Stability properties of infection diffusion dynamics over directed networks

We analyze the stability properties of a susceptible-infected-susceptible diffusion model over directed networks. Similar to the majority of infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the all-healthy state is globally asymptotically stable (GAS). Otherwise, an endemic state arises and the entire network could become infected. Using notions from positive systems theory, we prove that the endemic state is GAS in strongly connected networks. When the graph is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of weak and strong endemic states. Several simulations demonstrate our results.