Fig 1.
(a-c) Three one-day contact networks in (a) a primary school with 242 students, (b) a workplace with 217 workers, (c) a high school with 327 students. Node colors correspond to known groups (classes or department). We see that the majority of contacts happen within groups. (d) A synthetic random graph with 9 groups selected randomly.
Fig 2.
The infection model for SARS-CoV-2.
The incubation (Exposed, green) lasts on average 3.7 days [11–13] and is followed by an infectious period (Infectious, orange) of 9.5 days on average, consisting of a presymptomatic period of average 1.5 days [11] followed by a symptomatic period of average 8 days [14]. We assume that symptomatic individuals self-isolate after one day of symptoms, while asymptomatic individuals (40% of the infected [15]) do not isolate.
Fig 3.
Comparison of the effects on SARS-CoV-2 outbreak of containment strategies implemented in the contact graph of a high school when .
The three panels respectively correspond to three relevant metrics: (top) the probability that at least 5 people are infected besides patient 0 (which we define as ‘Outbreak’ event); (middle) the average number of days until 5 people are infected besides patient 0, conditioned on the occurrence of outbreak; (bottom) the average total number of people infected in the population in case of outbreak. That number is a random variable that has a large standard deviation, but with probability 95% its expectation lies within the error bars.
Fig 4.
Impact of the strategies for the high school contact graph.
The x-coordinate gives the value of the baseline reproduction number (mean number of persons infected by index case). For each strategy the y-coordinate gives the mean value of the effective reproduction number as a result of using the strategy. Thus, for our baseline value
(dotted vertical line), doing nothing leads to Re = 1.25 > 1, whereas, as long as
, all strategies lead to Re < 1. For each curve, the shaded areas correspond to 95% confidence intervals on the estimate of
(calculated as a function of the model parameters: horizontal error bar) and on the estimate of Re (as a function of the strategy: vertical error bar).
Fig 5.
Epidemic propagation in the high school contact graph for different strategies.
Each panel corresponds to an example simulation for a given strategy; strategies are sorted as by their effectiveness, from no strategy to full telecommuting. In each panel, the horizontal axis corresponds to time (day of infection) and each white or gray column corresponds to one week. The vertical axis shows the prevalence (percentage of infectious persons among the students), its evolution is plotted in grey. The epidemic propagation is shown as a tree, where each node represents an infected person and points to the persons it infects. Nodes corresponding to symptomatic (resp. asymptomatic) persons are circled in blue (resp. red). Similarly a blue (resp. red) arrow corresponds to a contamination by a symptomatic (resp. asymptomatic) person. The thickness of arrows indicates the super-spreading factor. The node color corresponds to the group of the person (class or department). The node size is linear in its degree in the graph. All the propagation trees are generated using the same realizations of the probabilistic events (run 15978 in our simulations), so that the differences between the trees are not artifacts of their randomness, but solely depend on the different strategies in place.
Table 1.
Contact graphs characteristics.
The three studied Sociopatterns contact networks and the synthetic random graph are detailed in the table. Averaging over the days on which the data was gathered, the high school, in which data was gathered over 5 school days, comprised 327 individuals (students and teachers), each of which was in contact with 35 persons on average (degree), and the student had 230 20-second contacts per day on average (cont./pers./day). The primary school has the highest number of contacts per person in a day, followed by the high school, and finally by the workplace. All graphs have around 10 groups (classes or work departments). The percentage of intra-group contacts (perc. intra) is at least 70% in real networks while it is around 10% in a random graph with close to ten groups.
Fig 6.
Impact of strategies on outbreak probability in a partially immunized population subject to the circulation of variants, simulated in the high school contact network.
In our baseline scenario with no strategy, the probability of outbreak equals 27.2%. Each row reproduces results under specific strategies. In the left column, we study the sensitivity of this quantity with respect to potential variants showing increased transmission capacity compared to the reference strain (multiplicative factor on the y-axis), by assuming that such variants would circulate in partially immune populations (percentage of immune people on the x-axis); immune people do not get infected nor transmit the virus. In the right column, we study the sensitivity with respect to vaccination: we assume that 40% of the population is partially immune and that partially immune individuals are less likely to be infected and to contaminate others when infected. On the x-axis, we vary the relative probability of becoming infected and, on the y-axis, the relative probability of transmission by an infected partially immune individual. These results assume that patient 0 is never immune.
Table 2.
The top table gives the parameter values used in our simulations, with the supporting references.
The bottom table summarizes some relevant quantities that can be computed from our simulations: their consistency with the literature is argued in the Discussion section.
Fig 7.
Relation between the baseline transmission parameter p0 and the network-dependent local reproduction number for our four contact networks.
We see that each curve is monotone increasing, as expected, but that the dependency is not quite linear.