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We construct a set of new epidemiological thresholds to address the generalproblem of spreading and containment of a disease with influx of infectedindividuals when the classic $\mathcal R_0$ is no longer meaningful. We provideanalytical properties of these indices and illustrate their usefulness in acompartmental model of COVID-19 with data taken from Chile showing a goodpredictive potential when contrasted with the recorded disease behaviour. Thisapproach and the associated analytical and numerical results allow us toquantify the severity of an immigration of infectious individuals into acommunity, and identification of the key parameters that are capable ofchanging or reversing the spread of an infectious disease in specific models.

Quantifying the spread of communicable diseases with immigration of infectious individuals