Quantifying the spread of communicable diseases with immigration of infectious individuals

We construct a set of new epidemiological thresholds to address the general problem of spreading and containment of a disease with influx of infected individuals when the classic $\mathcal R_0$ is no longer meaningful. We provide analytical properties of these indices and illustrate their usefulness in a compartmental model of COVID-19 with data taken from Chile showing a good predictive potential when contrasted with the recorded disease behaviour. This approach and the associated analytical and numerical results allow us to quantify the severity of an immigration of 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.