Spread of epidemics in time-dependent networks

We consider SIS models for the spread of epidemics. In particular we consider the so called nonhomogeneous case, in which the probability of infection and recovery are not uniform but depend on a neighborhood graph which describes the possibility of infection between individuals. In addition it is assumed, that infection, recovery probabilities as well as the interconnection structure may change with time. Using the concept of the joint spectral radius of a family of matrices conditions are provided that guarantee robust extinction of the epidemics

On the D-Stability of Linear and Nonlinear Positive Switched Systems

We present a number of results on D-stability of positive switched systems. Different classes of linear and nonlinear positive switched systems are considered and simple conditions for D-stability of each class are presented

Stability Analysis of Positive Systems with Applications to Epidemiology

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

Stability Analysis of Positive Systems with Applications to Epidemiology

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

Spread of epidemics in time-dependent networks

We consider SIS models for the spread of epidemics. In particular we consider the so called nonhomogeneous case, in which the probability of infection and recovery are not uniform but depend on a neighborhood graph which describes the possibility of infection between individuals. In addition it is assumed, that infection, recovery probabilities as well as the interconnection structure may change with time. Using the concept of the joint spectral radius of a family of matrices conditions are provided that guarantee robust extinction of the epidemics

Spread of epidemics in time-dependent networks

We consider SIS models for the spread of epidemics. In particular we consider the so called nonhomogeneous case, in which the probability of infection and recovery are not uniform but depend on a neighborhood graph which describes the possibility of infection between individuals. In addition it is assumed, that infection, recovery probabilities as well as the interconnection structure may change with time. Using the concept of the joint spectral radius of a family of matrices conditions are provided that guarantee robust extinction of the epidemics

On the D-Stability of Linear and Nonlinear Positive Switched Systems

We present a number of results on D-stability of positive switched systems. Different classes of linear and nonlinear positive switched systems are considered and simple conditions for D-stability of each class are presented

Correction to “D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems”

We correct some errors in the statements and proofs presented in Section V of the above mentioned manuscript

D-Stability and Delay-independent stability of homogeneous cooperative systems

We introduce a nonlinear definition of D-stability, extending the usual concept for positive linear time-invariant systems. We show that globally asymptotically stable, cooperative systems, homogeneous of any order with respect to arbitrary dilation maps are D-stable. We also prove a strong stability result for delayed cooperative homogeneous systems. Fi- nally, we show that both of these results also hold for planar cooperative systems without the restriction of homogeneity

Stability and positivity of equilibria for subhomogeneous cooperative systems

Building on recent work on homogeneous cooperative systems, we extend results concerning stability of such systems to subhomogeneous systems. We also consider subhomogeneous cooperative systems with constant input, and relate the global asymptotic stability of the unforced system to the existence and stability of positive equilibria for the system with input