The basic reproduction number plays an important role in exploring the dynamics of the epidemic models. Such value has been extensively used estimation of how severe an epidemic outbreak. Although several methods for calculating the basic reproduction number have been proposed, there isn’t an effectively universal method to estimate such value from mathematical view of points. In this paper, we propose a general approach to calculate the explicit formulation of the basic reproduction number by the renewal equation. We apply such method to estimate the basic reproduction number of many epidemic models on complex networks consisting of mean-field models, pairwise models and edge-based compartmental models.

This paper studies a two-strain SIS epidemic model with a competing mechanism and a saturating incidence rate on complex networks. This type of incidence rate can be used to reflect the crowding effect of the infective individuals. We first obtain the associated reproduction numbers for each of the two strains which determine the existence of the boundary equilibria. The stability of the disease-free and boundary equilibria are further examined. Besides this, we also show that the two competing strains can coexist under certain conditions. Interestingly, the saturating incidence rate can have specific effects on not only the stability of the boundary equilibria, but also the existence of the coexistence equilibrium. Numerical simulations are presented to support the theoretical results.