Discrete Model

Discrete Model: Concept Architecture

Orientation and learning roadmap

This page shows which part of a model may be represented in separate categories or steps. It then compares fixed-cycle and event-driven timing, works through a small health-state calculation, and explains how the chosen resolution affects costs, outcomes, and validation.

Identify what is discrete

A modeller can divide health into a finite set of states, advance time in fixed cycles, or process individual events one at a time. These are different modelling choices: a model with discrete health states may still place events at exact continuous times, while a model with fixed time steps may contain continuously valued costs or patient characteristics.

Specify the discrete element explicitly rather than inferring it from the label alone. For example, a cohort state-transition model commonly updates the proportions in mutually exclusive health states at the end of each monthly cycle; an individual discrete-event simulation jumps between event times that need not be equally spaced. ISPOR–SMDM guidance on state-transition modelling and its discrete-event simulation guidance describe these distinct implementations.

Fixed-cycle health-state calculation

In a cohort model, a vector records how much of the population occupies each state at a cycle boundary. A transition matrix gives the probability of moving from each current state to each next state over the chosen cycle length, including remaining in the same state.

Using a row vector $\boldsymbol{s}_t$ and transition matrix $P_t$, the next-cycle distribution is:

$$\boldsymbol{s}_{t+1}=\boldsymbol{s}_tP_t,\qquad \sum_j P_{t,ij}=1\quad\text{for every current state }i.$$

The matrix entries must be nonnegative, and each row must sum to one for mutually exclusive and exhaustive destinations. Transition probabilities must correspond to the model's cycle duration; an annual probability cannot simply be used as a monthly probability. Costs and health benefits accrued between cycle boundaries require an explicit timing convention, which may include a half-cycle correction or a more appropriate event-time calculation.

Worked two-state example

Suppose everyone begins alive, and the probability of dying during one monthly cycle is $0.02$. With states ordered alive and dead, the starting distribution is $(1,0)$ and the monthly transition matrix is $\begin{pmatrix}0.98&0.02\\0&1\end{pmatrix}$.

After one cycle, the model assigns $0.98$ of the cohort to alive and $0.02$ to dead; after two equal cycles with unchanged probabilities, alive occupancy is $0.98^2=0.9604$. If the cycle starts with 100 people, the expected alive counts at successive boundaries are 100, 98, and 96.04. These are cohort expectations, not a claim that a real population contains a fractional person.

Fixed cycles and event-driven timing

Fixed-cycle models update state membership on a shared time grid, which can simplify transparent accounting across treatment strategies. Event-driven models instead advance a simulation clock to the next scheduled event, which can represent irregular event times and queues more naturally.

Neither approach is automatically more accurate; adequacy depends on the decision question, available data, and clinically important timing. The ISPOR–SMDM model-selection overview treats model structure as a choice to be justified by the problem rather than a universal hierarchy of methods.

Consequences for economic evaluation

In a cost-effectiveness analysis, discrete health states can carry costs and health-related quality-of-life values while transitions change the time spent in each state. The selected cycle length or event resolution affects estimates when treatment effects, resource use, or mortality change rapidly.

Check whether recurrent events, treatment history, or competing risks require more states, individual memory, or event-driven logic. A very short cycle can reduce some timing approximations but increase computation and may not repair an inappropriate state structure. Test plausible cycle lengths and timing conventions when they can alter incremental cost or quality-adjusted life-years.

Validation and interpretation

Validate the state definitions against clinical pathways, check that transition probabilities match their time unit, and compare simulated survival or event counts with relevant external evidence. For a fixed-cycle cohort model, inspect initial occupancy, row sums, absorbing states, and whether accumulated time, costs, and outcomes use consistent within-cycle assumptions.

For an event-driven model, inspect the event calendar, ties, censoring, reproducibility, and simulated event-time distributions. Keep structural uncertainty separate from random simulation error and parameter uncertainty. The term “discrete model” by itself does not determine whether the process is deterministic or stochastic, whether time is continuous, or whether its outputs are integer-valued.