of a model's states are transient, and the time patients spend in them, and their movement between them, drives the accrual of outcomes. The transient states thus capture the substance of the disease course, with absorbing states marking its permanent conclusions.
Source: Sonnenberg & Beck 1993
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 7 Oct 2025
Content version: 1.0.0
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- Term code
- HE-EM-MM-025
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