We consider multiple diseases spreading in a static configuration model network. We make standard assumptions that infection transmits from neighbor to neighbor at a disease-specific rate and infected individuals recover at a disease-specific rate. Infection by one disease confers immediate and permanent immunity to infection by any disease. Under these assumptions, we find a simple, low-dimensional ordinary differential equations model which captures the global dynamics of the infection. The dynamics depend strongly on initial conditions. Although we motivate this Rapid Communication with infectious disease, the model may be adapted to the spread of other infectious agents such as competing political beliefs, or adoption of new technologies if these are influenced by contacts. As an example, we demonstrate how to model an infectious disease which can be prevented by a behavior change.