Parameterizing network graph heterogeneity using a modified Weibull distribution.

We present a simple method to quantitatively capture the heterogeneity in the degree distribution of a network graph using a single parameter σ . Using an exponential transformation of the shape parameter of the Weibull distribution, this control parameter allows the degree distribution to be easily interpolated between highly symmetric and highly heterogeneous distributions on the unit interval. This parameterization of heterogeneity also recovers several other canonical distributions as intermediate special cases, including the Gaussian, Rayleigh, and exponential distributions. We then outline a general graph generation algorithm to produce graphs with a desired amount of heterogeneity. The utility of this formulation of a heterogeneity parameter is demonstrated with examples relating to epidemiological modeling and spectral analysis.

Fundamental bound on epidemic overshoot in the SIR model.

We derive an exact upper bound on the epidemic overshoot for the Kermack-McKendrick SIR model. This maximal overshoot value of 0.2984 · · · occurs at [Formula: see text]. In considering the utility of the notion of overshoot, a rudimentary analysis of data from the first wave of the COVID-19 pandemic in Manaus, Brazil highlights the public health hazard posed by overshoot for epidemics with R0 near 2. Using the general analysis framework presented within, we then consider more complex SIR models that incorporate vaccination.