Dengue virus has four distinct serotypes whose cross-reactive immune responses contribute to increased disease severity following heterologous infections. It was proposed that non-protective cross-reactive antibodies may play a role in disease enhancement. In this study we develop a mathematical model of host-virus interaction and predict the mechanisms responsible for virus expansion and loss during primary and secondary dengue infections. We use the model to determine the role of cross-reactive antibodies during dengue fever and dengue hemorrhagic fever-inducing secondary infections, and then compare the model to published patient data. We predict that the cross-reactive antibodies interfere with the non-neutralizing antibody effects by reducing the phagocyte-mediated removal of antibody-virus immune complexes.