Stability of epidemic models over directed graphs: A positive systems approach

Abstract

We study the stability properties of a susceptible-infected-susceptible (SIS) diffusion model, so-called the n-intertwined Markov model, over arbitrary directed network topologies. As in the majority of the work on infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the disease-free state is the unique equilibrium over the network. Otherwise, an endemic equilibrium state emerges, where some infection remains within the network. Using notions from positive systems theory, we provide novel proofs for the global asymptotic stability of the equilibrium points in both cases over strongly connected networks based on the value of the basic reproduction number, a fundamental quantity in the study of epidemics. When the network topology is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of an endemic state, and we study the stability of the disease-free state. Finally, we demonstrate that the n-intertwined Markov model can be viewed as a best-response dynamical system of a concave game among the nodes. This characterization allows us to cast new infection spread dynamics; additionally, we provide a sufficient condition for the global convergence to the disease-free state, which can be checked in a distributed fashion. Several simulations demonstrate our results.

Stability properties of infection diffusion dynamics over directed networks

“…, then p = 0 is the only possible equilibrium over G. The complete proof of this theorem can be found in [20]. The proof relies on observing that the states of the nodes in G 1 are not affected by the nodes in G 2 1 .…”

“…We proved this result using the comparison lemma and the well-known fact that the state of an exponentially stable linear system converges to zero when its input converges to zero-see [20] for the complete proof.…”

“…Proof: Due to space limitation, we will show sufficiency only for the case when R o < 1; the complete proof can be found in [20]. …”

“…It is important to note that the endemic states q 2 resulting in cases (1), (2), and (3) are not necessarily the same. Having shown existence and uniqueness of weak and strong endemic equilibria, the next theorem characterizes their stability properties-for the complete proof, see [20].…”

Output-Feedback Control for Discrete-Time Spreading Models in Complex Networks

“…When the extinction state is not stable, the control must be designed in such a way that destabilizing components are compensated and self-stabilization mechanisms are enhanced. This can be accomplished in different ways, e.g., by changing the network structure [33][34][35] or node-specific parameters [27,[36][37][38]. As for complex networks, not all node parameters are subject to control, and a systematic adaptation of parameters also implies computational complexity and may imply necessary communication structures, a set of nodes whose parameters should be adapted in order to achieve stabilization must be chosen.…”