| We study the global dynamics of a disease model with both direct and indirect transmissions. Though the model system is nonlinear and it couples two transmission mechanisms, the basic reproduction number exhibits a nice linear property: it is simply the sum of two basic reproduction numbers for direct and indirect disease transmissions respectively. We further demonstrate that the basic reproduction number is a threshold parameter which characterizes the local and global dynamics of the model system. The key ideas (as well as main difficulties) in the proof will be explained in detail, and the presentation should be accessible to graduate and undergraduate students who have some basic knowledge of differential equations. |