Event-based+models

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 * TODO List**
 * REF Malthus
 * REF Verhulst
 * Add model of ribosome binding in yeast (Shah et al, 2013)

=Event-based models=

Single level (autonomous) dynamical systems can be classified by the way they implement time and variables: (continuous time requires a continuous variable) || MAP (next state function with a defined interval) || (time continuous but not everything changing together only certain events) || FSMs //n-FSMs, CAs, Bool-nets// (nominal states require discrete time) ||
 * || **continuous time** || **discrete time** ||
 * **continuous variables** || ODE
 * **discrete variables/**
 * nominal entities** || EVENT-BASED

So far we have taken FSMs as a center-stage and have shown how other model formalisms can be derived from them using short cuts on conditions defining FSMs. At this stage we will introduce [|event-based models] and compare them to ODEs. As a case example we use the logistic growth equation: dN/dt = aN - bN^2.

Consider for a moment the history of the interpretation of the terms of this classic equation. For instance the N^2 term can be interpreted as representing competition. Alternatively in the case of dN/dt = rN(1-N/K), K is the [|carrying capacity], which is a concept which was derived from the [|population] concept. These interpretations are in fact later derived, while the original interpretation was much more arbitrary and stems from a conference called by king [|Leopold I] of [|Belgium] (19th century?) to discuss the implications of [|Malthus]'s (REF?) prediction about the human overpopulation (population doubling every 30 years!) as derived from British parish records. At the conference, [|Verhulst] (REF?), a physicist, put forward a simple solution to the problem and was so able to calm the poor king's fears. Verhulst used the argument that any function (including a growth function) can be approximated by a [|Taylor expansion], and with a little reasoning a reasonable human population growth function can be derived from dN/dt = a + bN + cN^2 + ... + nN^y. The first term can obviously be dropped because there is no external influx into the human population. Moreover there should be at least one additional term other than aN and it should be negative, otherwise the population would grow to infinity, which is obviously a physical impossiblity. And so, obviously there was no reason to fear for infinite overpopulation! //Notwithstanding this reasoning, it still depends on the [|coefficients] whether overpopulation is alarming or not!// The main point here however, is that the **interpretations of model terms** can be very **minimalistic** indeed.

//[|Gillespie algorithm] for a birth/death process//
We can model a stochastic birth/death process using a stochastic ODE: dN/dt = aN - bN^2 + //noise// In the ODE-formalism, the variables change continously, i.e. something is happening at every time. However, we can also assume that things only happen at certain times, as events. In this event-based formalism each event is modelled using [|probabilities] which determine which event occurs when in continuous time. Since we use probabilities for the events, this automatically gives a **stochastic description** of the system. In this formalism each term needs to be explicitly interpreted:
 * aN = birth + death
 * bN^2 = extra death + reduced birth due to competition

Now, assume that a1 is the per capita birth rate, a2 the per capita death rate, b1 the reduction in births due to competition, and b2 the extra deaths due to competition (hence a = a1+a2 and b = b1+b2). There are two possible events that can take place: birth-events and death-events. Then, the frequency of all events (birth and death) is: E0 = (a1+a2)N - b1N^2 + b2N^2, and the time we have to wait to the next event is drawn as tau = 1/E0 * ln(1/rand1), where rand1 is a random number between 0 and 1. Hence, on average we have to wait 1/E0 time till the next event, but this can vary (stochasticity). Last, we have to determine whether the event at time T+tau is a birth of a death:
 * N -> N+1 (birth) if (a1N - b1N^2) < rand2*E0 where rand2 is again a random number between 0 and 1;
 * N -> N-1 (death) else.

The behaviour in this model allows for stochasticity which is comparable to the chaotic regime in MAPs (in ODEs such behaviour never occurs). However in the MAP the direction of change in the chaotic regime is always to the other side of the equilibrium. In the Event-based formalism the direction is not necessarily relative to the equilibrium. This can be considered to be more realistic for stochastic simulations where once in a while something happens.

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