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Understanding models which represent the invasion of network-based systems byinfectious agents can give important insights into many real-world situations,including the prevention and control of infectious diseases and computerviruses. Here we consider Markovian susceptible-infectious-susceptible (SIS)dynamics on finite strongly connected networks, applicable to several sexuallytransmitted diseases and computer viruses. In this context, a theoreticaldefinition of endemic prevalence is easily obtained via the quasi-stationarydistribution (QSD). By representing the model as a percolation process andutilising the property of duality, we also provide a theoretical definition ofinvasion probability. We then show that, for undirected networks, theprobability of invasion from any given individual is equal to the(probabilistic) endemic prevalence, following successful invasion, at theindividual (we also provide a relationship for the directed case). The total(fractional) endemic prevalence in the population is thus equal to the averageinvasion probability (across all individuals). Consequently, for such systems,the regions or individuals already supporting a high level of infection arelikely 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 froman emerging infectious disease.

An Exact Relationship Between Invasion Probability and Endemic Prevalence for Markovian SIS Dynamics on Networks