Bokharaie, Vahid Samadi
Stability Analysis of Positive Systems
with Applications to Epidemiology.
PhD thesis, National University of Ireland Maynooth.
In this thesis, we deal with stability of uncertain positive systems. Although in
recent years much attention has been paid to positive systems in general, there
are still many areas that are left untouched. One of these areas, is the stability
analysis of positive systems under any form of uncertainty. In this manuscript
we study three broad classes of positive systems subject to different forms of
uncertainty: nonlinear, switched and time-delay positive systems. Our focus
is on positive systems which are monotone. Naturally, monotonicity methods
play a key role in obtaining our results.
We start with presenting stability conditions for uncertain nonlinear positive
systems. We consider the nonlinear system to have a certain kind of parametric
uncertainty, which is motivated by the well-known notion of D-stability in
positive linear time-invariant systems. We extend the concept of D-stability
to nonlinear systems and present conditions for D-stability of different classes
of positive nonlinear systems. We also consider the case where a class of
positive nonlinear systems is forced by a positive constant input. We study
the effects of adding such an input on the properties of the equilibrium of the
We then present conditions for stability of positive time-delay systems, when
the value of delay is fixed, but unknown. These types of results are known
in the literature as delay-independent stability results. Based on some recent
results on delay-independent stability of linear positive time-delay systems,
we present conditions for delay-independent stability of classes of positive
nonlinear time-delay systems.
After that, we present conditions for stability of different classes of positive
linear and nonlinear switched systems subject to a special form of structured
uncertainty. These results can also be considered as the extensions of the
notion of D-stability to positive switched systems.
And finally, as an application of our theoretical work on positive systems, we
study a class of epidemiological systems with time-varying parameters. Most
of the work done so far in epidemiology has been focused on models with timeindependent
parameters. Based on some of the recent results in this area, we
describe the epidemiological model as a switched system and present some
results on stability properties of the disease-free state of the epidemiological
We conclude this manuscript with some suggestions on how to extend and
develop the presented results.
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