Epidemic dynamics in inhomogeneous populations and the role of superspreaders

A variant of the susceptible-infected-recovered model for an inhomogeneous population is introduced to account for the effect of variability in susceptibility and infectiousness across a population. An initial formulation of this dynamics leads to infinitely many differential equations. Our model, however, can be reduced to a single first-order one-dimensional differential equation. Using this approach, we provide quantitative solutions for different distributions. In particular, we use GPS data from $\ensuremath{\sim}{10}^{7}$ cell phones to determine an empirical distribution of the number of individual contacts and use this to infer a possible distribution of susceptibility and infectivity. We quantify the effect of superspreaders on the early growth rate ${\mathcal{R}}_{0}$ of the infection and on the final epidemic size, the total number of people who are ever infected. We discuss the features of the distribution that contribute most to the dynamics of the infection.