Fig 1.
Mechanistic comparison of the different offspring distribution models.
Under a Poisson model, each infectious individual generates a number of secondary cases, drawn from a Poisson distribution which is the same for all infectious individuals. Under a negative binomial model, each infectious individual is assigned an innate infectivity, drawn from a gamma distribution, which defines a Poisson distribution from which they draw their secondary cases. Under the zero-inflated Poisson model, each infectious individual either generates no secondary cases with probability σ, or else with probability 1 − σ generates a Poisson-distributed number of secondary cases, with the same Poisson parameter for all infectious individuals. Finally, under the beta-Poisson distribution each infectious individual is assigned their own infection probability p from a beta distribution, makes a Poisson-distributed number of contacts, and then infects each of these contacts with probability p.
Table 1.
Notation and interpretation of parameters of each model considered in our analysis.
Table 2.
Frequency of secondary case numbers by dataset.
Underlined entries denote the superspreading boundary in each dataset.
Fig 2.
Maximum likelihood estimates of the beta-Poisson model parameters by dataset: a) basic reproductive ratio λ; b) overdispersion parameter Φ; c) inverse contact parameter ν. Black lines are 95% confidence intervals. In plot c) MLEs and lower confidence bounds of 0 are not shown; these results should be compared with Table 4 which identifies these cases.
Fig 3.
Overdisperion of maximum likelihood offspring distributions fitted to reconstructed transmission trees from a) plague; b) Mpox; c) Ebola, Nigeria 2014; d) Ebola, Guinea 2014; e) SARS, Singapore 2003; f) MERS, South Korea 2015; g) MERS, Saudi Arabia 2015; h) Norovirus, Netherlands 2012. Black lines are 95% confidence intervals.
Table 3.
Maximum likelihood estimates of beta-Poisson model parameters by dataset, 95% confidence intervals in parentheses.
Values of 0.00 in the confidence intervals for Φ represent small non-zero values rounded for display purposes; values of 0 in the MLE or confidence intervals for ν represent “true” zeros corresponding to fits where the MLE is at the negative binomial limit.
Table 4.
Akaike information criterion at MLE by model and dataset.
The underlined entry of each row is the minimal value of AIC attained for that dataset. Values in brackets in column names give the number of parameters which are inferred for each model.
Table 5.
Likelihood ratio test statistics obtained by comparing beta-Poisson to other candidate models under each dataset.
Values in brackets in column names give the difference in number of parameters between each model and the beta-Poisson distribution.
Fig 4.
Lower portion of maximum likelihood offspring distributions fitted to secondary case data from a) plague; b) Mpox; c) Ebola, Nigeria 2014; d) Ebola, Guinea 2014; e) SARS, Singapore 2003; f) MERS, South Korea 2015; g) MERS, Saudi Arabia 2015; h) Norovirus, Netherlands 2012.
Fig 5.
Proportion of superspreaders in maximum likelihood offspring distributions fitted to reconstructed transmission trees from a) plague; b) Mpox; c) Ebola, Nigeria 2014; d) Ebola, Guinea 2014; e) SARS, Singapore 2003; f) MERS, South Korea 2015; g) MERS, Saudi Arabia 2015; h) Norovirus, Netherlands 2012. Black lines are 95% confidence intervals.
Fig 6.
Proportion of zeros in maximum likelihood offspring distributions fitted to reconstructed transmission trees from a) plague; b) Mpox; c) Ebola, Nigeria 2014; d) Ebola, Guinea 2014; e) SARS, Singapore 2003; f) MERS, South Korea 2015; g) MERS, Saudi Arabia 2015; h) Norovirus, Netherlands 2012. Black lines are 95% confidence intervals.