Memorylessness

Concept

Theoretically, Memorylessness is the defining property of a Markov process whereby the probability of a future transition depends only on the current state and not on the sequence of previous states or the time already spent in that state. This property is known as the Markov property and underpins conventional Markov models used in health economics. Memorylessness simplifies the representation of disease progression but may be inappropriate when transition risks depend on patient history or state duration.

Mathematically, memorylessness is expressed as a conditional independence property. The future state of the process is independent of all previous states once the current state is known. For continuous-time processes, the exponential distribution is the only continuous probability distribution possessing the memoryless property, making it fundamental to continuous-time Markov models.

In practice, memorylessness is assumed when transition probabilities are estimated solely according to the patient's current health state. When empirical evidence indicates that previous events, treatment history or time spent in a health state influence future risks, model extensions such as tunnel states, semi-Markov models or individual-level microsimulation are commonly employed to relax the assumption.

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Purpose

Used to simplify state-transition modelling by assuming that future transitions depend only on the current health state, facilitating estimation of long-term disease progression and economic outcomes.

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Mathematical Formulae

Primary Formula

Markov property:

P(X??? = j | X? = i, X???, ?, X?) = P(X??? = j | X? = i)

where:

• X? = state occupied at time t
• i, j = health states

Supporting Formulae

Memoryless property of the exponential distribution:

P(T > s + t | T > s) = P(T > t)

Exponential survival function:

S(t) = e????????

or equivalently,

S(t) = e???

where:

• T = time-to-event
• ? = constant hazard rate

Related Mathematical Methods

• Markov chains
• Continuous-time Markov processes
• Exponential distribution
• State-transition modelling
• Semi-Markov modelling
• Tunnel state modelling

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Example

A three-state Markov model for chronic heart failure assumes that patients in the stable disease state have the same annual probability of hospitalisation regardless of whether they entered the state one month or five years earlier. If clinical evidence demonstrates that hospitalisation risk increases with time since diagnosis, the memorylessness assumption is violated and a semi-Markov model or tunnel states may be introduced to represent duration-dependent risks.

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Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=EXP(-HazardRate*Time)	Calculate exponential survival under the memoryless assumption
IF	=IF(RAND()<TransitionProb,NextState,CurrentState)	Simulate state transitions based solely on the current state
RAND	=RAND()	Generate random transitions in stochastic Markov simulations
MMULT	=MMULT(StateVector,TransitionMatrix)	Update cohort state distributions under the Markov assumption

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VBA (Optional)

Automate repeated Markov state transitions while applying fixed transition probabilities that depend only on the individual's current health state.

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Sources

• Norris JR. Markov Chains. Cambridge University Press.
• Ross SM. Introduction to Probability Models. 12th ed. Academic 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.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press.