Concept Architecture
Concept
Theoretically, Tunnel State is a modelling construct used within a state-transition model to represent time spent within a health state when future transition probabilities, costs or health outcomes depend on the duration of occupancy rather than solely on the current state. Tunnel states partition a single clinical state into a sequence of temporary substates, thereby relaxing the Markov assumption of memorylessness while retaining the state-transition modelling framework. In health economics, tunnel states are commonly used to represent duration-dependent treatment effects, disease progression, recovery and post-event risks.
Mathematically, tunnel states are represented by expanding a single health state into an ordered sequence of substates connected by deterministic or probabilistic transitions. Each tunnel substate has its own transition probabilities and rewards, allowing risks and outcomes to vary according to time spent in the underlying clinical state. The expanded transition matrix therefore incorporates additional rows and columns corresponding to the tunnel substates.
In practice, tunnel states are implemented when empirical evidence indicates that transition risks or costs change with state duration. The number of tunnel substates is selected to reflect the period over which duration-dependent effects occur. Transition probabilities are estimated from longitudinal clinical data or survival analyses, and model validation confirms that the expanded state structure reproduces observed time-dependent disease progression and health outcomes.
Purpose
Used to represent duration-dependent transition probabilities, costs and health outcomes within state-transition models while preserving the overall Markov modelling framework.
Mathematical Formulae
Primary Formula
Expanded state update:
????? = ??????
where:
- ??? = expanded state occupancy vector at cycle t
- ??? = transition matrix containing tunnel substates
Supporting Formulae
Deterministic progression through tunnel substates:
P(T? ? T???) = 1
where:
- T? = tunnel substate representing duration interval k
Expected reward:
R? = ?????
where:
- ?? = vector of state or tunnel-state rewards
Related Mathematical Methods
- Markov modelling
- State-transition modelling
- Matrix algebra
- Semi-Markov approximation
- Cohort simulation
- Survival analysis
- Probabilistic sensitivity analysis
Example
A post-myocardial infarction model assumes that mortality risk is highest during the first year after the event and decreases thereafter. Rather than using a single Post-MI state, the model defines three tunnel states:
- Post-MI Year 1
- Post-MI Year 2
- Post-MI Year 3+
Patients move sequentially through these substates, with each having different mortality probabilities and healthcare costs. This structure allows mortality risk to depend on time since infarction while remaining within a Markov state-transition framework.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(StateVector,ExpandedTransitionMatrix) | Update occupancy across tunnel substates |
| INDEX | =INDEX(TransitionTable,TunnelState,Cycle) | Retrieve duration-specific transition probabilities |
| SUMPRODUCT | =SUMPRODUCT(StateVector,RewardVector) | Calculate costs or QALYs for tunnel substates |
| IF | =IF(Cycle<=3,EarlyRisk,LateRisk) | Apply duration-dependent transition probabilities |
VBA (Optional)
Automate progression through tunnel substates and apply duration-specific transition probabilities and rewards during model execution.
Sources
- 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.
- Caro JJ, Briggs AH, Siebert U, Kuntz KM. Modeling good research practices: overview. Medical Decision Making. 2012;32(5):667?677.
- 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 (3)
Library
Publications
2
State-Transition Modeling: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3 — Siebert, Alagoz, Bayoumi, Jahn, Owens, Cohen & Kuntz, Task Force Report 3 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance for cohort and individual-based state-transition (Markov) models, covering development, analysis, validation and reporting.
Journal ArticleView source →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 tunnel state?
A modelling technique creating temporary, time-specific states a patient passes through in sequence, letting a Markov model approximate time-dependent risk.
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.
Why are tunnel states a workaround for memorylessness?
Because a standard Markov state forgets how long a patient has been in it, a single state cannot represent a risk that changes with time since entry. Tunnel states get around this by replacing that one state with a fixed sequence of temporary states through which a patient must pass in order, one per cycle, each carrying its own transition risk. Which tunnel state a patient is in then records how long they have been there, restoring the missing memory. It is a device to mimic time dependence within a memoryless framework. Briggs and colleagues (2006) describe it.
Source: Briggs et al. 2006
Why are tunnel states used?
Tunnel states are used to represent time-dependent risk that depends on how long a patient has been in a state, which the standard memoryless Markov model cannot capture, since its transitions depend only on the current state. By splitting a state into a sequence of tunnel states representing successive periods, each with appropriate transition probabilities, the model encodes the elapsed time in the state structure. This lets risks that vary with time since an event, such as high risk soon after a procedure, be represented while keeping the Markov form.
Source: Sonnenberg & Beck 1993
How do tunnel states work?
Tunnel states work by replacing a single state with a sequence of states through which patients pass one per cycle, each representing a specific period since entering, with its own transition probabilities reflecting the risk at that time. Patients enter the first tunnel state, move to the next each cycle, and can transition out to other states with time-specific probabilities. Because the tunnel state a patient occupies indicates how long they have been in the sequence, elapsed time is captured, approximating time-dependent risk.
Source: Sonnenberg & Beck 1993
When are tunnel states needed?
Tunnel states are needed when the risk of a transition depends on the time since a patient entered a state, a form of time dependency that violates the memoryless Markov assumption, and when representing this with an individual simulation is not chosen. For example, the risk of complications may be highest just after surgery and fall over time, which a single state cannot capture. Tunnel states let a cohort Markov model represent such time-in-state dependency by encoding elapsed time in a sequence of states.
Source: Sonnenberg & Beck 1993
What are the limitations of tunnel states?
Tunnel states increase the number of states, since each period since an event requires its own state, which can make the model larger and more complex, and they only approximate time-dependent risk to the resolution of the cycle and the number of tunnel states used. Many periods require many states, adding to data and computational demands and reducing transparency. Where time dependency is extensive, an individual-level simulation that tracks time in state directly may be simpler, so tunnel states suit limited time-dependency within a cohort model.
Source: Sonnenberg & Beck 1993
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Sep 2026
Content version: 1.0.0
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- Term code
- HE-EM-MM-030
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