Basic reproduction number (${\mathcal{R}}_0$)

The basic reproduction number ${\mathcal{R}}_0$ is the expected number of secondary infections caused by one infective in an otherwise susceptible population.

The entry $K_{ij}$ is the expected number of newly infected type-$i$ individuals produced by a newly infected type-$j$ individual while the population is disease-free. In a compartmental linearization, $F$ describes new-infection production and $V$ transitions and removal. Provided the infection-lifetime subsystem is transient, its integrated occupation matrix is $V^{-1}$, yielding $K=F{V}^{-1}$. The basic reproduction number is its spectral radius. This construction is described in the original next-generation-matrix derivation.

For a heterogeneous SIS model with common recovery rate and a positive endemic equilibrium $I_{\ast }$, force balance gives $\mathrm{diag}(s_{\ast })K{I}_{\ast }=I_{\ast }$. When the transmission matrix is irreducible and nonnegative, the positive equilibrium vector and the Perron–Frobenius theorem imply the displayed spectral radius identity. It replaces the homogeneous scalar relation $s_{\ast }=1/R_0$: different recipient types generally have different susceptible fractions.

A rare infection grows initially when ${\mathcal{R}}_0>1$ and decays when ${\mathcal{R}}_0<1$.