In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We then study the stability of spreading processes in this extended family. We first show that a direct analysis based on It\^o's formula provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of edges. To overcome this limitation, we derive alternative stability conditions involving the eigenvalues of a matrix whose size grows linearly with the number of nodes. Based on our results, we also show that heuristics based on aggregated static networks approximate the epidemic threshold more accurately as the number of nodes grows, or the temporal volatility of the random graph process is reduced. Finally, we illustrate our findings via numerical simulations

This paper studies the mean stability of positive semi-Markovian jump linear systems. We show that their mean stability is characterized by the spectral radius of a matrix that is easy to compute. In deriving the condition we use a certain discretization of a semi-Markovian jump linear system that preserves stability. Also we show a characterization for the exponential mean stability of continuous-time positive Markovian jump linear systems. Numerical examples are given to illustrate the results. Introduction.The stability analysis of switched systems, a class of dynamical systems whose mathematical structure experiences abrupt changes, is one of the most fundamental problems in mathematical systems theory [13,29,37]. In particular, the stability of positive switched systems, whose state variables are constrained to be in positive orthants, has received considerable attentions over the past decade [3,17,27,32,35]. The study of positive switched systems is motivated by their possible application in pharmacokinetics. In the modern treatment of human immunodeficiency virus (HIV) infection, multiple drug regimens are employed to prevent the emergence of drug-resistant viruses [39]. The authors in [21] solve the minimization problem of such virus mutation by its reduction to the optimal control problem of a positive switched system under simplifying assumptions. The reduction was made possible by the switching nature of HIV treatments and the positivity constraint naturally placed on the population of virus. The importance of this class of switched systems also stems from the fact that such positivity constraints naturally arise in broad areas, including communication systems [36], formation flying [24], and multiagent systems [33].The stability of positive switched linear systems has been mainly studied by copositive Lyapunov functions [3,17,19,27,40]. A copositive Lyapunov function is a nonnegative linear form of state variables, which makes a contrast with general cases where the nonnegativity of Lyapunov functions forces us to use quadratic functionals of state variables. Its linearity often reduces the stability analysis of positive switched linear systems to linear problems. For example, a positive switched linear system is stable for all switching signals if a family of square matrices associated with the given system consists of matrices whose eigenvalues have only negative real parts [17,27,40].However, once a switched system is modeled as a stochastic switched system [28,34], the above-mentioned Lyapunov function approach fails to take the probability distribution into account appropriately because it treats any sample path in a rather equal manner. One of the natural notions of stability in this case is mean stability [28],

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In this paper, we study the dynamics of epidemic processes taking place in temporal and adaptive networks. Building on the activity-driven network model, we propose an adaptive model of epidemic processes, where the network topology dynamically changes due to both exogenous factors independent of the epidemic dynamics as well as endogenous preventive measures adopted by individuals in response to the state of the infection. A direct analysis of the model using Markov processes involves the spectral analysis of a transition probability matrix whose size grows exponentially with the number of nodes. To overcome this limitation, we derive an upper-bound on the decay rate of the number of infected nodes in terms of the eigenvalues of a 2 × 2 matrix. Using this upper bound, we propose an efficient algorithm to tune the parameters describing the endogenous preventive measures in order to contain epidemics over time. We confirm our theoretical results via numerical simulations.

In this paper, we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology. We focus our study on the adaptive susceptible-infected-susceptible (ASIS) model, where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection. In this paper, we derive a closed-form expression for a lower bound on the epidemic threshold of the ASIS model in arbitrary networks with heterogeneous node and edge dynamics. For networks with homogeneous node and edge dynamics, we show that the resulting lower bound is proportional to the epidemic threshold of the standard SIS model over static networks, with a proportionality constant that depends on the adaptation rates. Furthermore, based on our results, we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks. We confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures.

SUMMARYIn this paper, we study state-feedback control of Markov jump linear systems with partial information. In particular, we assume that the controller can only access the mode signals according to a hidden-Markov observation process. Our formulation generalizes various relevant cases previously studied in the literature on Markov jump linear systems, such as the cases with perfect information, no information, and cluster observations of the mode signals. In this context, we propose a Linear Matrix Inequalities (LMI) formulation to design feedback control laws for (stochastic) stabilization, H 2 , and H∞ control of discrete-time Markov jump linear systems under hidden-Markovian observations of the mode signals. We conclude by illustrating our results with some numerical examples.

This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a geometric program, which in turn can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the H 2 norm, H ∞ norm, Hankel norm, and Schatten p-norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tunning problem jointly constrained by the exponential decay rate, the L 1 -gain, and the L ∞ -gain can be solved by convex optimization. The assumption on the system parametrization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

In this paper, we propose an optimization framework to design a network of positive linear systems whose structure switches according to a Markov process. The optimization framework herein proposed allows the network designer to optimize the coupling elements of a directed network, as well as the dynamics of the nodes in order to maximize the stabilization rate of the network and/or the disturbance rejection against an exogenous input. The cost of implementing a particular network is modeled using posynomial cost functions, which allow for a wide variety of modeling options. In this context, we show that the cost-optimal network design can be efficiently found using geometric programming in polynomial time. We illustrate our results with a practical problem in network epidemiology, namely, the cost-optimal stabilization of the spread of a disease over a time-varying contact network.