Many disease transmission models exhibit a threshold behaviour based on the basic reproduction number [Formula: see text], where the disease-free equilibrium is locally asymptotically stable if [Formula: see text] and unstable if [Formula: see text]. However, if a system includes immigration of infected individuals, then there is no disease-free equilibrium. We consider how the disease-free equilibrium moves as the level of immigration of infected individuals is increased from 0, finding, under mild assumptions, that the disease-free equilibrium becomes an endemic equilibrium if [Formula: see text] and leaves the biologically relevant space (by having at least one coordinate become negative) if [Formula: see text].
