The combination of long-term concurrent sexual partnerships and high infectiousness early in HIV infection has been suggested as a key driver of the extensive spread of HIV in general populations in sub-Saharan Africa, but this has never been scientifically investigated. We use a mathematical model to simulate HIV spreading on sexual networks with different amounts of concurrency. The models show that if HIV infectiousness is constant over the duration of infection, the amount of concurrency has much less influence on HIV spread compared to when infectiousness varies over three stages of infection with high infectiousness in the first months. The proportion of transmissions during primary infection is sensitive to the amount of concurrency and, in this model, is estimated to be between 16 and 28% in spreading epidemics with increasing concurrency. The sensitivity of epidemic spread to the amount of concurrency is greater than predicted by models that do not include primary HIV infection.