Upon an outbreak of a dangerous infectious disease, people generally tend to reduce their contacts with others in fear of getting infected. Such typical actions apparently help to reduce the outbreak size. Thanks to today’s broad media coverage, the fear factor may also contribute to preventing an outbreak from happening at all. We are motivated to conduct a careful study on modeling and evaluating such effects with a complex network approach. As a first step of this study, we consider the relatively simple case where involved individuals randomly remove a certain fraction of links between them. Analytical and simulation results show that such an action cannot effectively prevent an epidemic outbreak from happening. However, it may significantly reduce the fraction of all the people ever getting infected when an outbreak does happen.