In a recent experiment, we found that mice previously infected with Bordetella pertussis were not protected against a later infection with Bordetella parapertussis, while primary infection with B. parapertussis conferred cross-protection. This challenges the common assumption made in most mathematical models for pathogenic strain dynamics that cross-immunity between strains is symmetric. Here we investigate the potential consequences of this pattern on the circulation of the two pathogens in human populations. To match the empirical dominance of B. pertussis, we made the additional assumption that B. parapertussis pays a cost in terms of reduced fitness. We begin by exploring the range of parameter values that allow the coexistence of the two pathogens, with or without vaccination. We then track the dynamics of the system following the introduction of anti-pertussis vaccination. Our results suggest that (1) in order for B. pertussis to be more prevalent than B. parapertussis, the former must have a strong competitive advantage, possibly in the form of higher infectivity, and (2) because of asymmetric cross-immunity, the introduction of anti-pertussis vaccination should have little effect on the absolute prevalence of B. parapertussis. We discuss the evidence supporting these predictions, and the potential relevance of this model for other pathogens.