Results for a single batch of vaccine

Because we are interested in investigating the optimal use of vaccine for a quick response, we concentrate most of this work on optimizing vaccine delivered in a single batch at the beginning of the epidemic when only a few million doses are available. We considered four vaccination days very early in transmission, at either 5, 10, 15, or 30 days after its start, and two vaccination days later on at either 60 or 90 days after the beginning of transmission. Since we cannot know how much vaccine production will be ready at the time of an epidemic, for each of these control days, we consider allocating two, four, five, six, seven or ten million doses corresponding to vaccinating 2.8, 5.6, 7.0, 8.5, 9.9 or 14.1% of the total population, respectively (table 2).

For each possible vaccination day and coverage combination, we compare the best vaccine allocation given by the genetic algorithm, denoted as the optimal strategy, to a baseline scenario, where no vaccine is available, and two other possible allocations. The first is the pro rata strategy, where we distribute vaccine to each age-group in each city proportional to the age-group's size. For example, if adults in city correspond to 20% of the total population in the network, then we assign 20% of the available resources to the adults in city . The second strategy is the children-only pro rata strategy, where vaccine is distributed only to the children in each city proportional to the children's population size.

All the simulations presented below started in the same city, Jakarta, with 10 infectious people. Fig. S1 shows the epidemic curves when no vaccination is applied. We also started our simulations in Hong Kong and Taipei. The transmissibility of an infectious disease is often characterized by the basic reproduction number, , defined as the expected number of secondary infections resulting from a single typical infectious person in a completely susceptible population. In the results presented here, , corresponding to a virus roughly as transmissible as the 2009H1N1P [24], [25]. We also considered a less ( ) and a more ( ) transmissible virus.

The optimal strategy shows a modest decrease in the attack rate when very little vaccine is available, provided vaccination occurs during the first five days of the epidemic (i.e., 27% reduction in the attack rate relative to the baseline for two million doses, fig. 2A). As more vaccine becomes available, the optimal strategy greatly reduces the attack rate, although this effect is attenuated as vaccination starts later in the epidemic. For example, if vaccination occurs on day five with five million doses, 3.5% of the population would become infected, with an 85% reduction in the attack rate compared to the baseline case. But if vaccination occurs on day 30, then 14.9% of the population would become infected with only 41% reduction compared to baseline (fig. 2C). Once 10 million doses are available, the optimal strategy can interrupt transmission as long as vaccination occurs before or on day 30 (fig. 2F).

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 2. Attack rate with 95% bootstrapped CI with the epidemic starting in Jakarta. This figure shows the results for a single intervention for six different vaccination days considered and six different vaccination coverages. Each panel represents a given number of vaccine doses available to distribute in the entire network: A) Two million doses. B) Four million doses. C) Five million doses. D) Six million doses. E) Seven million doses. F) Ten million doses. For each panel, each point in the graph corresponds to the attack rate for a single vaccination day, either on day 5, 10, 15, 30, 60, or 90 after the beginning of the epidemic. The optimal, pro rata and children-only pro rata strategies are shown in blue, green and orange respectively. The baseline scenario (red) indicates no vaccination. For each vaccination coverage and day combination, the optimal strategy considerably outperformed the pro rata strategy. When vaccination occurs early in the epidemic, and few doses are available, the optimal strategy also outperforms the children-only pro rata strategy. https://doi.org/10.1371/journal.pcbi.1002964.g002

The optimal strategy outperforms the pro rata strategy for all scenarios considered. With only two million doses available, there is a slight difference in the attack rate with early vaccination (5% difference), but this difference tends to disappear as we start vaccination later in the epidemic (fig. 2A). As more vaccine becomes available, this difference becomes more noticeable, peaking when five and 10 million doses are available (17% and 16% difference in the attack rate, respectively, fig. 2C and 2F). Our results suggest that for the vaccine coverages considered, the pro rata strategy is somewhat insensitive to the timing of the intervention, while the other two strategies considered are not. This would further imply that the pro rata strategy has little indirect effects of herd immunity, but still protects the individuals being vaccinated.

When compared to the children-only pro rata strategy, the optimal strategy performs better only when vaccination occurs early in the epidemic and there are few doses of vaccine available (fig. 2B–D). As more vaccine becomes available, the optimal strategy and the children-only pro rata strategy yield very similar attack rates, and for some cases the optimal strategy is in fact the children-only pro rata strategy (for example fig. 2F, vaccination on day 30, 60 or 90).

We next investigate the capacity of a strategy to mitigate a large epidemic (where more than 1% of the population in each city got infected and ill) from occurring. To this end, we calculate for each solution the epidemic prevention potential (EPP) [26], defined as one minus the ratio of the probability of an epidemic given a particular intervention over the probability of an epidemic given no intervention (fig. 3). Mathematically, (1)

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 3. Epidemic prevention potential (EPP) with 95% bootstrapped CI with the epidemic starting in Jakarta. Each panel represents a given number of vaccine doses available to distribute in the network. A) Two million doses. B) Four million doses. C) Five million doses. D) Six million doses. E) Seven million doses. F) Ten million doses. Each point in each graph corresponds to the EPP for a single vaccination day, either on day 5, 10, 15, 30, 60, or 90 after the beginning of the epidemic. The optimal, pro rata and children-only pro rata strategies are shown in blue, green and orange respectively. When fewer than 10 million doses of vaccine are available, only the optimal strategy is capable of mitigating a significant fraction of the epidemics if vaccination starts early. With as few as 4 million doses, the optimal strategy can mitigate as many as 57% of the epidemics. https://doi.org/10.1371/journal.pcbi.1002964.g003

The optimal strategy is able to mitigate epidemics with very low quantities of vaccine, provided that vaccination starts very early in the epidemic. With as little as four million doses, the optimal strategy mitigates 58% of the epidemics if vaccination starts on day 5, but only 21% if vaccination starts on day 10 (fig. 3B). As more vaccine becomes available, the optimal strategy is able to mitigate a higher proportion of epidemics (fig. 3C–F). In contrast, the other two strategies considered fail to mitigate the epidemics in all of the scenarios considered when less than 10 million doses are available.With 10 million doses, the optimal strategy mitigates over 90% of the epidemics as long as vaccination occurs during the first 15 days. If vaccination occurs during the first ten days, children-only pro rata is also able to mitigate the majority of the epidemics with ten million doses.

Our results suggest that with limited quantities of vaccine, the geographical allocation of vaccine is key in stopping the epidemic if vaccination occurs early on, with most of the vaccine going only to a few cities (figs. S2A, S3A). However, as the epidemic progresses, or more vaccine becomes available, allocating resources more evenly, to the high transmission groups, here the children, becomes a predominant feature (figs. 4C–E and 5C–E). For late vaccination, allocating vaccine in children becomes less relevant and again, geographical location is more important, with the optimal strategy favoring cities where either where the epidemic has not peaked yet or it is possible to reach a high proportion of children vaccinated (figs. 4F, 5F). This is in agreement with previous work [27], [28], [20], which have suggested that vaccinating children is important early in the epidemic, but that there is a threshold after which other groups might benefit more from limited quantities of vaccine.

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 4. Optimal vaccine distribution when four million doses are available and the epidemic is started in Jakarta. Each panel corresponds to allocating vaccine on a different day: either 5 days (A), 10 days (B), 15 days (C), 30 days (D), 60 days (E), or 90 days (F) after the beginning of the epidemic. Each bar corresponds to the percentage of children (red) or adults (blue) vaccinated in each city. These results suggest that the geographical allocation of vaccine is important early in the beginning of the epidemic, when the optimal strategy allocates most of vaccine in Jakarta, but then it is better to distribute vaccine evenly among children up to a certain threshold in time when it becomes important to allocate the vaccine to those cities where either the epidemic has not peaked yet or it is possible to reach a higher proportion of children vaccinated. https://doi.org/10.1371/journal.pcbi.1002964.g004