## Significance

The ongoing pandemic of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) raises an important question: who should we vaccinate first? Answering this question requires an analysis of both the short-term (epidemiological) and the long-term (evolutionary) consequences of targeted vaccination strategies. We analyze the speed of pathogen adaptation and the cumulative number of deaths in heterogeneous host populations to shed light on the effects of alternative vaccination strategies. This analysis shows that minimizing the speed of pathogen adaptation does not always minimize the number of deaths. This evaluation of both the epidemiological and evolutionary consequences of public health policies provides a practical tool to identify the best vaccination strategy.

## Abstract

The limited supply of vaccines against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) raises the question of targeted vaccination. Many countries have opted to vaccinate older and more sensitive hosts first to minimize the disease burden. However, what are the evolutionary consequences of targeted vaccination? We clarify the consequences of different vaccination strategies through the analysis of the speed of viral adaptation measured as the rate of change of the frequency of a vaccine-adapted variant. We show that such a variant is expected to spread faster if vaccination targets individuals who are likely to be involved in a higher number of contacts. We also discuss the pros and cons of dose-sparing strategies. Because delaying the second dose increases the proportion of the population vaccinated with a single dose, this strategy can both speed up the spread of the vaccine-adapted variant and reduce the cumulative number of deaths. Hence, strategies that are most effective at slowing viral adaptation may not always be epidemiologically optimal. A careful assessment of both the epidemiological and evolutionary consequences of alternative vaccination strategies is required to determine which individuals should be vaccinated first.

The development of effective vaccines against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) raises hope regarding the possibility of eventually halting the ongoing pandemic. However, vaccine supply shortages have sparked a debate about the optimal distribution of vaccination among different categories of individuals. Typically, infections with SARS-CoV-2 are far more deadly in older individuals than in younger ones (1). Prioritizing vaccination for older classes may thus provide a direct benefit in terms of mortality (2, 3). Yet, younger individuals are usually more active, and consequently, they may contribute more to the spread of the epidemic. Prioritizing vaccination for younger and more active individuals may thus provide an indirect benefit through a reduction of the epidemic size (4, 5). Earlier studies have compared alternative ways to deploy vaccination in heterogeneous host populations and showed that recommendation varies with the choice of the quantity one is trying to minimize (e.g., the cumulative number of deaths, the remaining life expectancy, or the number of infections) (3, 6, 7). The recommendation also varies with the properties of the pathogen and the efficacy of the vaccine (3, 4, 8). For SARS-CoV-2, the increase in mortality with age is such that the direct benefit associated with vaccinating more vulnerable individuals tends to overwhelm the indirect benefits obtained from vaccinating more active individuals (2, 3, 9, 10). However, some studies challenge this view and identified specific conditions where vaccinating younger and more active classes could be optimal (5, 7, 11, 12). A similar debate emerges over the possibility to delay the second vaccination dose to maximize the number of partially vaccinated individuals. A quantitative exploration of alternative vaccination strategies can help provide useful recommendations: a two-dose strategy is recommended when the level of protection obtained after the first dose is low and/or when vaccine supply is large (13⇓⇓–16).

Vaccine-driven evolution, however, could erode the benefit of vaccination and alter the above recommendations which are based solely on the analysis of epidemiological dynamics. Given that hosts differ both in their sensitivity to the disease and in their contribution to transmission, who should we vaccinate first if we want to minimize the spread of vaccine-adapted variants? The effect of alternative vaccination strategies on the speed of pathogen adaptation remains unclear. Previous studies of adaptation to vaccines focused on long-term evolutionary outcomes (17, 18). These analyses are not entirely relevant for the ongoing pandemic because what we want to understand first is the short-term consequence of different vaccination strategies (19). A few studies have discussed the possibility of SARS-CoV-2 adaptation following different targeted vaccination strategies but did not explicitly account for evolutionary dynamics (12, 20). A recent simulation study explored the effect of a combination of vaccination and social distancing strategies on the probability of vaccine-driven adaptation (21). This model, however, did not study the impact of targeted vaccination strategies on the speed of adaptation.

Here we develop a theoretical framework based on the analysis of the deterministic dynamics of multiple variants after they successfully managed to reach a density at which they are no longer affected by the action of demographic stochasticity. We study the impact of different vaccination strategies on the rate of change of the frequency of a novel variant, which allows us to quantify the speed of virus adaptation to vaccines. Numerical simulations tailored to the epidemiology of SARS-CoV-2 confirm the validity of our approximation of the strength of selection for vaccine-adapted variants.

## Results

We are interested in tracking the frequency *p _{m}* of hosts infected by the vaccine-adapted variant among all the infected hosts. It is possible to show that under a broad range of conditions one can approximate the dynamics of the vaccine-adapted variant frequency asp˙m≈pm(1−pm)S(t),[1]where S(t) is the selection coefficient on the vaccine escape mutation. This selection coefficient measures the rate of change of the logit of the frequency of the vaccine-adapted variant [i.e., ln (pm/(1−pm))] and provides a relevant measure for the speed at which the viral population is adapting (

*Materials and Methods*).

Targeted vaccination strategies aim to preferentially vaccinate hosts according to specific epidemiological characteristics. For instance, we could target hosts that have more contacts or are more at risk for a severe disease. In our model, we therefore introduce some heterogeneity among hosts. As a result, from the point of view of the parasite, the quality of the host may differ among infected hosts, and this variation is likely to affect the dynamics of vaccine-adapted variants. To quantify host quality, we use the concept of reproductive value, a key concept in demography and evolutionary biology (22⇓–24). Reproductive value measures how much a virus infecting a given class of hosts will contribute to the future of the viral population. Our general mathematical analysis allows us to take the difference in host quality into account when calculating the selection coefficient S(t) (*Materials and Methods*).

We use this approach to analyze the speed of adaptation during the ongoing pandemic of SARS-CoV-2 under different scenarios. We use an epidemiological model tailored to the biology of SARS-CoV-2 (*Materials and Methods*). However, it is important to keep in mind that due to simplifying assumptions and uncertainty about parameter values, our results cannot be translated directly into public health recommendations without further investigations (*Discussion*). Nonetheless, our theoretical framework gives clear foundations for future applied work and captures some of the most salient features of the COVID-19 pandemic. In particular, we introduce a time-varying parameter *c*(*t*), which measures the intensity of nonpharmaceutical interventions (NPI). We assume that the epidemic is initially controlled by NPI, which yields successive epidemic waves before the deployment of vaccination at *t* = 150 d. We use this model to explore the effect of two different forms of heterogeneity on the speed of SARS-CoV-2 adaptation.

### Heterogeneity in Contact Numbers and Vulnerability.

In the first scenario we assume that hosts differ in their ability to mix and thus to transmit the disease. More specifically, following the model used by ref. 12, we assume that some hosts (*L*) have a low number of social interactions, while other hosts (*H*) have a higher number of contacts. These two types of hosts can be thought as corresponding to the older and younger halves of the population. The increased rate of social interactions among *H* hosts is captured by a parameter M≥1. Susceptible hosts are initially naive (*S ^{L}* and

*S*), but they can become vaccinated (S^L and S^H) at rates

^{H}*ν*and

^{L}*ν*, respectively. When vaccinated, hosts have a lower probability to become infected (rσ measures the efficacy at blocking infection), and if they become infected, they have a lower probability to transmit the virus (rτ measures the efficacy at blocking transmission) and to die from the infection (rμ measures the efficacy at reducing mortality). Viral adaptation, however, can erode these benefits. We consider different viral strains characterized by an escape trait

^{H}*e*which takes values between 0 (no escape) and 1 (full escape). The capacity of a variant to reduce the effect of the vaccine on transmissibility (infectivity) is captured by a function Eτ(e) for transmissibility (and Eσ(e) for infectivity), which allows us to quantify the overall ability of the virus to escape the protective effects of the vaccine as E(e)=Eτ(e)Eσ(e). Note that the capacity of a variant to reduce mortality does not affect the strength of selection in our model (i.e., the duration of infection is affected neither by the variant nor by the vaccine).

In *Materials and Methods*, we derive a simple approximation for the strength of selection acting on the vaccine-adapted variant:S(t)=(1−c(t)) β ΔE (S^L+M2S^H),[2]where ΔE refers to the change in vaccine escape ability caused by the mutation. This tells us that the intensity of selection depends on 1) the ability of the virus mutant to escape the protective effects of vaccine, 2) the densities of uninfected hosts (both *L* and *H*) who have been vaccinated, and 3) the relative number of contacts of each class of hosts. Note that the epidemiological impact of a higher contact rate (M>1) translates into a magnified selective impact (M2). Thus, if we have to choose between vaccinating *L* and *H* hosts, targeting *H* hosts is expected to select more strongly for the vaccine-adapted variant. Fig. 1*B* confirms that this approximation captures very well the temporal dynamics of the vaccine-adapted variant. In particular, the simulations confirm that targeted vaccination of the *L* hosts slows down the rate of adaptation of the virus.

*A*) A graphical presentation of the epidemiological life cycle with

*L*hosts who are more vulnerable to the disease and

*H*hosts who have a higher number of contacts. Infected hosts are indicated with a light red shading, and vaccination is indicated with a bold circle border. The force of infection on naive hosts is noted Λi=hi+h^i (see

*Materials and Methods*and Table 1 for additional details on this model). (

*B*) Dynamical change of the frequency

*p*of the vaccine-adapted mutant for two distinct targeted vaccination strategies: 1) mostly

_{m}*L*hosts are vaccinated (blue lines) and 2) mostly

*H*hosts are vaccinated (red lines). The full lines indicate the exact numerical computation, and the dashed lines indicate the approximation obtained from Eq.

**2**. The gray areas indicate the period where NPI were used to control the epidemic (c(t)=0.7 with NPI). (

*C*) Incidence of the epidemic (fraction of the total host population that is infected) in the absence of vaccination (dotted black line) or under the two alternative vaccination strategies used in

*B*(blue and red lines). (

*D*) Cumulative number of deaths (fraction of the total host population) in the absence of vaccination (dotted black line) or under the two alternative vaccination strategies used in

*B*(blue and red lines).

” data-icon-position=”” data-hide-link-title=”0″>

Of course, the choice of the vaccination strategy should not be based solely on the reduction of the speed of adaptation to vaccines. Indeed, the best way to limit the spread of vaccine escape mutations would be to adopt the worst epidemiological strategy: avoiding the use of vaccines. Yet, we urgently need vaccines to save lives and halt the current pandemic. We can use our numerical simulations to study the consequences of distinct targeted vaccination strategies on the total number of cases and on mortality (*Materials and Methods*). Fig. 1*C* shows that targeting *L* hosts is expected to increase the number of cases because *H* hosts contribute more to the spread of the disease. Yet, Fig. 1*D* shows that targeting *L* hosts is expected to decrease the cumulative number of deaths after some time because *L* hosts (i.e., older individuals) are also associated with higher risks of dying from the infection. Hence, targeting *L* hosts makes sense both for epidemiological and evolutionary reasons.

We explored the robustness of the above results for a range of alternative scenarios. First, we note that as expected from our analytic approximation, the use of a transmission-blocking vaccine (instead of an infection-blocking vaccine) yields very similar outcomes (compare Fig. 1 and *SI Appendix*, Fig. S1). Second, we show in *SI Appendix*, Fig. S2 that evolution amplifies the increase in the cumulative number of deaths when *H* hosts are vaccinated compared to a scenario without viral adaptation. Indeed, the spread of a vaccine-adapted variant drives a large epidemic wave in vaccinated populations. This evolutionary effect is maximized for intermediate values of the speed of the vaccination rollout because when vaccine rollout is very fast, the vaccine-adapted variant is rapidly favored, whatever the targeted vaccination strategy (*SI Appendix*, Fig. S3). Finally, we note that maintaining social distancing for longer can substantially decrease the speed of adaptation (*SI Appendix*, Fig. S4).

### Heterogeneity in the Number of Vaccination Doses.

In our second scenario, we assume that the heterogeneity among hosts is determined by differential strength of immunity induced by distinct vaccination status. We distinguish between unvaccinated hosts (*S*), hosts partially vaccinated with one dose (S^I), and hosts “fully vaccinated” with two doses (S^II). Using the same approach as before, we obtain the following expression for the strength of selection acting on the strength of selection on the vaccine-adapted variant:S(t)=(1−c(t)) β (ΔEI S^I+ ΔEII S^II).[3]

Eq. **3** is very similar to Eq. **2**, but now we have to account for the fact that the escape mutation has different effects in each class. Hence, the influences of an increase in the densities of hosts vaccinated by a single or two doses of vaccines are weighted by ΔEI and ΔEII, respectively. A single vaccine dose is likely to induce a lower protection against the virus (i.e., EI>EII), but this does not necessarily imply that ΔEI>ΔEII. In fact, we can show that if the vaccine is acting on a single step of the virus’ life cycle (e.g., only blocking infection), we expect ΔEII>ΔEI. Delaying the acquisition of the second dose will have two effects: 1) a lower density S^II of fully vaccinated hosts decreases the more intense selection imposed by these hosts but 2) delaying the second dose allows for more hosts to be vaccinated, and the increase in S^I may result in stronger selection for the vaccine-adapted variants. We show in Fig. 2*B* that this second effect can be more important than the first one, and delaying the second dose can result in faster adaptation. However, Fig. 2*D* shows that delaying the second dose may reduce the cumulative number of deaths because a larger fraction of the population would benefit from the protection of the vaccine (but higher rates of vaccination rollout can reverse this effect on mortality; *SI Appendix*, Figs. S5 and S6). Hence, in contrast to the previous scenario, the strategy that maximizes the speed of adaptation may result in a lower mortality. The contrast between our two scenarios illustrates the necessity to quantify both the epidemiological and the evolutionary consequences of different targeted vaccination strategies to identify the optimal way to distribute vaccines.

*A*) A graphical presentation of the epidemiological life cycle where the superscripts I and II refer to the first and second doses of vaccine. Infected hosts are indicated with a light red shading, and vaccination is indicated with a bold circle border. The force of infection on naive hosts is noted Λi=hi+h^i (see

*Materials and Methods*and Table 2 for additional details on this model). (

*B*) Dynamical change of the frequency

*p*of the vaccine-adapted mutant for two distinct targeted vaccination strategies: 1) vaccinated hosts receive two doses sequentially (purple lines) and 2) a single dose is used for each host (orange lines). The full lines indicate the exact numerical computation, and the dashed lines indicate the approximation obtained from Eq.

_{m}**3**. The gray areas indicate the period where NPI were used to control the epidemic (c(t)=0.7 with NPI). (

*C*) Incidence of the epidemic (fraction of the total host population that is infected) in the absence of vaccination (dotted black line) or under the two alternative vaccination strategies used in

*B*(purple and orange lines). (

*D*) Cumulative number of deaths (fraction of the total host population) in the absence of vaccination (dotted black line) or under the two alternative vaccination strategies used in

*B*(purple and orange lines).

” data-icon-position=”” data-hide-link-title=”0″>