Long-term susceptible fractions in networked epidemic models and their relation to the basic reproduction number

Compartmental epidemic models with dynamics that evolve over a graph network have gained considerable importance in recent years. Fundamental to these models is an important threshold known as the basic reproduction number (BRN) that aims to capture, on average, the tendency of a communicable disease to spread. In this paper, we develop two complementing frameworks that provide insights into the long term evolution of a wide range of compartmental epidemic models, including group and networked processes, exploring the positive feedback that is inherent in such models. Specifically, for the case of a group (resp. networked) process, we show that the proportion of the population that is susceptible to a disease (resp. the susceptible proportion in at least one subgroup) tends to a limit that is bounded from above by the reciprocal of the BRN of the respective model, thereby establishing that the BRN encodes critical information on the level of penetration of the disease into a subpopulation. The two substantially distinct scenarios, where the disease remains always present in the population or not, are discussed and the significance of the bound explained. To verify the validity of our conclusions, we apply the developed frameworks to examining various networked epidemic models, including a model that was recently introduced for a bi-virus process.