  Understanding models which represent the invasion of network-based systems by
infectious agents can give important insights into many real-world situations,
including the prevention and control of infectious diseases and computer
viruses. Here we consider Markovian susceptible-infectious-susceptible (SIS)
dynamics on finite strongly connected networks, applicable to several sexually
transmitted diseases and computer viruses. In this context, a theoretical
definition of endemic prevalence is easily obtained via the quasi-stationary
distribution (QSD). By representing the model as a percolation process and
utilising the property of duality, we also provide a theoretical definition of
invasion probability. We then show that, for undirected networks, the
probability of invasion from any given individual is equal to the
(probabilistic) endemic prevalence, following successful invasion, at the
individual (we also provide a relationship for the directed case). The total
(fractional) endemic prevalence in the population is thus equal to the average
invasion probability (across all individuals). Consequently, for such systems,
the regions or individuals already supporting a high level of infection are
likely to be the source of a successful invasion by another infectious agent.
This could be used to inform targeted interventions when there is a threat from
an emerging infectious disease.
