Spectral radius and network processes with spontaneous infection/failure rate

We relate the time-limiting behavior of a network epidemics process to the spectral radius of the underlying network. The process we study is the scaled SIS network process, a continuous-time Markov process on a static network. Our analysis differs from previous work in that the scaled SIS process accounts for the possibility that a healthy individual has a nonzero probability of becoming infected even when all of its neighbors are healthy. For example, the source of infection may be outside a human only contact network for diseases with animal to human transmissions such as Ebola. We show that the sufficient condition for infection to become extinct not only depends on the ratio of infection and healing rates but also on N, the size of the network, whereas previous models, assuming no exogenous infection, showed dependency only on the infection and healing rates.