Dynamic Model

Concept

Theoretically, a Dynamic Model is a mathematical model that explicitly represents how the state of a healthcare system evolves over time as a consequence of interactions between its components. Unlike static models, dynamic models allow current outcomes to influence future system behaviour, thereby capturing feedback mechanisms, changing population structures and time-dependent processes. In health economics, dynamic models are used to evaluate interventions where disease transmission, population behaviour or resource utilisation changes over time, such as vaccination programmes, infectious disease control and long-term healthcare planning.

Mathematically, a dynamic model is represented using systems of differential equations, difference equations or state-transition equations that update the system through time. The future state depends on both the current state and model parameters, allowing feedback effects and temporal interactions to be incorporated. The mathematical framework supports deterministic or stochastic formulations depending on the objectives of the analysis.

In practice, dynamic models are parameterised using epidemiological, clinical and demographic data and calibrated against observed outcomes. They are implemented as dynamic transmission models, compartmental models, dynamic state-transition models or system dynamics models. Outputs are used to estimate long-term costs, quality-adjusted life years, budget impact and population health effects of healthcare interventions while accounting for indirect effects and temporal changes.

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Purpose

Used to model time-dependent healthcare systems, estimate the long-term health and economic consequences of interventions that alter future disease or resource dynamics, and support policy decisions where feedback effects are important.

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Mathematical Formulae

Primary Formula

General dynamic system:

dx(t)/dt = f(x(t), u(t), t)

where:

• x(t) = system state at time t
• u(t) = external inputs or interventions
• f(�) = function governing system dynamics

For discrete-time models:

x??? = f(x?)

Supporting Formulae

State-transition equation:

????? = ?????(t)

where:

• ??(t) = time-dependent transition probability matrix

Related Mathematical Methods

• Dynamic transmission modelling
• Compartmental modelling
• System dynamics modelling
• Differential equation modelling
• Difference equation modelling
• Dynamic programming
• Model calibration

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Example

A vaccination programme is evaluated using a dynamic transmission model.

Initially:

• Susceptible population = 900,000
• Infectious population = 5,000

Vaccination reduces transmission over successive years, causing the effective force of infection to decline. Unlike a static model, the probability of future infection changes as fewer infectious individuals remain in the population. The model therefore estimates both the direct health gains experienced by vaccinated individuals and the indirect benefits arising from reduced disease transmission, leading to lower long-term healthcare costs and greater QALY gains.

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Excel Implementation

Function	Example Formula	Health Economics Application
MMULT	=MMULT(B2:D2,B5:D7)	Updates state distributions in dynamic state-transition models.
EXP	=B2*EXP(GrowthRate*A2)	Models continuously changing population or disease dynamics.
SUMPRODUCT	=SUMPRODUCT(B2:B20,C2:C20)	Calculates expected costs and health outcomes over time.
Solver	Solver Optimisation	Calibrates dynamic model parameters to observed epidemiological data.

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VBA (Optional)

Automate iterative updating of dynamic health economic models across multiple time periods and generate projected costs, health outcomes and cost-effectiveness results.

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Sources

• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Pitman R, Fisman D, Zaric GS, et al. Dynamic Transmission Modeling: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force. Value in Health. 2012.
• Keeling MJ, Rohani P. Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.