Discrete vs continuous models
The terms discrete and continuous describe how time, state, or both are represented. This distinction is separate from whether a model is deterministic or stochastic.
Discrete time
A discrete-time model observes or updates the system at separated times
\[t_0,t_1,t_2,\ldots\]A population model might be written
\[N_{n+1}=f(N_n),\]where \(n\) labels successive time steps.
Continuous time
In continuous time, the system is defined for every time \(t\) in an interval. A deterministic example is
\[\frac{dN}{dt}=f(N).\]Continuous time does not necessarily mean continuous state. A continuous-time Markov chain, for example, can evolve continuously in time while its state is an integer-valued population count.
Discrete and continuous state
A discrete state takes values from a separated set, such as
\[I\in\{0,1,2,\ldots,N\}.\]A continuous state can take values throughout an interval, for example a concentration \(C\geq0\).
| Feature | Discrete | Continuous |
|---|---|---|
| Time | Specified time steps | Every time in an interval |
| State | Separated values | Values over a continuum |