Fig. 1: Outbreak risks with time-dependent and heterogeneous transmission.

A–C Inputs to the outbreak risk calculation. A Instantaneous reproduction number, \({R}_{t}\), varying periodically over calendar time, \(t\). B Heterogeneity in infectiousness, as characterised by a gamma distribution of relative individual infectiousness factors with mean 1 and dispersion (shape) parameter \(k\) (with a lower value of \(k\) corresponding to a greater degree of heterogeneity). This distribution is shown for a range of values of \(k\) (\(k=0.1\): red; \(k=0.41\): blue; \(k=1\): green; \(k=10\): orange; \(k=100\): light blue; \(k=\infty\): black dashed). C Generation time distribution³². D Illustration of how the outbreak risk at a given calendar time can be calculated either through repeated stochastic simulation of a generalised renewal equation model (in this case, the outbreak risk is defined to be the proportion of simulations in which the daily incidence of new infections ever exceeds a specified threshold, here taken to be 30 daily infections; red curves show example simulations in which incidence exceeds the threshold, and blue curves show simulations in which the threshold is never exceeded), or by solving a system of analytically derived equations numerically. E Values of the outbreak risk, \({p}_{t}\), for the inputs shown in (A–C) (considering the same values of \(k\) as in (B), with the same colour key), for both the analytical approach (lines) and simulation-based approach (dots).