Fig 1.
Poisson and NB-distributed secondary infections.
Figure shows a schematic of an NB branching process and example distributions of NB and Poisson distributions of secondary infections. Both distributions have R0 = 2.6 [28], and the NB has dispersion parameter k = 0.16 [25] with 100,000 draws. The code required to generate this figure can be found in S1 Data. NB, negative binomial.
Fig 2.
Example trajectories of NB and Poisson branching processes.
Figure shows example trajectories (in number of active infections versus generation) of NB and Poisson branching processes and cumulative infection sizes after 6 generations of spread. Both simulations start with 1 infection and have the same R0 = 2.6. For NB branching process, we assume dispersion parameter k = 0.16, same as SARS-CoV-1 [24,33]. We run all simulations 10,000 times. Dashed red lines represent theoretical values in the large-population limit , where I is number of active infections, and n is number of generations. Solid blue lines are the mean values of all simulations including those that have not taken off, which overlap with the theoretical values when the susceptibles are not depleted. Solid orange lines are the mean value for simulations that took off, and the outbreaks appear more explosive in the first few generations in the NB simulations. Both number of active cases and cumulative infections are in log10 scale. The code required to generate this figure can be found in S1 Data. NB, negative binomial; SARS-CoV-1, Severe Acute Respiratory Syndrome Coronavirus 1.
Fig 3.
Controlling outbreaks and the effect of “cutting the tail”.
(A) Values of Reff under different thresholds of maximum number of secondary infections and the probability of keeping the maximum number of secondary infection thresholds. We compare 3 scenarios with R0 = 1.5, 2.0, and 3.0, where lower R0 represents the consequence of population-wide mild to moderate NPIs. The NB distribution is truncated at the max number of secondary cases, and the distribution is renormalized after the truncation. Contours highlight Reff = 1 boundary. Lower R0 facilitates the extinction of outbreak, and increases the probability of success controlling the outbreak, especially when the probability of reducing hotspot transmission is low. If the vast majority of transmission inside hotspots can be eliminated under all R0 scenarios, a control target of less than 10 secondary infections can bring Reff to close to unity. (B) Probability of an outbreak successfully taking off with 1 seed infection under NB and Poisson distributions, as well as an NB distribution with cutoff number of secondary infections of 10 and with 100% efficiency. The stochastic branching process simulations are run 10,000 times with R0 ranging from 0.8 to 3.0. The success of outbreak is defined as having more than 20 infections at generation 6. Dispersion parameter k for NB is 0.16. Data are calculated using methods based on probability-generating functions [50]. The code required to generate this figure can be found in S2 Data. NB, negative binomial; NPI, non-pharmaceutical intervention.