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Continuous-Time Discrete-State Stochastic Decay

Continuous-Time Discrete-State Stochastic Decay

Continuous-Time Markov Chain of Degradation, Decay, or Departure

Introduction

A simple homogeneous degradation, decay (e.g. memory decay), death, or departure process can be described as a directed graphical model (stoichiometric equation in chemics):

NAλ(1)

where:

A :=  is a species of interest

NA :=  is the number of particles of species of interest

λ :=  is the rate constant of degradation events

 :=  is a species of no special interest

In a homogeneous process the rate is defined as a constant λ so that λdt gives the probability that a randomly chosen particle of A reacts, degrades, dies, decays, or departs during the time interval [t,t+dt) where t are time and dt an (infinitesimally) small time step.

In an in- or nonhomogeneous process the rate is dependent on time or state  λ(t) or λ(NA(t)) so that λ(t)dt or λ(NA(t))dt gives the probability that a randomly chosen particle of A reacts, degrades, dies, decays, or departs during the time interval [t,t+dt) where t are time and dt an (infinitesimally) small time step.

The stoichiometric equations or the DAGs are now:

NAλ(t) with time-dependent rate (2.1)

NAλ(NA) with state-dependent rate (2.2)

The Definition of the (In-)Homogeneous Counter Decay Process

The "naive" implementation of the solution of the decay counter process is based not on dt but on a discrete approximation Δt>>dt so it is wise to reuse and modify the second definition of the Poisson process. The decay process NA(t),t[0,) is called a homogeneous counter decay process with rate λ>0 if all of the following conditions hold:

1.NA(0)=NA0>0 ; 

2.NA(t) has independent and stationary increments

then we have

P(NA(Δt)=0)1λΔt(3.1.1)

P(NA(Δt)=1)λΔt(3.1.2)

P(NA(Δt)=2)0(3.1.3)

or more precisely

P(NA(Δt)=0)=1λΔt+o(Δt)(3.2.1)

P(NA(Δt)=1)=λΔt+o(Δt)(3.2.2)

P(NA(Δt)=2)=o(Δt)(3.2.3)

where "little o"  denotes a function g(.) o(Δt):=g(Δt) that 'vanishes' faster than Δt when

limΔt0g(Δt)Δt=0,(4)

or short

o(Δt)0 as Δt0.

WebPPL-Script: Stochastic Simulation of (In-)Homogeneous Process with Constant Time Increment Delta

Simulation Runs as an Inhomogeneous Pure Decay Process

Fig. 1: Parameters of Inhomogeneous Process with State-dependent Rates

Fig. 2 - 11: Sequence of 10 Independent Simulation Runs of an Inhomogeneous Pure Decay Process

Simulation Runs as an Homogeneous Pure Decay Process

Fig. 1: Parameters of Homogenous Process

Fig. 2-11 Independent Runs

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