The Basic Reproduction Number as a Loop Gain Matrix

The COVID-19 pandemic and the disordered reactions of most governments made the importance of mathematical modelling and model-based predictions evident, even outside the scientific community. The basic reproduction number
$\mathcal {R}_{0}$
quickly entered the common jargon, as a concise but effective tool to communicate the spreading power of a disease and estimate, at least roughly, the possible outcomes of the epidemic. However, while
$\mathcal {R}_{0}$
is easily defined for simple models, its proper definition is more subtle for larger, state-of-the-art models. Here we show that it is nothing else than the spectral radius of the gain matrix of a linear system, and that this matrix generalizes
$\mathcal {R}_{0}$
in the computation of the vector-valued final epidemic size and epidemic threshold, in a large class of finite-dimensional SIR-like models.