We construct a set of new epidemiological thresholds to address the general problem of spreading and containment of a transmissible disease with influx of infected individuals (i.e., when the classic R0 is no longer meaningful). We provide analytical properties of these indices and illustrate their usefulness in compartmental model of COVID-19 with data taken from Chile showing a good predictive potential when contrasted with the recorded disease behavior. This geometric approach and the associated analytical and numerical results break new ground in that they allow us to quantify the severity of an immigration infectious individuals into a community, and identification of the key parameters that are capable of changing or reversing the spread of an infectious disease in specific models.