Biology Reference
In-Depth Information
These are PDEs with stochastic terms that can be used to model proba-
bilistic processes such as the spread of epidemics, neuronal signal trans-
duction,
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or evolution theory. In discrete stochastic models, probabilistic
effects mostly pertain to discrete random events. These events are
characterized by their probability density functions. Examples include
population dynamics (individuals have certain probabilities to be born,
die, eat, or be eaten), random walks of diffusing molecules, or stochasti-
cally occurring chemical reactions. Several methods are also available for
combining stochastic and deterministic models into hybrid stochastic-
deterministic models.
59,60
4. Spatiotemporal Simulation Methods
Depending on the modeling technique chosen for a given system (Fig. 2),
different numerical methods exist for simulating the resulting model in a
computer. While it is impossible to give an exhaustive list of all available
methods, we will highlight the most important ones for each category of
models. The same numerical methods can be used for both physical and
phenomenological models.
4.1. Methods for Discrete Stochastic Models
Discrete stochastic models as formulated by events occurring according
to certain probability distributions can be simulated using stochastic
simulation algorithms (SSAs) such as Gillespie's algorithm
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or the
Gibson-Bruck algorithm.
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While most of these algorithms were origi-
nally developed for temporal dynamics only, they have since been gen-
eralized to spatiotemporal models such as reaction-diffusion
models.
64,65
Monte Carlo methods
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provide the basis for most of
these algorithms. Simulating probabilistic trajectories of the model thus
amounts to sampling from the model probability distributions. In
order to estimate the average trajectory or the standard deviation,
ensemble averages over many simulations must be computed. This fun-
damentally limits the convergence properties of these methods to
O
N
), where
N
is the number of simulations performed.
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Agent-
based methods with probabilistic agents are also frequently used to
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