Fig 1.
Probability mass functions (PMF) of Poisson (in orange), negative binomial (in blue), and Delaporte (in purple) distributions.
In each panel, the dispersion parameter k is fixed at 0.5, and the fraction of fixed component ρ is fixed at 0.3.
Fig 2.
Simulation results of the proportion (P) of the most infectious cases that cause (Q =) 80% of secondary cases as a function of fraction of fixed component (ρ) generated from Delaporte distributions.
The ‘NB’ in the horizontal axis label stands for negative binomial (distribution).
Fig 3.
Simulation results of the proportion of cases, i.e., fD(0), that cause 0 secondary case as a function of fraction of fixed component (ρ) generated from Delaporte distributions.
The ‘NB’ in the horizontal axis label stands for negative binomial (distribution).
Fig 4.
Simulation results of the expected proportion of secondary cases (Q) due to the proportion of the most infectious cases (P), i.e., Lorenz curve, generated from Poisson (in orange), negative binomial (in blue), and Delaporte (in purple) distributions.
In each panel, the diagonal line shows the scenario of perfect homogeneity (i.e., uniform distribution). In each panel label, ‘fixed frac.’ is the fraction of fixed component (ρ), and ‘disp.’ is the dispersion parameter (k).
Fig 5.
Fitting results of offspring distributions using the medians of posterior distributions for model parameters.
In each panel, probability mass functions (PMF) of negative binomial (NB, in blue), and Delaporte (in purple) distributions are shown in dots and lines, and the observations of number of secondary cases per infector (in grey) are in histogram. Note: The PMFs of NB and Delaporte distributions were shifted horizontally in each panel with slight jitters at −0.05 and +0.05, respectively to aid visualization and comparison.
Table 1.
The summary of parameter estimates of offspring distribution in the existing literature and this study.
The ‘−2∙log(L)’ denotes twice of the negative log-likelihood. The highlighted estimates are considered as main results for Delaporte distribution (in red) and negative binomial (NB) distribution (in blue).
Fig 6.
The power and type I error rate of the likelihood ratio (LR) test for Delaporte distribution against negative binomial (NB) distribution.
Panels (A) and (D) show the test statistics (dots) from LR test, and the critical threshold (red horizontal dashed line) for p-value < 0.05. In panel (A), the ‘+’ dots are 10000 pseudo datasets generated by random sampling with replacement from the real-world datasets, and the circle dots represent datasets #1-#5. Panels (B) and (E) summarized the power and type I error rate of LR test for Delaporte distribution against NB distribution as a function of sample size. Panels (C) and (F) summarized the power and type I error rate of LR test with sample size reciprocal-distributed from 30 to 3000. In panel (D), the ‘×’ dots are generated by 10000 datasets generated by Monte Carlo sampling from NB distributions. In panels (B) and (C), the horizontal dashed line is the threshold of power at 0.80. In panels (E) and (F), the horizontal dashed line is the threshold of type I error rate at 0.05.
Fig 7.
The relative risk reduction (RRR) of outcome (I): Having superspreading event as a function of the relative reduction in reproduction number (ξ).
The RRR of control scheme (I) RRR(1)(ξ) is dashed cyan curve, and the RRR of control scheme (II) RRR(2)(ξ) is bold orange curve. In each panel, the dispersion parameter k is fixed at 0.2, and the shading region indicates the situation that ξ ≥ 1 − ρ. In each panel label, ‘R’ is the reproduction number, and ‘fixed frac.’ is the fraction of fixed component (ρ).
Fig 8.
The relative risk reduction, RRR(2,1)(ξ), of outcome (I): Having superspreading event under control scheme (II) against scheme (I) as a function of the fraction of fixed component (ρ).
In each panel, the dispersion parameter k is fixed at 0.2, the shading region indicates the situation that ξ ≥ 1 − ρ, and the bold red segment highlights the range of ρ from 0.1 to 0.5, which characterizes the feature of COVID-19. In each panel label, ‘R’ is the reproduction number, and ‘reduction in R’ is the relative reduction in reproduction number (ξ). The ‘NB’ in the horizontal axis label stand = s for negative binomial (distribution).
Fig 9.
The relative risk reduction (RRR) of outcome (II): Having outbreak with final size c > 100 as a function of the relative reduction in reproduction number (ξ).
The RRR of control scheme (I) RRR(1)(ξ) is dashed cyan curve, and the RRR of control scheme (II) RRR(2)(ξ) is bold orange curve. In each panel, the dispersion parameter k is fixed at 0.2, and the shading region indicates the situation that ξ ≥ 1 − ρ. In each panel label, ‘R’ is the reproduction number, and ‘fixed frac.’ is the fraction of fixed component (ρ).
Fig 10.
The relative risk reduction, RRR(2,1)(ξ), of outcome (II): Outbreak with final size c > 100 under control scheme (II) against scheme (I) as a function of the fraction of fixed component (ρ).
In each panel, the dispersion parameter k is fixed at 0.2, the shading region indicates the situation that ξ ≥ 1 − ρ, and the bold red segment highlights the range of ρ from 0.1 to 0.5, which characterizes the feature of COVID-19. In each panel label, ‘R’ is the reproduction number, and ‘reduction in R’ is the relative reduction in reproduction number (ξ). The horizontal dashed grey line marked the level of RRR = 0. The ‘NB’ in the horizontal axis label stand = s for negative binomial (distribution).