Concept Architecture
Concept
Theoretically, Cycle is the fixed interval of time over which transitions between health states are evaluated in a cohort-based state-transition model. During each cycle, all individuals are assumed to occupy a defined health state before moving to subsequent states according to specified transition probabilities. Cycles provide the temporal framework for modelling disease progression, intervention effects, costs and health outcomes over the chosen time horizon.
Mathematically, a cycle represents one iteration of the state-transition process. At the end of each cycle, the cohort state vector is updated by multiplication with the transition probability matrix. Repeated application of this process generates the evolution of the cohort over successive cycles and enables estimation of cumulative costs and health outcomes.
In practice, the cycle is selected to reflect the natural timing of disease progression, treatment administration or clinical follow-up. Common cycle lengths include one week, one month, three months or one year. Costs, utilities and transition probabilities are applied within each cycle, with half-cycle correction frequently used to reduce bias arising from the assumption that transitions occur only at cycle boundaries.
Purpose
Used to divide the model time horizon into discrete intervals over which health state transitions, costs and health outcomes are evaluated in cohort-based state-transition models.
Mathematical Formulae
Primary Formula
State update for one cycle:
????? = ?????
where:
- ??? = cohort state vector at cycle t
- ?? = transition probability matrix
Supporting Formulae
State distribution after n cycles:
??? = ?????�
Total expected cost:
C = ????? ?????
Total expected health outcome:
E = ????? ?????
where:
- ?? = vector of state-specific costs
- ?? = vector of state-specific health outcomes
Related Mathematical Methods
- Markov modelling
- Matrix algebra
- State-transition modelling
- Cohort simulation
- Half-cycle correction
Example
A Markov model evaluating chronic heart failure uses monthly cycles over a five-year horizon.
Number of cycles:
5 ? 12 = 60
At the end of each monthly cycle, the cohort distribution is updated using the transition probability matrix, and monthly costs and quality-adjusted life years are accumulated to estimate the intervention's cost-effectiveness.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(StateVector,TransitionMatrix) | Update the cohort after each cycle |
| SEQUENCE | =SEQUENCE(60,1,0,1) | Generate cycle numbers across the model horizon |
| SUMPRODUCT | =SUMPRODUCT(StateVector,CostVector) | Calculate expected cost per cycle |
| SUMPRODUCT | =SUMPRODUCT(StateVector,UtilityVector) | Calculate expected QALYs per cycle |
| ROW | =ROW()-1 | Index successive model cycles |
VBA (Optional)
Automate repeated cycle calculations, updating cohort distributions and accumulating health economic outcomes throughout the model horizon.
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.
- 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
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 cycle in a Markov model?
A discrete unit of time in a Markov model during which patients may transition between health states, dividing the whole time horizon.
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 does a Markov model advance in cycles rather than continuously?
A Markov model divides the whole time horizon into equal cycles and updates the cohort only once per cycle, applying the transition probabilities at each step. This discrete stepping suits transition data, which are usually estimated as a probability of moving over a fixed period, and it keeps the calculation to a manageable sequence of updates. The alternative of tracking change at every instant would demand continuous-time methods and rarely adds accuracy worth the complexity. Sonnenberg and Beck (1993) describe this cyclic operation.
Source: Sonnenberg & Beck 1993
What happens during a cycle?
During a cycle, patients may move between health states according to the transition probabilities, so the distribution of the cohort across states is updated, and the costs and health effects associated with the states occupied are accrued for that period. Transitions are treated as occurring within the cycle, and the state values apply over its duration. At the end of the cycle, the model advances to the next, repeating the process, so each cycle represents one step of the cohort's progression through the model.
Source: Sonnenberg & Beck 1993
How do cycles represent time in a Markov model?
Cycles represent time in a Markov model by dividing the time horizon into equal discrete periods, with the model advancing one cycle at a time. At each cycle, transitions occur and costs and effects accrue, so the passage of time is represented as a sequence of these steps. The number of cycles times the cycle length equals the horizon. This discrete-time representation, advancing cycle by cycle, is how the Markov model tracks the cohort's changing state and accumulating outcomes over time.
Source: Sonnenberg & Beck 1993
Why is the timing of events within a cycle addressed?
The timing of events within a cycle is addressed because transitions are usually modelled as occurring at a single point, such as the cycle's start or end, whereas in reality they occur throughout, which can introduce error in the accrual of costs and effects. The half-cycle correction is commonly applied, treating transitions as occurring on average at the middle of the cycle, to reduce this error. Addressing within-cycle timing improves the accuracy of the model, particularly when cycles are long relative to the rate of change.
Source: Sonnenberg & Beck 1993
How does the cycle relate to cycle length?
The cycle is the discrete time step of a Markov model, and cycle length is the duration that one cycle represents, such as one month or one year. The cycle length determines how finely time is divided and how many cycles span the horizon. A shorter cycle length gives more cycles and finer temporal resolution, while a longer one gives fewer, coarser steps. The cycle is the unit of advance, and its length sets the scale of the time step, so the two are directly linked.
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
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/cycle
- Term code
- HE-EM-MM-004
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