<title>Discrete Stochastic Simulations</title>
<h1>Discrete Stochastic Simulations</h1>
We consider discrete stochastic simulations that are modelled with a set
of species and a set of reactions that transform the species' amounts.
Instead of using a
continuum approximation and dealing with species mass or concentration,
the amount of each species is a non-negative integer which is the population.
Depending on the species, this could be the number of molecules or the
number of organisms, etc. Reactions transform a set reactants into a set of
products, each being a linear combination of species with integer coefficients.
Consider a system of <em>N</em> species represented by the state vector
<em>X(t) = (X<sub>1</sub>(t), ... X<sub>N</sub>(t))</em>.
<em>X<sub>n</sub>(t)</em> is the population of the
<em>n<sup>th</sup></em> species at time <em>t</em>.
There are <em>M</em> reaction channels
which change the state of the system. Each reaction is characterized by
a propensity function <em>a<sub>m</sub></em> and a state change vector
<em>V<sub>m</sub> = (V<sub>m1</sub>, ..., V<sub>mN</sub>)</em>.
<em>a<sub>m</sub> dt</em> is the
probability that the <em>m<sup>th</sup></em> reaction will occur in the
infinitesimal time interval <em>[t .. t + dt)</em>. The state
change vector is the difference between the state after the reaction and
before the reaction.
To generate a trajectory (a possible realization of the evolution of the
system) one starts with an initial state and then repeatedly fires reactions.
To fire a reaction, one must answer the two questions:
<li> When will the next reaction fire?
<li> Which reaction will fire next?
Let the next reaction have index μ and fire at time <em>t + τ</em>.
Let α be the sum of the propensities. The time to the
next reaction is an exponentially distributed random variable with mean
1 / α ; the probability density function is
<em>P(τ = x) = α e<sup>- α x</sup></em>.
The index of the next reaction to fire is a discrete random variable
with probability mass function <em>P(μ = m) = a<sub>m</sub> / α</em>.