Abstract
This paper concerns the SIR dynamics with two types of spreading mechanism, i.e., local spreading through network contacts and global spreading through casual contacts. A low-dimensional edge-based model of ordinary differential equations incorporating arbitrary heterogeneous number of individual contacts is formulated. The basic reproduction number is obtained, and on networks of Poisson type, it is the sum of the basic reproduction numbers for global and local spreading pathways; however, on networks of other type, it is a nonlinear function of the basic reproduction numbers for global and local spreading pathways. To measure the control efforts imposed on one specific transmission pathway, type reproduction numbers for global and local transmission pathways are calculated, respectively. Equations of the final epidemic size are analytically derived. Finally, the numerical solutions to our model are compared with the ensemble averages of the stochastic simulations. Simulations have shown that casual contacts in the population may trigger large stochastic fluctuations, which may cause huge variances around their mean; thus, in this scenario the ensemble mean is not a good representation of the behavior of the stochastic epidemic process. However, increasing the local infection rate or the connectedness of networks yields better predictions. The results presented provide insights in setting a framework for the analysis and containment of multiple routes of epidemic transmission in reality.