Concept Architecture
Concept
Theoretically, a Continuous Model is a mathematical model in which time, state variables or both are represented as continuous quantities that change without discrete intervals. Continuous models describe dynamic processes using continuous functions and are particularly suited to biological, epidemiological and physiological systems where changes occur continuously rather than at fixed time points. In health economics, continuous models are used to represent disease progression, population dynamics and optimisation problems where discrete-cycle approximations may introduce unnecessary error.
Mathematically, continuous models are represented using differential equations that describe the instantaneous rate of change of one or more state variables. Solutions may be obtained analytically for simple systems or numerically for more complex models. Costs and health outcomes are commonly accumulated by integrating continuous functions over time, avoiding the need for discrete cycle corrections.
In practice, continuous models are parameterised using clinical, epidemiological and demographic data and implemented using numerical methods and specialised modelling software. They are applied in infectious disease transmission modelling, pharmacokinetic and pharmacodynamic modelling, continuous-time Markov models and health economic optimisation problems where continuous representation improves accuracy or computational efficiency.
Purpose
Used to represent continuously evolving health processes, estimate long-term health and economic outcomes without discrete time cycles, and support economic evaluations requiring continuous-time modelling.
Mathematical Formulae
Primary Formula
Continuous-time state equation:
dx(t)/dt = f(x(t), t)
where:
- x(t) = state variable at time t
- f(�) = function describing the rate of change
Supporting Formulae
Continuous accumulation of outcomes:
Y = ??? y(t) dt
Discounted continuous outcome:
Y = ??? y(t)e??? dt
where:
- Y = cumulative outcome
- y(t) = outcome rate at time t
- T = time horizon
- r = continuous discount rate
Related Mathematical Methods
- Ordinary differential equations
- Continuous-time Markov modelling
- Numerical integration
- Dynamic systems modelling
- Continuous optimisation
- Survival analysis
Example
An infectious disease model assumes that the number of infectious individuals changes continuously according to:
dI/dt = 0.25I ? 0.10I
The net growth rate is:
0.25 ? 0.10 = 0.15
Starting with 500 infectious individuals, the model continuously estimates disease prevalence over time. The predicted numbers of infections are then combined with treatment costs and QALYs to evaluate alternative intervention strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =B2*EXP((GrowthRate-RecoveryRate)*A2) | Calculates continuous exponential growth or decline in disease models. |
| LN | =LN(B2/B1)/(A2-A1) | Estimates continuous growth or transition rates from observed data. |
| Solver | Solver Optimisation | Estimates continuous-time model parameters from clinical or epidemiological data. |
| SUMPRODUCT | =SUMPRODUCT(B2:B101,C2:C101) | Approximates continuously accumulated costs or health outcomes using numerical integration. |
VBA (Optional)
Automate numerical solution of continuous-time health economic models and generate discounted costs and health outcomes over the specified time horizon.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Siebert U, Alagoz O, Bayoumi AM, et al. State-Transition 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.
- Boyce WE, DiPrima RC. Elementary Differential Equations and Boundary Value Problems. Wiley.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
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 (6)
What is a continuous model?
A mathematical model in which time or another key variable flows continuously rather than advancing in discrete steps.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
What is an example of a continuous model in health?
Continuous models let a quantity change smoothly at every instant rather than only at set intervals. Compartmental models of infection written as differential equations are a common example, since they express the rate at which people move between susceptible, infected, and recovered groups as a continuous flow. Pharmacokinetic models of how a drug concentration rises and falls over time are another. Such models suit processes that genuinely unfold without natural steps. Sterman (2000) describes continuous formulations.
Source: Sterman 2000
How does a continuous model represent change?
A continuous model represents change through rates that apply at every instant, usually expressed as differential equations giving the rate at which quantities change with time. Solving these equations, analytically or numerically, gives the values of the quantities at any point in time. Because change is treated as smooth and continuous, the model does not need to choose a cycle length, and it can capture events occurring at any moment rather than only at fixed intervals.
Source: Briggs, Claxton & Sculpher 2006
How does a continuous model differ from a discrete model?
A continuous model treats time as flowing smoothly, with change occurring at every instant, whereas a discrete model advances time in fixed steps, updating the state only at each step. Continuous models suit processes where events can happen at any time and avoid the approximation of grouping events into cycles, but they can be harder to formulate and solve. Discrete models are simpler to implement and often adequate, so the choice depends on whether the timing of events matters finely.
Source: Briggs, Claxton & Sculpher 2006
When is a continuous model appropriate?
A continuous model is appropriate when the timing of events matters finely and treating time as flowing smoothly better represents the process than fixed cycles, for instance in infectious-disease transmission or where rates change continuously. It avoids the errors that arise when a discrete cycle is too long to capture rapid change. Where events are naturally periodic or the extra precision is unnecessary, a discrete model may be simpler and sufficient, so continuous representation is chosen when it materially improves the model.
Source: Briggs, Claxton & Sculpher 2006
What are the challenges of continuous models?
Continuous models can be harder to formulate, since they require expressing the process as differential equations, and harder to solve, often needing numerical methods that introduce their own approximation. They can be less transparent than discrete models and more demanding to build and check. The continuous treatment of time may also exceed the precision the data support. These challenges mean continuous models are used where their advantages in representing timing justify the added complexity over a discrete approach.
Source: Briggs, Claxton & Sculpher 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 29 Sep 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-EM-DM-014
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