Stability Analysis of Positive Systems with Applications to Epidemiology
Bokharaie, Vahid Samadi (2012) 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 system. 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 model. We conclude this manuscript with some suggestions on how to extend and develop the presented results.
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