من ينشر العدوى ومن يصاب بها: الحياة الخفية للوباء سريع

الملخص الأصلي

كما كتبه المؤلفون، بالإنجليزية arXiv:2607.03979

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.