What $R_0$ Deletes: Eigenvectors, Non-Normality, and the Social Content of the Basic Reproduction Number

Abstract

The basic reproduction number is the spectral radius of a matrix, $R_0=\rho (K)$. Taking that definition literally, we ask what $K\mapsto \rho (K)$ discards. A matrix carries three kinds of information: its dominant eigenvalue, its dominant eigenvectors, and its departure from normality. $R_0$ keeps only the first; the other two are where the epidemic's social structure lives. The right eigenvector is the burden distribution, the left the source distribution; they coincide when the system is normal and diverge under heterogeneity. Across the $177$ national contact matrices of Prem et al., the operator is \emph{never} normal, and once age-specific susceptibility is included, its source and burden eigenvectors are misaligned by a median of $26^{\circ }$, exceeding $40^{\circ }$ in some countries: the groups that drive transmission are systematically not those that bear it. We prove that under reciprocal contact this misalignment obeys a Kantorovich bound set by the susceptibility contrast $q_{\max }/q_{\min }$ alone, and zero when susceptibility is uniform, with the excess in real, non-reciprocal matrices contributed by contact asymmetry. Transient amplification, by contrast, stays small, so the operative social content is the misalignment, not transient blow-up. The omission also has teeth: because minimizing $R_0$ protects those who \emph{spread} infection, while minimizing deaths protects those who \emph{die} from it, the two target different age groups; the former sometimes raises average infection fatality even as it lowers the scalar. When contact is strongly structured and susceptibility is heterogeneous, we suggest reporting $R_0$ along with its eigenvectors rather than reporting it alone.

14 pages, 1 figure