The evolution of a pathogen's host range is shaped by the ecology of its hosts and by the physiological traits that determine host specificity. For many pathogen traits, there is a trade-off: a phenotype suitable for infecting one set of hosts poorly infects another. Introducing and analysing a simple evo-epidemiological model, here we study how such a trade-off is expected to affect evolution of the host ranges of influenza viruses. We examine a quantitative trait underlying host specificity, given by an influenza virus's degree of adaptation to certain conformations of sialic acid receptors, and investigate how this receptor preference evolves in a minimal network of host species, including humans, that differ in life history and receptor physiology. Using adaptive dynamics theory, we establish thresholds in interspecific transmission rates and host population sizes that govern the emergence and persistence of human-adapted viruses. These ecological thresholds turn out to be largely independent of the strength of the evolutionary trade-off, underscoring the importance of ecological conditions in determining a disease's host range.