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 $\sim {10}^7$ 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.

Figure 1

Comparison of empirical, log-normal, and gamma distributions with the same average infectivity $\mathbb{E}\left[s\right]=0.6{\mathrm{day}}^{-1/2}$ and variance ${\zeta }^2$ with $\zeta =4.16{\mathrm{day}}^{-1/2}$.

Figure 2

Comparison of epidemics spread for gamma [panels (a), (b)] and log-normal [panels (c), (d)] distributions of infectivity and susceptibility with standard deviation $\zeta$ and parameters in Eq. (31). The cases of independent or completely correlated $\sigma$ and $s$ are shown.

Figure 3

Dependence of final epidemic size ${\mathrm{\Omega }}_{\infty }$ on $\rho$ where $[ln(s),ln(\sigma \left)\right]$ is a Gaussian vector with mean $\mathbb{E}\left[s\right]=\mathbb{E}\left[\sigma \right]=0.6{\text{day}}^{-1/2}, \text{Var}\left(s\right)=\text{Var}\left(\sigma \right)={\zeta }^2$ and correlation $\mathbb{E}\left[s\sigma \right]=\mathbb{E}\left[s\right]\mathbb{E}\left[\sigma \right]exp\left(\rho {\varsigma }^2\right)$ with ${\varsigma }^2\equiv ln[1+{\zeta }^2/\left(\mathbb{E}\left[s\right]\mathbb{E}\left[\sigma \right]\right)]$. Note that $\rho$ is the correlation coefficient for $ln\left(s\right)$ and $ln\left(\sigma \right)$ rather than for $s$ and $\sigma$.

Figure 4

Epidemics progression for the distributions shown in Fig. 1 with parameters in Eq. (31). A classical SIR solution for the same susceptibility and infectivity is also shown

Figure 5

Final epidemic size for a mix of normal individuals (same distribution as in Fig. 2) and superspreaders described by Eq. (33). The effect of superspreaders is at most linear in their proportion.