System Dynamics Model

System Dynamics Model for Feedback and Capacity in Health Systems

A system dynamics model represents a health care problem as quantities that build up over time, such as patients on a waiting list, people with a chronic condition or trained staff, together with the rates that fill and drain them. Feedback links each accumulation back to those rates, and the model is solved by simulation, usually for the population in aggregate, to project the course of a policy once the system starts to respond. Health economists use it when feedback and delay drive the result, such as a workforce shortfall that takes years to correct, which a decision tree or a Markov model normally leaves out. An ISPOR task force names it as one of three dynamic simulation methods for health care delivery problems, alongside discrete event simulation and agent-based models.

Stocks, flows, feedback and delay

Marshall and colleagues, reporting for the ISPOR Simulation Modeling Emerging Good Practices Task Force, name feedback, accumulations (stocks), rates (flows) and time delays as the core elements of system dynamics. Stocks are accumulations of something such as people or beds, and flows feed in and out of them with the units of the stock per unit of time. The quantified core is a stock and flow model, in which each stock obeys a balance equation.

$$\frac{dS(t)}{dt} = I\big(S(t), X(t), \theta\big) - O\big(S(t), X(t), \theta\big)$$

where $S(t)$ is the vector of stocks at time $t$, $I$ and $O$ are the inflow and outflow rates, $X(t)$ holds external inputs such as a policy change, and $\theta$ is the vector of parameters. Feedback enters because the flows are functions of the stocks, so a change in any stock alters the rates that will change it next.

The task force ties nonlinearity to feedback, meaning an effect is seldom proportional to its cause, and states the central assumption of the method, that the behaviour of the system is due to its structure and not to external forces. The loops are usually mapped first in a causal loop diagram, which records the sign of each link and whether a loop reinforces or balances change.

Origins in industrial management

Marshall and colleagues trace the method to Jay Forrester at the Massachusetts Institute of Technology in the 1950s. Working with General Electric, he simulated by hand the stock, flow and feedback structure of its production plants, including the rules for hiring and layoffs, and showed that employment instability came from the internal structure of the firm rather than from the business cycle. His team built DYNAMO, the first computer simulator, and his book Industrial Dynamics appeared in 1961.

Davahli, Karwowski and Taiar reviewed 235 health care papers using system dynamics, published between 2000 and 2019, and found significant attention from 2013 onwards. Patient flow, obesity, workforce demand and HIV/AIDS were the most popular research areas.

Building and solving the equations

Davahli and colleagues describe the usual building sequence. The problem is stated, a qualitative diagram of the system is drawn, the diagram is converted into a quantitative simulation model, the model is tested, and its implications inform policy.

Crielaard and colleagues set out the conversion step. Each variable in the diagram is labelled as a stock, a constant or an auxiliary. Stocks accumulate, constants stay approximately fixed, and auxiliaries change their value instantly when the variables feeding them change. Time is then treated as discrete with a step chosen with the stocks in mind, and each stock is updated as follows.

$$S_{t+\Delta t} = S_t + \big(I_t - O_t\big)\,\Delta t$$

where $S_t$ is the stock at time $t$, $I_t$ and $O_t$ are the inflow and outflow rates at time $t$, and $\Delta t$ is the time step. Crielaard and colleagues note that such a difference equation implicitly uses Euler method integration with a fixed step, and that the step size and solver can significantly affect numerical error.

The ISPOR-SMDM task force report on dynamic transmission modelling by Pitman and colleagues notes that spreadsheets often use the Euler method although it is less accurate than other numerical techniques, whereas dedicated packages mostly offer several options for numerical integration.

An aggregate view of the population

Marshall and colleagues describe system dynamics models in health care as traditionally aggregate, counting the sizes of subpopulations over time rather than following individuals, with patterns, trends and mean values as outputs. Breeze and colleagues, in guidance on complex systems models for public health economic evaluation, likewise say that system dynamics typically takes an aggregate perspective, whereas agent-based models allow individual-level simulation of behaviour, heterogeneity and interaction.

In infectious disease work, Pitman and colleagues report that most dynamic transmission models have been built with system dynamics, with transitions between compartments represented by differential equations. Deterministic compartmental models of this kind suit the average behaviour of epidemics in large populations, while stochastic agent-based approaches may be preferred when chance effects or nonrandom mixing matter.

Worked example of a waiting list with a delayed capacity response

All figures are illustrative. A health board has 1,200 patients waiting for an elective procedure, receives 400 referrals a month and carries out 400 procedures a month, so the list is steady. It sets a target of 800 and a rule that capacity should cover referrals plus enough to remove the excess over four months. New theatre sessions take time to staff, so actual capacity closes the gap to the desired level with an average delay of three months. The model has two stocks, the list and the capacity, and runs in monthly steps.

$$C^{*}_t = R + \frac{W_t - W^{*}}{\tau}$$

where $C^{*}_t$ is the capacity the board wants at month $t$, $R$ is referrals of 400 a month, $W_t$ is the number waiting at month $t$, $W^{*}$ is the target of 800 and $\tau$ is the clearance time of 4 months.

$$W_{t+\Delta t} = W_t + \big(R - C_t\big)\,\Delta t$$

where $C_t$ is the number of procedures carried out per month at month $t$ and $\Delta t$ is the step of 1 month.

$$C_{t+\Delta t} = C_t + \frac{C^{*}_t - C_t}{D}\,\Delta t$$

where $D$ is the capacity delay of 3 months, so each month capacity closes one third of the gap between the desired and the actual level.

1. First month. Desired capacity is 400 + (1,200 − 800) / 4 = 500. The list does not move, since 1,200 + (400 − 400) × 1 = 1,200. Capacity rises to 400 + (500 − 400) / 3 = 433.33.

2. Second month. Desired capacity is still 500. The list falls to 1,200 + (400 − 433.33) = 1,166.67, and capacity rises to 433.33 + (500 − 433.33) / 3 = 455.56.

3. Third month. Desired capacity is now 400 + (1,166.67 − 800) / 4 = 491.67. The list falls to 1,166.67 + (400 − 455.56) = 1,111.11, and capacity rises to 455.56 + (491.67 − 455.56) / 3 = 467.59.

4. Three years of the same rules. The table continues the path and adds two comparisons, the list when capacity responds at once ($C_t = C^{*}_t$ every month) and the list under the delayed rule with a step of 0.05 months.

5. Interpretation. With an instant response the list settles on the target. With the delay, the extra sessions are still running when the list reaches 800 around month 8, so the list falls to about 721 at month 12, and capacity is then cut below the referral rate, to about 386 a month in months 15 and 16, before the list climbs back. Over 36 months both versions carry out about 14,800 procedures and record about 30,400 patient-months of waiting, so a model reporting only totals would miss the cycle of hiring and cutting sessions that the delay creates. The monthly step also exaggerates the dip. With a step of 0.05 months the low point is about 755 at around 13 months, so the coarse Euler step overstates the undershoot by about three quarters, 79 patients against 45.

Linking simulated stocks to costs and QALYs

An economic evaluation attaches values to the simulated trajectories. Costs and health effects can accrue per unit of time spent in a stock, such as a month on a waiting list, and per unit of flow, such as a procedure performed. In continuous time the discounted total over the time horizon is an integral.

$$E = \int_0^T \Big( \sum_k v_k\, S_k(t) + \sum_j g_j\, F_j(t) \Big) e^{-\rho t}\, dt$$

where $E$ is the discounted total cost or health outcome, $T$ is the time horizon, $S_k(t)$ is stock $k$ at time $t$, $v_k$ is the cost or QALY weight per unit of that stock per unit of time, $F_j(t)$ is flow $j$, $g_j$ is the value per unit of that flow, and $\rho = \ln(1 + r)$ is the continuous rate equivalent to the annual discount rate $r$, with $t$ in years. The integral is evaluated numerically, so the time step affects costs and QALYs as well as the stocks.

Marshall and colleagues describe the ReThink Health model, which tracks health status, utilisation and costs and was used by Milstein and colleagues to compare five health reform proposals. Breeze and colleagues note that there are few examples of economic evaluations that use complex systems models, and most applications above are planning and policy studies rather than formal cost-effectiveness analyses.

Choosing system dynamics over other model types

The SIMULATE checklist of Marshall and colleagues lists eight elements that mark a problem as needing dynamic simulation rather than a Markov model or a decision tree, one of which asks whether feedback loops change future interactions and their consequences for the delivery system. It contrasts dynamic simulation models, generally nonlinear with open populations, with linear Markov cohort models.

The same report places discrete event simulation in problems of resource use and queues, where entities with attributes consume resources. Its system dynamics examples are high-level, such as an Alberta model for system-wide planning of hip and knee osteoarthritis care with feedback loops across the whole care pathway. Davahli and colleagues list the inability to model low-level interactions as a main disadvantage, which is where agent-based models take over. Because flows are written as rates in differential equations, system dynamics shares the continuous-time logic of a continuous model, while the broader dynamic model label covers any model whose state changes over time through feedback and interaction.

Validation, calibration and uncertainty

Davahli and colleagues grouped validation into direct structural tests, structure-oriented behavioural tests such as extreme-condition and sensitivity tests, and behaviour pattern tests against data, which they treat as calibration. Of the 187 quantitative studies in their review, 115 discussed calibration, 57 reported direct structural tests and 104 ran sensitivity tests. They call heavy reliance on qualitative validation a weakness of the method.

Breeze and colleagues add that Bayesian calibration gives distributions compatible with probabilistic sensitivity analysis, and that calibrating many parameters risks overfitting and makes model validation against external data harder. They also note that complex systems models can need more assumptions not closely tied to strong experimental evidence, and that model structures as well as parameters may be uncertain, so structural uncertainty and scenario analysis of alternative loop structures belong beside parameter variation. Pitman and colleagues ask that the differential equations be published with the results.

Where system dynamics models are misread

Several errors recur when system dynamics results reach decision makers. Most ask the model for more precision or detail than its aggregate structure can give.

1. Reading a trajectory as a point forecast. The outputs are patterns, trends and means, and a month-by-month figure depends on the assumed delays and on the solver.
2. Ignoring the time step. The worked example shows a monthly Euler step overstating an undershoot, so results need checking with a smaller step or a better solver.
3. Assuming the aggregate is enough. Averaging over a subpopulation hides heterogeneity and chance effects.
4. Treating the diagram as the model. A causal loop diagram shows which way loops push, but only quantified stocks, flows and delays show how large and fast the effects are.

Sources

• Breeze PR, Squires H, Ennis K, Meier P, Hayes K, Lomax N, et al. Guidance on the use of complex systems models for economic evaluations of public health interventions. Health Economics. 2023;32(7):1603-1625. https://doi.org/10.1002/hec.4681
• Crielaard L, Uleman JF, Châtel BDL, Epskamp S, Sloot PMA, Quax R. Refining the causal loop diagram: a tutorial for maximizing the contribution of domain expertise in computational system dynamics modeling. Psychological Methods. 2024;29(1):169-201. https://doi.org/10.1037/met0000484
• Davahli MR, Karwowski W, Taiar R. A system dynamics simulation applied to healthcare: a systematic review. International Journal of Environmental Research and Public Health. 2020;17(16):5741. https://doi.org/10.3390/ijerph17165741
• Marshall DA, Burgos-Liz L, IJzerman MJ, Osgood ND, Padula WV, Higashi MK, Wong PK, Pasupathy KS, Crown W. Applying dynamic simulation modeling methods in health care delivery research: the SIMULATE checklist. Report of the ISPOR Simulation Modeling Emerging Good Practices Task Force. Value in Health. 2015;18(1):5-16. https://doi.org/10.1016/j.jval.2014.12.001
• Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health. 2012;15(6):828-834. https://pmc.ncbi.nlm.nih.gov/articles/PMC7110742/