System Dynamics Model

Concept

Theoretically, System Dynamics Model is a continuous-time simulation modelling approach used to represent the behaviour of complex systems through the interaction of stocks, flows, feedback loops and time delays. Originating from control theory and system science, it represents the dynamic evolution of populations, resources or processes over time. In health economics, system dynamics models are used to analyse healthcare systems, disease progression, workforce planning and resource allocation where feedback mechanisms influence long-term outcomes.

Mathematically, a system dynamics model is represented as a system of coupled ordinary differential or difference equations describing changes in stock variables over time. Stocks accumulate inflows and outflows, while auxiliary variables and feedback relationships determine flow rates. The mathematical framework estimates the evolution of the system under alternative assumptions, interventions or policies.

In practice, system dynamics models are developed by defining stocks, flows, feedback structures and governing equations using specialist simulation software or numerical methods. Parameters are estimated from epidemiological, demographic, clinical and economic data. Model validation typically includes structural validation, behavioural validation, sensitivity analysis and calibration before application to health policy or economic evaluation.

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Purpose

Used to simulate the dynamic behaviour of complex healthcare systems, evaluate long-term policy interventions, analyse feedback effects and estimate the health and economic consequences of system-level change.

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Mathematical Formulae

Primary Formula

For stock S(t):

dS(t)/dt = I(t) ? O(t)

where:

• S(t) = stock at time t
• I(t) = inflow rate
• O(t) = outflow rate

Equivalent discrete-time representation:

S??? = S? + (I? ? O?)?t

Supporting Formulae

General state equation:

d??(t)/dt = ??(??(t), ??(t), ?)

Feedback relationship:

I(t) = f(S(t), X(t), ?)

Related Mathematical Methods

• Ordinary differential equations
• Difference equations
• Numerical integration
• Feedback system analysis
• System simulation
• Calibration
• Sensitivity analysis

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Example

A health authority models hospital bed occupancy.

Initial occupied beds:

S? = 500

Daily admissions:

I = 45

Daily discharges:

O = 40

After one day:

S? = 500 + (45 ? 40) = 505

The model links bed occupancy to delayed discharge rates, emergency admissions and staffing levels through feedback loops to estimate long-term healthcare costs and capacity requirements under alternative policies.

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Excel Implementation

Function	Example Formula	Health Economics Application
SUM	=B2+C2-D2	Update stock level each time period
IF	=IF(B2>Capacity,HighRate,BaseRate)	Model feedback triggered by capacity constraints
EXP	=EXP(-Rate*Time)	Represent exponential adjustment processes
INDEX	=INDEX(ParameterTable,MATCH(Scenario,ScenarioList,0),2)	Retrieve scenario-specific parameters
Data Table	What-If Analysis	Perform deterministic sensitivity analysis across policy scenarios

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VBA (Optional)

Automate repeated simulation runs across multiple intervention scenarios and export comparative health and economic outcomes.

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Sources

• Forrester JW. Industrial Dynamics. MIT Press; 1961.
• Forrester JW. Principles of Systems. MIT Press; 1968.
• Sterman JD. Business Dynamics: Systems Thinking and Modeling for a Complex World. McGraw-Hill; 2000.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
• ISPOR-SMDM Modeling Good Research Practices Task Force reports.
• Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.