RNA viruses exist in large intra-host populations which display great genotypic and phenotypic diversity. We analyze a model of viral competition between two viruses infecting a constantly replenished cell pool. We assume a trade-off between the ability of the virus to colonize new cells (cell killing rate or virulence) and its local competitiveness (replicative success within coinfected cells). We characterize the conditions that allow for viral spread by means of the basic reproductive number and show that a local coexistence equilibrium exists, which is asymptotically stable. At this equilibrium, the less virulent competitor has a reproductive advantage over the more virulent colonizer reflected by a larger equilibrium population size of the competitor. The equilibria at which one virus outcompetes the other one are unstable, i.e., a second virus is always able to permanently invade. We generalize the two-virus model to multiple viral strains, each displaying a different virulence. To account for the large phenotypic diversity in viral populations, we consider a continuous spectrum of virulences and present a continuum limit of this multiple viral strains model that describes the time evolution of an initial continuous distribution of virulence without mutations. We provide a proof of the existence of solutions of the model equations, analytically assess the properties of stationary solutions, and present numerical approximations of solutions for different initial distributions. Our simulations suggest that initial continuous distributions of virulence evolve toward a distribution that is extremely skewed in favor of competitors. At equilibrium, only the least virulent part of the population survives. The discrepancy of this finding in the continuum limit with the two-virus model is attributed to the skewed equilibrium subpopulation sizes and to the transition to a continuum. Consequently, in viral quasispecies with high virulence diversity, the model predicts collective virulence attenuation. This result may contribute to understanding virulence attenuation, which has been reported in several experimental studies.