Time is one of the main factors in any kind of real-life systems. When a certain system is
analysed one is often interested in its evolution with respect to time. Various phenomena
can be described using a form of time-dependency. The difference between load in
call-centres is the example of time-dependency in queueing systems. The process of
migration of biological species in autumn and spring is another illustration of changing
the behaviour in time. The ageing process in critical infrastructures (which can result in
the system component failure) can also be considered as another type of time-dependent
Considering the variability in time for chemical and biological systems one comes to
the general tasks of systems biology . It is an inter-disciplinary study field which
investigates complex interactions between components of biological systems and aims
to explore the fundamental laws and new features of them. Systems biology is also
used for referring to a certain type of research cycle. It starts with the creation a
model. One tries to describe the behaviour in a most intuitive and informative way
which assumes convenience and visibility of future analysis. The traditional approach
is based on deterministic models where the evolution can be predicted with certainty.
This type of model usually operates at a macroscopic scale and if one considers chemical
reactions the state of the system is represented by the concentrations of species and
a continuous deterministic change is assumed. A set of ordinary differential equations
(ODE) is one of the ways to describe such kind of models. To obtain a solution numerical
methods are applied. The choice of a certain ODE-solver depends on the type of the
ODE system. Another option is a full description of the chemical reaction system where
we model each single molecule explicitly operating with their properties and positions
in space. Naturally it is difficult to treat big systems in a such way and it also creates
restrictions for computational analysis.
However it reveals that the deterministic formalism is not always sufficient to describe
all possible ways for the system to evolve. For instance, the Lambda phage decision
circuit  can be a motivational example of such system. When the lambda phage
virus infects the E.coli bacterium it can evolve in two different ways. The first one is
lysogeny where the genome of the virus is integrated into the genome of the bacterium.
Virus DNA is then replicated in descendant cells using the replication mechanism of
the host cell. Another way is entering the lytic cycle, which means that new phages
are synthesized directly in the host cell and finally its membrane is destroyed and new
phages are released. A deterministic model is not appropriate to describe this process of
choosing between two pathways as this decision is probabilistic and one needs a stochastic
model to give an appropriate description.
Another important issue which has to be addressed is the fact that the state of the
system changes discretely. It means that one considers not the continuous change of
chemical species concentrations but discrete events occuring with different probabilities
(they can be time-dependent as well).
We will use the continuous-time Markov Population Models (MPMs) formalism in
this thesis to describe discrete-state stochastic systems. They are indeed continuous-
time Markov processes, where the state of the system represents populations and it is
expressed by the vector of natural numbers. Such systems can have innitely many
states. For the case of chemical reactions network it results in the fact that one can
not provide strict upper bounds for the population of certain species. When analysing
these systems one can estimate measures of interest (like expectation and variance for
the certain species populations at a given time instant). Besides this, probabilities for
certain events to occur can be important (for instance, the probability for population to
reach the threshold or the probability for given species to extinct).
The usual way to investigate properties of these systems is simulation  which means
that a large amount of possible sample trajectories are generated and then analysed.
However it can be difficult to collect a sufficient number of trajectories to provide statistical
estimations of good quality. Besides the simulation, approaches based on the
uniformization technique have been proven to be computationally efficient for analysis
of time-independent MPMs. In the case of time-dependent processes only few results
concerning the performance of numerical techniques are known .
Here we present a method for conducting an analysis of MPMs that can have possibly
infinitely many states and their dynamics is time-dependent. To cope with the problem
we combine the ideas of on-the-y uniformization  with the method for treating timeinhomogeneous
behaviour presented by Bucholz.