Concept Architecture
Concept
Theoretically, the Stock and Flow Model is a mathematical modelling framework used to represent dynamic systems by describing how quantities accumulate over time and how they are increased or decreased by inflows and outflows. It is founded on system dynamics and conservation principles, whereby stocks represent state variables and flows represent rates of change. In health economics, stock and flow models are used to analyse disease progression, healthcare workforce dynamics, patient pathways, resource utilisation and population health over time.
Mathematically, a stock and flow model is represented by a system of differential or difference equations in which each stock changes according to the net balance of inflows and outflows. The mathematical framework estimates the evolution of system states over time and captures feedback mechanisms, delays and nonlinear relationships. Continuous-time models are commonly expressed as ordinary differential equations, while discrete-time implementations use recursive update equations.
In practice, stock and flow models are implemented by defining stocks, specifying flow equations, estimating model parameters from empirical data and numerically integrating the resulting equations over time. In health economics they are applied to model disease transmission, chronic disease progression, hospital occupancy, healthcare workforce planning, pharmaceutical supply chains and population health interventions.
Purpose
Used to model dynamic healthcare systems by representing the accumulation and depletion of populations or resources over time, thereby supporting forecasting, policy evaluation and economic analysis.
Mathematical Formulae
Primary Formula
Continuous-time formulation:
dS(t)/dt = ? Inflows ? ? Outflows
where:
- S(t) = stock at time t
Discrete-time formulation:
S??? = S? + (? Inflows ? ? Outflows)?t
Supporting Formulae
General flow function:
F = f(S, ?, t)
where:
- F = flow rate
- S = stock variables
- ? = model parameters
Related Mathematical Methods
- System dynamics
- Ordinary differential equations
- Difference equations
- Numerical integration
- Feedback modelling
- Compartmental modelling
- Sensitivity analysis
Example
A health authority models hospital bed occupancy.
Initial occupied beds:
B? = 250.
Daily admissions:
25 patients/day.
Daily discharges:
20 patients/day.
Using the stock update equation:
B? = 250 + (25 ? 20) = 255.
After one day, the stock of occupied beds increases to 255 beds. Repeated numerical integration of the stock equation forecasts future bed occupancy under alternative admission and discharge scenarios.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:C2)-SUM(D2:E2) | Calculates net flow into a stock. |
| SUMPRODUCT | =SUMPRODUCT(B2:B10,C2:C10) | Computes flow rates influenced by multiple parameters. |
| IF | =IF(B2>Capacity,Overflow,B2) | Represents capacity constraints within the system. |
| Solver Add-in | Optimises flow parameters to minimise shortages or costs | Supports calibration and policy optimisation. |
VBA (Optional)
A VBA procedure can automate time-step simulation by repeatedly updating stocks from inflows and outflows and generating long-term projections under alternative policy scenarios.
Sources
- Forrester JW. Industrial Dynamics. MIT Press.
- Sterman JD. Business Dynamics: Systems Thinking and Modeling for a Complex World. McGraw-Hill.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on decision-analytic modelling.
Related Concepts (2)
Library
Publications
1
A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →
Frequently Asked Questions (7)
What is a stock and flow model?
A system dynamics structure representing accumulations, called stocks, such as infected individuals, and the rates of change into and out of them, called flows.
Source: Sterman 2000
What is a stock-and-flow model?
A stock-and-flow model is a system dynamics structure representing accumulations, called stocks, such as the number of infected individuals, and the rates of change into and out of them, called flows. Stocks are quantities that build up or deplete over time, and flows are the rates that add to or subtract from them. Described by Sterman, stock-and-flow models capture how quantities accumulate and change through their inflows and outflows, forming the basis of system dynamics simulation of complex systems over time.
Source: Sterman 2000
What system-dynamics tool is a stock-and-flow model?
A stock-and-flow model is a system-dynamics tool that represents quantities accumulating in stocks and the rates, called flows, that raise or lower them. Stocks capture the state of a system at a moment, such as the number of people with a condition, while flows describe how quickly that number changes. Feedback loops link stocks and flows, so that the level of a stock can influence the rates acting on it. This makes the approach suited to problems with delays and accumulation over time. It has been applied to chronic disease and service capacity.
Source: Sterman 2000
What are stocks and flows?
In a stock-and-flow model, stocks are accumulations, quantities that exist at a point in time and change only through their flows, such as a population, the number of infected people, or a budget. Flows are the rates of change, the quantities per unit time that add to a stock, inflows, or remove from it, outflows, such as infection and recovery rates. Stocks integrate their flows over time, so the level of a stock reflects the accumulated difference between what has flowed in and out.
Source: Sterman 2000
How do stock-and-flow models represent system behaviour?
Stock-and-flow models represent system behaviour by tracking how stocks change over time through their flows, with the flows often depending on the stocks and on other variables, creating feedback. As a simulation runs, the stocks accumulate their net inflows, and the resulting levels influence the flows, so the system evolves dynamically. This structure captures accumulation, delay, and feedback, generating the patterns of growth, decline, and oscillation that characterise complex systems, which is why stock-and-flow models underlie system dynamics analysis.
Source: Sterman 2000
How do stock-and-flow models relate to feedback?
Stock-and-flow models relate to feedback because the flows that change stocks frequently depend on the stocks themselves, creating feedback loops: a stock influences a flow, which changes the stock. For example, more infected individuals raise the infection flow, increasing the infected stock further, a reinforcing loop, until susceptibles deplete, a balancing effect. The combination of stocks, flows, and the feedback between them produces the system's dynamic behaviour, so stock-and-flow structures are the means by which system dynamics represents feedback quantitatively.
Source: Sterman 2000
How are stock-and-flow models used in health?
In health, stock-and-flow models are used in system dynamics analysis to represent how quantities accumulate and change over time with feedback, such as the numbers of susceptible, infected, and recovered people in an epidemic, patients accumulating in a care pathway, or the size of a waiting list. By capturing stocks, flows, and feedback, they simulate how such systems behave and respond to interventions over time. They support understanding of complex, feedback-rich health problems and testing policies before implementation.
Source: Sterman 2000
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Verified by Dr Darrin Baines
British health economist
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Verification date: 3 Oct 2025
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