Cameron Nowzari

Stability analysis of generalized epidemic models over directed networks

In this paper we propose a generalized version of the Susceptible-Exposed-Infected-Vigilant (SEIV) disease spreading model over arbitrary directed graphs. In the standard SEIV model there is only one infectious state. Our model instead allows for the exposed state to also be infectious to healthy individuals. This model captures the fact that infected individuals may act differently when they are aware of their infection. For instance, when the individual is aware of the infection, different actions may be taken, such as staying home from work, causing less chance for spreading the infection. This model generalizes the standard SEIV model which is already known to generalize many other infection spreading models available. We use tools from nonlinear stability analysis to suggest a coordinate transformation that allows us to study the stability of the origin of a relevant linear system. We provide a necessary and sufficient condition for when the disease-free equilibrium is globally exponentially stable. We then extend the results to the case where the infection parameters are not homogeneous among the nodes of the network. Simulations illustrate our results.

Periodic event-triggered average consensus over directed graphs

This paper considers a multi-agent consensus problem over strongly connected and balanced directed graphs. Unlike many works that consider continuous or periodic communication and control strategies, we are interested in developing an event-triggered algorithm to reduce the overall load of the network in terms of limited communication and control updates. Furthermore, we focus on a sampled-data implementation that allows agents in a communication network to determine whether locally sampled information should be discarded or broadcasted to neighbors. This formulation allows us to automatically rule out Zeno behavior that is often a challenge in distributed event-triggered systems. We show that all agents eventually rendezvous at the centroid of their initial formation given an appropriate selection of the local sampling period and event-triggering parameters. We demonstrate the effectiveness of the proposed communication and control law through simulations.

Optimal Resource Allocation for Control of Networked Epidemic Models

This paper proposes and analyzes a generalized epidemic model over arbitrary directed graphs with heterogeneous nodes. The proposed model, called the generalized–susceptible exposed infected vigilant, subsumes a large number of popular epidemic models considered in the literature as special cases. Using a mean-field approximation, we derive a set of ODEs describing the spreading dynamics, provide a careful analysis of the disease-free equilibrium, and derive necessary and sufficient conditions for global exponential stability. Building on this analysis, we consider the problem of containing an initial epidemic outbreak under budget constraints. More specifically, we consider a collection of control actions (e.g., administering vaccines/antidotes, limiting the traffic between cities, or running awareness campaigns), for which we are given suitable cost functions. In this context, we develop an optimization framework to provide solutions for the following two allocation problems: 1) find the minimum cost required to eradicate the disease at a desired exponential decay rate, and 2) given a fixed budget, find the resource allocation to eradicate the disease at the fastest possible exponential decay rate. Our technical approach relies on the reformulation of these problems as geometric programs that can be solved efficiently in polynomial time using tools from graph theory and convex optimization. In contrast with previous works, our optimization framework allows us to simultaneously allocate different types of control resources over heterogeneous populations under budget constraints. We illustrate our results through numerical simulations.

Team-Triggered Coordination for Real-Time Control of Networked Cyber-Physical Systems

This paper studies the real-time implementation of distributed controllers on networked cyber-physical systems. We build on the strengths of event- and self-triggered control to synthesize a unified approach, termed team-triggered, where agents make promises to one another about their future states and are responsible for warning each other if they later decide to break them. The information provided by these promises allows individual agents to autonomously schedule information requests in the future and sets the basis for maintaining desired levels of performance at lower implementation cost. We establish provably correct guarantees for the distributed strategies that result from the proposed approach and examine their robustness against delays, packet drops, and communication noise. The results are illustrated in simulations of a multi-agent formation control problem.

Optimal resource allocation for competing epidemics over arbitrary networks

This paper studies an SI1SI2S spreading model of two competing behaviors over a bilayer network. In particular, we address the problem of determining resource allocation strategies that ensure the extinction of one behavior while not necessarily ensuring the extinction of the other, and pose a marketing problem in which such a model can be of use. Our discussion begins by extending the SI1SI2S model to node-dependent infection and recovery parameters and generalized graph topologies, contrasting prior work. We then find conditions under which a chosen epidemic becomes extinct. We show that a distribution of resources which realizes this goal always exists for some budget under mild assumptions. We address the case in which the available budget is not sufficient for extinction by establishing analytic means for mitigating the spreading rate of the unwanted behavior. We demonstrate a method for tractably computing solutions to each problem via geometric programming. Our results are validated through simulation.

Data-Driven Network Resource Allocation for Controlling Spreading Processes

We propose a mathematical framework, based on conic geometric programming, to control a susceptible-infected-susceptible viral spreading process taking place in a directed contact network with unknown contact rates. We assume that we have access to time series data describing the evolution of the spreading process observed by a collection of sensor nodes over a finite time interval. We propose a data-driven robust optimization framework to find the optimal allocation of protection resources (e.g., vaccines and/or antidotes) to eradicate the viral spread at the fastest possible rate. In contrast to current network identification heuristics, in which a single network is identified to explain the observed data, we use available data to define an uncertainty set containing all networks that are coherent with empirical observations. Through Lagrange duality and convexification of the uncertainty set, we are able to relax the robust optimization problem into a conic geometric program, recently proposed by Chandrasekaran and Shah [1], which allows us to efficiently find the optimal allocation of resources to control the worst-case spread that can take place in the uncertainty set of networks. We illustrate our approach in a transportation network from which we collect partial data about the dynamics of a hypothetical epidemic outbreak over a finite period of time.

Team-triggered coordination of networked systems

This paper proposes an approach for improved methods of performing event- and self-triggered communication and control on networked systems. Current self-triggered strategies are known to be quite conservative whereas event-triggered approaches are costly to implement on distributed systems that rely on wireless communication for information transmission. To overcome these limitations, we propose a novel class of team-triggered coordination laws that combine ideas from event- and self-triggered control, are implementable on networked systems, and maintain desired levels of performance. We characterize the asymptotic convergence properties of team-triggered strategies and show that they perform no worse than self-triggered approaches in terms of required communication. Simulations on a multi-agent formation control problem illustrate our results.

Self-triggered time-varying convex optimization

In this paper, we propose a self-triggered algorithm to solve a class of convex optimization problems with time-varying objective functions. It is known that the trajectory of the optimal solution can be asymptotically tracked by a continuous-time state update law. Unfortunately, implementing this requires continuous evaluation of the gradient and the inverse Hessian of the objective function which is not amenable to digital implementation. Alternatively, we draw inspiration from self-triggered control to propose a strategy that autonomously adapts the times at which it makes computations about the objective function, yielding a piece-wise affine state update law. The algorithm does so by predicting the temporal evolution of the gradient using known upper bounds on higher order derivatives of the objective function. Our proposed method guarantees convergence to arbitrarily small neighborhood of the optimal trajectory in finite time and without incurring Zeno behavior. We illustrate our framework with numerical simulations.

A general class of spreading processes with non-Markovian dynamics

In this paper we propose a general class of models for spreading processes we call the SI*V * model. Unlike many works that consider a fixed number of compartmental states, we allow an arbitrary number of states on arbitrary graphs with heterogeneous parameters for all nodes and edges. As a result, this generalizes an extremely large number of models studied in the literature including the MSEIV, MSEIR, MSEIS, SEIV, SEIR, SEIS, SIV, SIRS, SIR, and SIS models. Furthermore, we show how the SI*V * model allows us to model non-Poisson spreading processes letting us capture much more complicated dynamics than existing works such as information spreading through social networks or the delayed incubation period of a disease like Ebola. This is in contrast to the overwhelming majority of works in the literature that only consider dynamics that can be captured by Markov processes. After developing the stochastic model, we analyze its deterministic mean-field approximation and provide conditions for when the disease-free equilibrium is stable. Simulations illustrate our results.

Optimal resource allocation for containing epidemics on time-varying networks

This paper studies the Susceptible-Infected- Susceptible (SIS) epidemic model on time-varying interaction graphs in contrast to the majority of other works which only consider static graphs. After presenting the mean-field model and characterizing its stability properties, we formulate and solve an optimal resource allocation problem. More specifically, we first assume that a cost can be paid to reduce the amount of interactions certain nodes can have with others (e.g., by imposing travel restrictions between certain cities). Then, given a budget, we are interested in optimally allocating the budget to best combat the undesired epidemic. We show how this problem can be equivalently formulated as a geometric program and solved in polynomial time. Simulations illustrate our results.