key: cord-0775154-i3u9y1sj
authors: Geoffroy, F.; Traulsen, A.; Uecker, H.
title: Vaccination strategies when vaccines are scarce: On conflicts between reducing the burden and avoiding the evolution of escape mutants
date: 2021-05-08
journal: nan
DOI: 10.1101/2021.05.04.21256623
sha: 212e05b46acf7a9e1f778b48d085bb35d05e1f1b
doc_id: 775154
cord_uid: i3u9y1sj

When vaccine supply is limited but population immunisation urgent, the allocation of the available doses needs to be carefully considered. One aspect of dose allocation is the time interval between the primer and the booster injections in two-dose vaccines. By stretching this interval, more individuals can be vaccinated with the first dose more quickly. Even if the level of immunity of these 'half-vaccinated' individuals is lower than that of those who have received both shots, delaying the second injection can be beneficial in reducing case numbers, provided a single dose is sufficiently effective. On the other hand, there has been concern that intermediate levels of immunity in partially vaccinated individuals may favour the evolution of vaccine escape mutants. In that case, a large fraction of half-vaccinated individuals would pose a risk - but only if they encounter the virus. This raises the question whether there is a conflict between reducing the burden and the risk of vaccine escape evolution or not. We develop a minimal model to assess the population-level effects of the timing of the booster dose. We set up an SIR-type model, in which more and more individuals become vaccinated with a two-dose vaccine over the course of a pandemic. As expected, there is no trade-off when vaccine escape evolves at equal probabilities in unvaccinated and half-vaccinated patients. If vaccine escape evolves more easily in half-vaccinated patients, the presence or absence of a trade-off depends on the reductions in susceptibility and transmissibility elicited by the primer dose.

Many vaccines are administered in two doses, a primer shot and -after a certain time interval -a 20 booster shot. The booster shot increases the strength and duration of protection. However, the primer 21 shot on its own already establishes some immunity. In a pandemic -such as in the current Covid 22 pandemic -when population immunisation is urgent but vaccine doses are scarce, the question arises 23 whether the second shot should be delayed at the benefit of administering the first vaccine dose to 24 more people more quickly. Even if half-vaccinated individuals are only partially immune, the overall 25 reduction in infections may be greater in a population in which many people have some immunity than 26 in a population in which fewer individuals have stronger immunity. In the current Covid pandemic, 27 such a delay strategy has been adopted by the UK (Campbell, 2020 Dec 30), while several other 28 countries such as the US stick to the interval between injections that has been applied in the original 29 clinical trials and is therefore recommended by the manufacturer (U.S. Food and Drug Administration). 30 In early January 2021, the WHO recommended to stretch the dosing interval of the first approved The primer dose reduces susceptibility by a factor x and transmissibility by a factor y. Vaccination with both doses provides perfect protection from the virus. Since vaccine supply is limited, there is a trade-off: either administering the booster shots after the minimally required prime-boost interval ω min or delaying the booster injection and giving the primer dose to more people more quickly (ω > ω min ). The model describes the dynamics in the absence of vaccine escape mutants. The rate at which such mutants emerge can be obtained from the number of infected individuals, where vaccine escape may possibly evolve at different per-capita rates in unvaccinated and half-vaccinated patients. The allocation strategy affects the total number of individuals that become infected throughout the pandemic, but also the risk of vaccine escape evolution across the population. It is apriori not clear whether the same strategy minimises both quantitites. (b-c) Exemplary dynamics of infection and vaccination with and without a delay in the booster dose. In Panel (b), the booster dose is administered as soon as possible (ω = ω min ). In this case, the fraction of half-vaccinated individuals is always low. In Panel (c), the booster dose is delayed (ω = ω max ), leading to a much higher fraction of half-vaccinated individuals over time. The vaccination campaign starts during the pandemic, when a considerable fraction of the population has already been affected by the virus (I(0) = 2 × 10 −3 and R(0) = 4 × 10 −2 ). For illustrative purposes, we use a large reproductive number R C = 2 here.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) 

Before the escape variant arises, the dynamics of the epidemic is described by the differential 109 equations:

where the first three equations describe individuals that are uninfected and either unvaccinated (S), . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 8, 2021. ; https://doi.org/10.1101/2021.05.04.21256623 doi: medRxiv preprint infections, while the qualitative results in our model depend on the ratio µ V /µ I (see below for details).

To account for the uncertainty in this ratio, we consider µ V /µ I = 1, 10, 100. We set µ I = 10 −6 . Our 133 primary focus is a scenario in which the vaccination campaign starts several months into the pandemic 134 when a noticeable fraction of the population has already been affected by the virus. We define t = 0 135 as the start of the vaccination campaign. As initial conditions, we then set I(0) = 2 × 10 −3 and 136 R(0) = 4 × 10 −2 (Roser et al., 2020), which corresponds approximately to twice the number of 137 confirmed cases in Germany in January with the assumption that only 50% of the cases are detected 138 (Backhaus et al., 2021). 139 We numerically integrate the differential equations using python, see our Jupyter notebook that is 140 available with this manuscript. Exemplary dynamics are shown in Fig. 1b We aim to determine how the strategy affects the number of cases and the risk of vaccine escape.

The first quantity that we consider is therefore the cumulative fraction of infected individuals from the 145 start of the vaccination campaign until the end of the pandemic, provided no vaccine escape mutants 146 evolve:

We refer to this as the burden. The total burden can be decomposed into the burden B S = 

In a non-deterministic world, these mutants may or may not evolve. The sum µ I I(t)

can also be interpreted as a stochastic rate, and the integral quantifies the total risk over the course of 158 the pandemic (not taking into account the risk prior to the vaccine roll-out, which is independent of 159 the chosen vaccination strategy). More precisely, the probability that vaccine escape mutants evolve is 160 given by

where N is the total population size. It should be noted that this is the probability of their mere 162 appearance. That does not mean that they will spread. E.g., some of those patients in whom vaccine 163 escape mutants evolve may not infect anyone, in which case the mutation is lost again from the 164 population. For shortness, we often refer to M as the risk of vaccine escape in the following, but it 165 should be kept in mind that the probability of vaccine escape is not directly given by M but by Eq. (4) 166 and that this does not involve any probability of establishment of the vaccine escape mutant in the 167 population.

168

When can conflicts between reducing the burden and the escape risk appear? injections reduces the burden (again, this is not true for higher R C , see below). For the escape risk, the 208 picture is different. With a strong reduction in susceptibility, a delay of the booster shot reduces the 209 escape risk. In that case, the effects of a delay on the burden and on the vaccine risk align. In contrast, 210 if the vaccine has no effect on susceptibility (x = 1), delaying the second dose increases the risk of 

We therefore can conclude from these limiting cases that conflicts between reducing the burden and 215 the risk of vaccine escape can exist (see x = 1, y = 0), but the reduction in cases can also outweigh 216 the increased risk of within-host evolution of vaccine escape (see y = 1, x = 0).

When do reductions in both transmissibility and susceptibility lead to trade-offs?

To investigate more closely under which circumstances the effect of the strategy on the burden and the 219 fraction of escape mutants diverge, we proceed to explore the entire range of reductions in susceptibility 220 and transmissibility and vary other parameters as well. As for the limiting cases, we found that the 221 optimal strategy is either no delay or a maximal delay. We therefore focus on the burden and the 222 escape risk with the recommended interval of ω min = 20 days and a maximally stretched interval of 223 ω max = 100 days. To determine which strategy is optimal under the respective criterion, we consider 224 6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Susceptibility reduction x 1 (no effect) 0.5 0 (full effect)

Effect of a reduction in transmissibility y (with x = 1) Transmissibility reduction y 1 (no effect) 0.5 0 (full effect) . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 8, 2021. ; https://doi.org/10.1101/2021.05.04.21256623 doi: medRxiv preprint the differences ∆B = B ω=100 − B ω=20 and ∆ M = M ω=100 − M ω=20 and ask when they are larger than 225 zero ('No delay') or smaller than zero ('Delay the 2 nd dose'), see Fig. 3 . For reference, the absolute 226 burden and fraction of new mutant infections with our default parameter set is given in Fig. 4 for both 227 values of ω. 228 We already know from the analysis of the limiting cases that neither strategy is optimal in minimizing 229 the burden across the entire range of x and y. This can be further seen in Fig. 3a- 

The differences ∆B and ∆M between the two strategies can be substantial, such that they become and ∆M = M ω=100 − M ω=20 , depending on the reductions in susceptibility x and transmissibility y elicited by the primer dose. If the difference is positive, a delay increases the burden/escape risk ('No delay'). If it is negative, a delay reduces the burden/escape risk ('Delay the 2 nd dose'). The three columns correspond to our baseline scenario with the default parameter set (left column), a high infection scenario with a higher value of R C (middle column), and a high vaccination rate scenario with a larger value of λ (right column). We consider three different ratios of the per-capita mutation rates in unvaccinated and half-vaccinated individuals. As expected, for µ V /µ I = 1 (Panels d-f) the same strategy minimizes both the burden and the risk of vaccine escape, irrespective of x and y. For µ V > µ I , there is a parameter range, in which a delay reduces the burden but increases the risk of vaccine escape.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (Panels b and d) , depending on the reductions in susceptibility x and transmissibility y elicited by the primer dose. Reducing the susceptibility or transmissibility reduces the total burden and the escape risk, but in quantitatively different ways such that a trade-off between them regarding the optimal ω can emerge. The figure shows results for our default parameter set with µ V /µ I = 10.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 3.6 ×10 8 Figure 5 : Comparison between a scenario, in which vaccines become available during a pandemic, (Panels a and c) and a scenario, where they are available right from the start (Panels b and d). The figure shows the differences in the burden and in the number of new mutants infections between the strategies with a minimal and a maximal interval between the two vaccine doses, ∆B = B ω=100 −B ω=20 and ∆M = M ω=100 −M ω=20 . When vaccines are available from the start, differences between strategies are much smaller. (Note the different scales of the color gradients.)

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. to the disease burden. Matters are much more complicated when it comes to assessing the risk of 315 vaccine escape, which requires to make predictions about evolution in a highly complex and dynamic 316 environment. In our simple model, the range of primer dose effects for which a reduction in burden comes 317 at an increased risk of vaccine escape depends on the relative probabilities of within-host evolution of 318 escape mutants in half-vaccinated and unvaccinated individuals. There is concern that vaccine escape 319 mutants evolve more easily in partially immune individuals, but we do not know whether this is really are key parameters. It could also provide a starting point for more detailed models that take further 341 complications into account. We hope that our model helps to provide a more solid foundation for the 342 discussion on vaccination strategies. 13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 8, 2021. ; https://doi.org/10.1101/2021.05.04.21256623 doi: medRxiv preprint

In this section, we show that there are at all times enough doses available to administer the second 437 dose after the chosen time interval ω, as long as S(t) > 0, i.e. there never arises a situation in which 438 the second dose needs to be further delayed due to vaccine shortage. For this, we need to show that 439 λ > 1 ω V H . 440 We first consider a disease-free population (I(t) = V H I (t) = 0 for all t). We denote the fraction of 16 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

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