key: cord-0759205-68ekfn3g
authors: Bushman, Mary; Kahn, Rebecca; Taylor, Bradford P.; Lipsitch, Marc; Hanage, William P.
title: Population impact of SARS-CoV-2 variants with enhanced transmissibility and/or partial immune escape
date: 2021-11-19
journal: Cell
DOI: 10.1016/j.cell.2021.11.026
sha: d7783d1f811d29a7747cd3670582bf50355922a0
doc_id: 759205
cord_uid: 68ekfn3g

SARS-CoV-2 variants of concern exhibit varying degrees of transmissibility and, in some cases, escape from acquired immunity. Much effort has been devoted to measuring these phenotypes, but understanding their impact on the course of the pandemic – especially that of immune escape – has remained a challenge. Here, we use a mathematical model to simulate the dynamics of wildtype and variant strains of SARS-CoV-2 in the context of vaccine rollout and nonpharmaceutical interventions. We show that variants with enhanced transmissibility frequently increase epidemic severity, whereas those with partial immune escape either fail to spread widely, or primarily cause reinfections and breakthrough infections. However, when these phenotypes are combined, a variant can continue spreading even as immunity builds up in the population, limiting the impact of vaccination and exacerbating the epidemic. These findings help explain the trajectories of past and present SARS-CoV-2 variants and may inform variant assessment and response in the future.

The second year of the COVID-19 pandemic has been dominated by variants of concern -SARS-27

CoV-2 lineages that have driven resurgent waves of the disease, often more severe than earlier waves. 28

The World Health Organization currently recognizes four variants of concern (VOCs): Alpha (lineage 29 B.1.1.7), which was first identified in the United Kingdom; Beta (B.1.351), first reported in South Africa; 30 Gamma (P.1), believed to have originated in Brazil; and Delta (B.1.617.2 and AY.x sub-lineages), first 31 detected in India. Initially reported in late 2020 or early 2021, these four variants have now reached 32 every continent except Antarctica (World Health Organization, 2021). 33

The spread of these variants has been attributed to some combination of enhanced 34 transmissibility and partial immune escape. Alpha is estimated to be 43-100% more transmissible than 35

wildtype (Davies et al., 2021; Volz et al., 2021) , but is similarly neutralized by convalescent sera (Planas 36 et al., 2021a; Supasa et al., 2021; Wang et al., 2021b) and is not associated with increased risk of 37 reinfection (Graham et al., 2021) . There is some uncertainty regarding the transmissibility and immune 38 escape of Beta because estimates of these quantities are inversely correlated. If immune escape is 39 minimal, Beta may be up to 50% more transmissible than WT (Tegally et al., 2021) . However, there is 40 considerable evidence suggesting a moderate degree of immune escape, with significantly reduced 41 neutralization by convalescent sera ( Zhou et al., 2021) . However, T cell responses may remain largely intact even when antibody 45 responses are compromised (Geers et al., 2021; Tarke et al., 2021) . Gamma is believed to be 70-140% 46 more transmissible than WT (Faria et al., 2021) , and perhaps has some degree of immune escape, with 47 a modest reduction in neutralization by convalescent sera (Wang et al., 2021a) . Gamma may reduce 48 protection against reinfection by 21-46%, although as with Beta, estimates of transmissibility and 49

immune escape are correlated (Faria et al., 2021) . Finally, multiple analyses suggest that Delta is at least 50 60% more transmissible than Alpha (Allen et al., 2021; Dagpunar, 2021) , and preliminary results suggest 51 moderately reduced neutralization by convalescent serum (Hoffmann et al., 2021; Planas et al., 2021b) . 52

The appearance of these variants has taken place against a backdrop of accelerated vaccine 53 development and rollout. Numerous candidate COVID-19 vaccines entered Phase III trials in the latter 54 half of 2020, and many began distribution in late 2020 or early 2021. As of November 7, 2021, over 7 55 billion vaccine doses have been administered, but coverage remains highly variable: over 70% of people 56

in high-and upper-middle income countries have received at least one dose, vs. 40% of people in lower-57 middle-income countries and 4% of those in low-income countries (Mathieu et al., 2021) . In the first 58 several months after vaccination, most authorized vaccines demonstrate at least 70% efficacy against 59 symptomatic COVID-19, and a few approach or exceed 90% efficacy ab severe disease are relatively rare in vaccinated individuals, and deaths exceedingly rare, so estimates of 62 efficacy against severe outcomes tend to be imprecise, but true efficacy against severe disease and 63 death is likely very high for most vaccines. 64

Along with evidence for partial escape from naturally acquired immunity, there is mounting 65 evidence that vaccines may offer reduced protection against some variants. There is little evidence to 66

suggest that Alpha evades vaccine-induced immunity; neutralization by post-vaccination sera is similar 67

to WT (Collier et For individuals, the implications of variants with partial vaccine escape are straightforward: 78 protection decreases linearly with the product of the reduction in efficacy and the variant's frequency in 79 the population. Estimating the risk to an entire population is more complex. Due to the nonlinearity of 80 epidemiological dynamics, population outcomes do not simply mirror those of individuals; a 30% 81 reduction in vaccine efficacy does not translate to 30% fewer infections prevented, or 30% more 82

infections in vaccinated individuals. The same goes for transmissibility: a 50% increase in the average 83 number of secondary infections from each case does not simply produce a proportional increase in the 84 total number of infections. 85

The difficulty of predicting population-level outcomes from variant phenotypes has limited our 86 ability to distinguish between variants of greater and lesser impact. "Variant of concern" is an umbrella 87 designation which indicates the possibility of adverse consequences but not the probability or 88 magnitude of those consequences. This may be a reflection of how challenging it is to quantify 89 phenotypes like transmissibility, immune escape, and disease severity. Even with precise estimates, 90 however, the leap from phenotype to population impact can be complex. 91

The situation would be more straightforward if variants existed in a vacuum, but they are part of 92 a complex dynamical system. 

A full description of the model can be found in the Methods, but we include a short overview 116

here to give context for the results. The following describes the default model conditions and 117

assumptions, but analyses varying many of these assumptions are included in the Results. 118

The model is an extended Susceptible-Infected-Recovered (SIR) compartment model, which 119 includes two strains -wildtype (WT) and variant -as well as vaccination. The WT is assumed to have a 120 J o u r n a l P r e -p r o o f basic reproduction number of 2.5, which is reduced to 1.5 by nonpharmaceutical interventions (NPIs) 121 that remain in place throughout the simulation . Both the variant and the vaccine are introduced at  122  specified times midway through the epidemic, and the vaccine is distributed at a constant rate until  123 100% coverage is reached. The vaccine is assumed to be 95% effective against WT, with efficacy against 124 the variant proportional to cross-reactivity between the strains. Infection is assumed to confer sterilizing 125 immunity against the infecting strain; protection against the other strain is proportional to the degree of 126 cross-reactivity, which is assumed to be symmetric. The model is run for a simulated duration of three 127 years (except where noted otherwise) and we assume no waning of immunity over this period (a key 128 reason why the results of this model should not be extrapolated beyond the epidemic phase). All 129 simulations assume a population size of 100 million individuals. 130

The purpose of the model is to examine the impact of two key phenotypes -transmissibility and 131

immune escape -on the population dynamics of emerging variants and the course of SARS-CoV-2 132 epidemics. In order to disentangle the effects of these traits, and to explore their interactions, we 133 consider three hypothetical variants: Model behavior is analyzed in terms of numbers of infections, which when summed over the 146 course of a simulation, we denote by epidemic size. We emphasize numbers of infections, rather than 147 cases or individuals infected; the former denotes only those infections which are detected, while the 148 latter ignores the fact that individuals may be infected more than once. We assume that epidemic size is 149 roughly proportional to key outcomes such as total hospitalizations and total deaths, and therefore use 150 this outcome as a principal metric for comparison, although in some cases we distinguish between 151 infections in susceptible and recovered/vaccinated individuals, as the latter are less likely to suffer 152 severe disease and death. We also measure the population impact of vaccination by calculating the 153 difference in epidemic size between simulations with and without vaccination. 154 155

Ability to increase to high frequency is driven primarily by transmissibility, not immune escape 156 157

We start by examining the dynamics that result when variants are introduced into WT epidemics 158 in the absence of vaccination. As expected, variant 0 (the neutral variant) behaves identically to WT: its 159 growth and decline occur at the same times and at the same rates, only in smaller numbers ( Fig 1A) . In 160 contrast, variant 1 and variant 3, which both have enhanced transmissibility, spread faster than WT and 161 quickly rise to high frequency. Variant 2, with partial immune escape, does not, and infects far fewer 162 people than WT over the course of the simulation (Fig 1B) . When vaccination is added, variant 1 is 163 rapidly controlled, as is variant 2, despite being partially refractory to the vaccine ( Fig 1A) . Although 164 vaccination "flattens the curve" of variant 3 by reducing its transmission rate, it is unable to completely 165 control this variant, which has a combination of enhanced transmissibility and immune escape. 166 167

Variants do not necessarily undermine the population-level impact of vaccination 168 169

Calculating the numbers of infections averted by vaccination in the simulations from Fig 1A, we 170 find that, surprisingly, variants do not necessarily reduce the population-level impact of vaccination. 171

Compared to simulations with the neutral variant, vaccination averts as many or more variant infections, 172 as well as more infections overall, in simulations with all three of the other variants ( Fig 1C-D) . For 173 variants 1 and 2, vaccination also averts a higher percentage of variant infections ( Fig 1E) and an equal 174 or greater percentage of total infections ( Fig 1F) Epidemic size is affected more by transmissibility than by partial immune escape 180 181

We next vary the timing of vaccine rollout -changing both the time at which vaccination begins 182 and the rate at which the rollout proceeds -to examine how variants with different phenotypes behave 183 across a range of vaccination scenarios. We find that the total numbers of infections are nearly identical 184 between variant 0 (the neutral variant) and variant 2, which has a partial immune escape (Fig 2A) ; the 185 number and percentage of infections averted by vaccination are likewise similar (Fig 2B-C increase is particularly great for variant 3, which also has partial immune escape (Fig 2A) . As noted 190 above, the number of infections averted by vaccination is sometimes greater for variants 1 and 3 than 191 for the neutral variant ( Fig 2B) . This occurs because these variants generate substantially more 192

infections in the absence of vaccination; if a large enough fraction of these are prevented by vaccination, 193 then more infections are averted in total. However, the percentage of total infections averted by 194 vaccination is not much higher than with the neutral variant in any of the scenarios examined, and is 195 sometimes markedly lower, especially for variant 3 (Fig 2C) . 196 197

Population-level outcomes are more sensitive to vaccination start time than duration of vaccine rollout 198 duration 199 200 Naturally, epidemic size increases when the start of vaccination is delayed or the pace of vaccine 201 rollout slows (Fig 2A) , and the reverse is true of vaccination impact: as vaccine rollout is delayed or 202 slowed, the number and proportion of infections averted decrease (Fig 2B-C) . These outcomes appear to 203 be more sensitive to vaccination start time than the pace of vaccine distribution; a one-month delay in 204 starting vaccination impacts outcomes more than a one-month increase in the duration of vaccine 205 rollout. The sensitivity to vaccination start time is particularly pronounced for variants 1 and 3, which 206 have increased transmissibility. Higher transmissibility increases the rate at which cases grow, which 207 means the epidemic peaks earlier. A delay in starting vaccination can result in the peak of the epidemic 208

being missed entirely, even if the pace of vaccine distribution is high ( Fig 2D) . However, if vaccination 209 begins early, the effect on the final size of the epidemic can be considerable because it starts to take 210 effect when the epidemic is still smaller, even if the rollout (the time before vaccination becomes fully 211 effective) is slow..

Reinfections and breakthrough infections remain rare with moderate immune escape unless aided by 214 transmissibility 215 216

In any scenario involving variants, especially variants with immune escape, it is important to 217 consider how many infections occur in people with immunity from prior infection or vaccination, as a 218 means of distinguishing between the potential to infect people with acquired immunity and the actual 219 occurrence of such infections. In addition, infections in those with immunity are likely to be less severe 220 than infections in susceptible (immunologically naïve) individuals, so distinguishing between infections 221 in susceptible and recovered/vaccinated hosts may better reflect the severity of an epidemic. We 222 therefore separated infections into those occurring in recovered/vaccinated individuals and those in 223 individuals with no history of infection or vaccination (primary infections). Total numbers of primary 224

infections exhibit the same patterns as overall epidemic size (Fig 2A, Fig 3A) , but infections in recovered 225 and vaccinated individuals exhibit different behavior. 226

Reinfections and breakthrough infections (which we define as active, transmissible infections in 227 previously infected and vaccinated individuals, respectively) are negligible in simulations with variants 1 228 and 2 (Fig 3B-C) . Reinfections and breakthrough infections do occur with these variants (Fig 3D) the potential to produce widespread reinfections and breakthrough infections under some 233 circumstances.) However, a combination of partial immune escape and increased transmissibility 234 produces significant numbers of reinfections and breakthrough infections, accounting for up to 80% of 235 all infections (WT + variant) following emergence ( Fig 3C) .

When control measures are weakened, the impact of variants with enhanced transmissibility and partial 238

immune escape is even greater 239 240

In the default model, aside from vaccine rollout, we assume optimal control measures: 241 vaccination coverage reaches 100%, vaccine efficacy against WT is 95%, and NPIs are maintained 242

indefinitely. We now explore how the impacts of different variants are affected by weakening control 243 measures in various ways. We again run simulations in which we vary the timing of vaccine rollout, and 244

consider three changes to control measures: lifting NPIs once vaccination coverage reaches 50% (Fig 4A) , 245

reducing the final vaccination coverage to 50% (Fig 4B) , and reducing the vaccine efficacy vs. WT to 70% 246 ( Fig 4C) . (In the following section, we discuss a fourth scenario in which these three changes are 247 implemented simultaneously.) In all three scenarios, we find that variant 2 still has a negligible impact 248 on epidemic size, while epidemic size is markedly increased for variant 1 and especially variant 3. 249

However, the difference between these variants is larger than in the default model, suggesting that 250 partial immune escape has a greater impact in these scenarios. Unlike in the default model (Fig 2A) , 251 epidemic size with variant 3 is considerable even when vaccine rollout is early and fast. Variant 3 has the 252 highest threshold for control: increased transmissibility means that transmission must be reduced by a 253

larger amount to bring it under control, and partial immune escape means that a higher level of 254 vaccination is required to achieve a given reduction in transmission. This is especially true early on, 255

when there is less infection-induced immunity, which also helps to control transmission. Thus, the 256 impact of weakening these control measures is particularly pronounced with earlier and faster vaccine 257 rollout (dynamics shown in Fig 4D) .

Sufficiently weak control measures can lead to a second wave of infections with immune escape variants 260 261

We now consider the first of three scenarios in which the behavior of these hypothetical 262 variants differs from the general findings above. For each, we highlight the differences and consider the 263 underlying dynamics. 264

Above, we describe simulations in which control measures are weakened by lifting NPIs, 265 reducing vaccination coverage, or decreasing vaccine efficacy. We now consider simulations in which 266 these three changes are implemented simultaneously (Fig 5) , and the results differ from previous 267 findings in two ways. The first is that the timing of vaccine rollout has no clear impact on epidemic size, 268

suggesting that the weakened control measures are not able to bring any of the variants fully under 269

control. Indeed, for variants 0, 1, and 3, the dynamics look similar to those observed in the absence of 270 vaccination (Fig 6 A,B ,D,V,W,Y). Second, the epidemic size is dramatically increased in simulations with 271 variant 2, in many cases exceeding the levels seen in the absence of vaccination ( Fig 5A; also refer to Fig  272  2A ). The increase is caused by a second wave of variant infections following the initial WT epidemic (Fig  273  6X ). This second wave occurs in a "Goldilocks zone" in which the frequency of recovered/vaccinated 274 hosts is sufficient to control the WT but not the variant. Higher levels of acquired immunity -induced 275 either by vaccination or by widespread infection -are able to control the spread of both strains (Fig  276  6C ,G,K,P,T). 277

The second wave -and indeed any second wave in a population with high levels of acquired 278

immunity -is necessarily comprised predominantly of reinfections and breakthrough infections (Fig 5B) , 279 which tend to be mild. When WT and variant waves occur simultaneously, as seen with variant 3 ( Degree of immune escape affects the propensity for a second wave of variant infections 284 285

As mentioned in the introduction, variants with less than 40% immune escape have almost 286 certainly arisen already , while variants with higher degrees of immune escape are at least a theoretical 287 possibility. It stands to reason that these differences would affect the dynamics of immune escape 288

variants, and so we now consider simulations in which we assign higher or lower values of immune 289 escape to variants 2 and 3. In one set of scenarios, we assume 20% escape (or 80% cross-protection), 290

while in the other we assume 80% escape (20% cross-protection). In both cases, we still assume a 60% 291 increase in transmissibility for variants 1 and 3. 292

In general, a lower degree of immune escape does not qualitatively change our findings ( In the default model, we assume the WT strain has R 0 = 2.5, similar to the SARS-CoV-2 virus that 308 dominated through most of 2020. However, variants with significantly increased transmissibility -first 309

Alpha, followed by Delta -subsequently replaced these less transmissible strains, effectively becoming -310 at least transiently -new wildtypes. We now consider a scenario in which the WT is highly transmissible, 311 similar to Delta, with R 0 = 6. (To avoid confusion with the default WT, we designate this alternative strain 312 WT Δ and denote the associated variants by 1 Δ , 2 Δ , and so on.) To better approximate the circumstances 313 in which these highly transmissible variants emerged, we assume that vaccine rollout begins before or 314

shortly after WT Δ appears, and because the spread of highly transmissible strains is accelerated 315

(occurring over shorter timeframes), we assume that variants emerge three months after WT Δ . Variant 316 phenotypes are the same as in the default model, but defined with respect to WT Δ ; for instance, variant 317 1 has R 0 = 9.6. 318

The key difference from earlier scenarios is that vaccination and/or a large WT Δ epidemic always 319 generate high levels of immunity in the population prior to emergence of the variant. The resulting lack 320 J o u r n a l P r e -p r o o f of susceptible hosts is sufficient to limit the spread of variant 1 Δ (Fig 7A) but not variant 2 Δ or 3 Δ . The 321

latter variants are able to spread even when the entire population is vaccinated prior to their 322 emergence (Fig 7B-C) , although these infections are necessarily all breakthrough infections. For these 323 variants, early and fast vaccine rollout has less impact on the total epidemic size ( Fig 7D) but leads to an 324 epidemic in which the majority of infections are breakthroughs and reinfections, which are generally 325 mild ( Fig 7E) . As a result, earlier/faster vaccine rollout does reduce the total number of primary 326

infections, a possible proxy for disease burden (Fig 7F) . However, this is mainly attributable to the 327 impact of vaccination on WT Δ , which means that vaccination must be timed to avert the WT Δ epidemic, 328 rather than subsequent waves of immune escape variants. 329 330

Qualitative findings are largely robust to changing structural model assumptions 331 332

Finally, we re-examine our findings after varying two aspects of the model structure. which this occurs in practice will depend on the strength of NPIs, the responsiveness to case numbers 343 (e.g. thresholds and lags for implementation), and the transmissibility of the variant. 344

Lastly, the default model assumes that immunity reduces the probability of infection by 345

decreasing the rate of movement from uninfected to infected states (so-called "leaky" immunity). An 346 alternative construction assumes that a given individual either does or does not develop immunity to a 347 given strain following infection or vaccination (termed "all-or-nothing" immunity). Using an alternate 348 model with all-or-nothing immunity (see Methods), we obtain results that are qualitatively 349

indistinguishable from leaky immunity, although epidemic size is lower with all-or-nothing immunity ( In this work, we use a mathematical model to characterize the population-level impact of SARS-355

CoV-2 variants with different phenotypes across a wide range of scenarios. We find that variants with 356 enhanced transmissibility invade easily in susceptible populations, while variants with partial immune 357 escape do not; the latter can sometimes produce a second wave of infections, but these primarily occur 358

in recovered and vaccinated individuals, who typically experience mild disease. Although the impact of 359 partial immune escape on its own is relatively mild, variants with a combination of enhanced 360 transmissibility and immune escape increase not just the total size of the epidemic but also the number 361 of primary infections in susceptible hosts, who are more likely to suffer severe illness or death. certain strains as "variants of concern" using similar criteria, which include evidence of immune escape, 374

vaccine escape, and/or enhanced transmissibility. In general, variants are not assigned threat levels or 375 differentiated status, although the US CDC would recognize variants with clear evidence of significant 376 vaccine escape as "variants of high consequence." Our findings suggest that variants with enhanced 377

transmissibility have a strong tendency to invade and can significantly worsen an epidemic. Partial 378 immune escape, in the absence of enhanced transmissibility, becomes a substantial proportion of the 379 epidemic only when population immunity is in a "Goldilocks zone" -strong enough to impose effective 380 selection but not so strong as to control transmission -and when invasion does occur, the variant 381 primarily causes breakthrough infections and reinfections, which are usually mild. 382

These results provide a theoretical basis to understand the behavior of existing variants, 383

anticipate the behavior of future variants, and develop appropriate strategies to mitigate the impact of 384 variants of concern in populations across the world. Our findings are consistent with the global sweeps 385 by highly transmissible variants Alpha and Delta, as well as the failure of Beta (which shows evidence of 386 partial immune escape) to reach high frequency in most areas. The ability to find patterns of risk by 387 modeling different variants across a wide range of scenarios suggests that this is a useful approach to 388 identify variant phenotypes of particular concern. Lastly, our work underscores the importance of 389 vaccination on a global scale, as quickly as possible, to mitigate the impact of present and future 390 variants. 391 392

Limitations of the study 393 394

An obvious limitation of our model is that it does not allow for circulation of multiple variants. In 395 reality, variants compete with one another as well as WT, and this may lead to complex behavior that is 396 difficult to predict from pairwise interactions. It is particularly challenging to anticipate the dynamics of 397

a system with multiple variants that each have some degree of enhanced transmissibility and immune 398 escape, as the relative fitness of these strains may change as the level of immunity in the population 399 increases. Additional modeling is therefore required to understand the risks associated with multiple 400 variants circulating simultaneously. 401 We also use epidemic size as a rough proxy for the burdens of disease and mortality, stratifying only by 402 immune status (susceptible vs. recovered/vaccinated Default model with "leaky" immunity. (B) Alternative model with "all-or-nothing" immunity. Numbered 635 terms at right give rates of movement associated with numbered arrows in each diagram. 636

Lead contact 641 642

Requests for further information should be directed to and will be answered by the lead contact, Mary 643

Bushman ( vaccination. Infection is a mass-action process resulting from contact between infected individuals and 687

individuals with partial or total susceptibility to infections. All individuals infected with a given strain are 688 assumed to be equally infectious, with transmission rate for WT and (1 + ) for the variant (see 689

Variant phenotypes). All infected individuals also have the same average duration of infectiousness, 690

which is the inverse of the recovery rate, . Individuals may have varying susceptibility to each strain, 691

depending on prior infection and/or vaccination. Upon recovery, individuals develop sterilizing immunity 692 against the infecting strain and partial protection against the non-infecting strain; the degree of cross-693 protection is given by . 694

Once vaccine rollout is initiated (see Simulation Overview), a fixed number of individuals are 695 vaccinated per day, but these individuals are drawn only from eligible compartments. By default, all 696 unvaccinated individuals are eligible. The parameters and control the vaccine eligibility of infected 697 and recovered individuals, respectively. Vaccination takes place in a single dose and is assumed to take 698 effect immediately, unless the recipient is currently infected, in which case the vaccine takes effect upon 699 recovery. The vaccine is assumed to have maximum efficacy against a perfect antigenic match. The 700 degree of antigenic mismatch between the vaccine and strain is given by . Both infection-induced 701 and vaccine-induced immunity are assumed to be durable, lasting longer than the duration of the 702 simulations; waning of immunity is not included in the model. Variants are characterized in terms of two phenotypes: transmissibility and immune escape. The 708

increase in transmissibility relative to WT is denoted by , giving a transmission rate of (1 + ) . 709

Immune escape is the complement of cross-reactivity between WT and variant (1 − ). ( ) = + ( 1 + 2 ) + ( 1 + 2 + 12 ) + ( 1 + 2 )

[1] 726 727

We then define ( ) as follows: 728 729 In addition, because there are six infected compartments for each strain, we simplify the equations by  732 defining new state variables representing the total number of infected individuals for each strain: 733 734 1 ( ) = 1 + 1 ( ) + 1 + 1 ( ) + 1 (2) + 1 (0)

[3] 735 736 2 ( ) = 2 + 2 ( ) + 2 + 2 ( ) + 2 (1) + 2 (0)

[4] 737 738

The rates of movement between compartments are shown alongside the model diagram in Fig NPIs are assumed to be maintained at a constant level throughout the simulation, with two 797 exceptions: scenarios in which NPIs are lifted once vaccination coverage reaches 50% (see Control  798 measures) and scenarios in which NPIs switch between high-and low-intensity states (see Rolling  799 lockdowns). These control measures are assumed to reduce transmission of both strains by a factor . 800 801

Core set of simulations 802 Simulations were run with each variant in combination with WT; each pair was simulated with 804 and without vaccination (Table S1 ). For each set of simulations, the start of vaccine rollout ( ) and 805 the duration of vaccine rollout ( 1 ) were varied in one-month increments over the ranges given in Table  806 S1. 807 808

Model outcomes 809 810

Epidemic size 811 812

The main outcome compared across simulations is the cumulative number of infections, which is 813 modeled as the sum of flows into all infected compartments. Similar quantities are defined for infections 814 in naïve individuals (previously belonging to compartment ) and in recovered/vaccinated individuals 815 (coming from and compartments). The relative frequency of reinfections and breakthrough 816

infections is obtained by dividing the number of infections in recovered/vaccinated individuals by the 817 total number of infections in each simulation. 818 819

Impact of vaccination 820 821

Vaccine impact is obtained by comparing numbers of infections between simulations with 822 vaccination (sets 5-8 in Table S1) Control measures 845 846

The default model configuration, with indefinite continuation of NPIs, complete vaccination 847 coverage, and high vaccine efficacy, represents the best-case scenario in each of these areas; we 848 therefore vary these assumptions to simulate realistic -not pessimistic -shortcomings. We consider 849 three ways in which control measures might suffer (Table S2) : lifting NPIs when vaccine coverage 850 reaches 50%, decreasing the final vaccination coverage to 50%, and lowering the peak vaccine efficacy 851 (against WT) to 70%. We repeat simulation sets 5-8 (Table S1 ) under each of these alternative 852 parameterizations, as well as the combination of all three. 853 854

Lower and higher degrees of immune escape 855 856

The default model assumes that immune escape manifests as a 40% reduction in cross-reactivity 857 with WT. This value is on the high end of escape estimates for existing variants, but lower degrees of 858 escape almost certainly exist, and higher degrees of immune escape are theoretically possible. We 859 therefore ran simulations with a lower level of immune escape (20% escape or 80% cross-protection 860

with WT) and a higher level (80% escape or 20% cross-protection). For variants with enhanced 861 transmissibility, we still assumed a 60% increase in R 0 . We repeated the simulations listed in Tables S1-862 S2 for both scenarios; simulations with a higher level of immune escape were run for an extended 863 duration (six years) because the occurrence of a second wave of variant infections in some scenarios 864 increased the time frame over which epidemic dynamics were occurring. 865 866

Increased transmissibility of all strains 867 868

The default model assumes that the WT strain has an R 0 of 2.5, which approximates the original 869 SARS-CoV-2 virus. However, this version has since been replaced by more transmissible variants. We 870 therefore ran simulations in which the WT has R 0 = 6; the variant phenotypes are the same but are 871 defined with respect to the new WT (i.e. all strains have R 0 ≥ 6). We re-ran the simulations listed in Table  872 S1 (but not those in Table S2 ) under this alternative parameterization; however, the timing of events 873 was shifted, with the WT and variant strains introduced at 3 and 6 months, respectively, and vaccine 874 rollout beginning between 0 and 9 months. 875 876

Varying model structure 877 878

Finally, we run simulations under two alternative versions of the model which make different 879 assumptions about nonpharmaceutical interventions and immunity, respectively. We repeat the 880 simulations in Tables S1-S2 under both alternative model versions.  881  Rolling lockdowns  883  884 In the default model, we assume that nonpharmaceutical interventions (NPIs) are fixed at an 885 intensity that reduces transmission by 40%. This configuration generates dynamics that are easy to 886 understand and analyze, but is not broadly representative of the dynamics of the COVID-19 pandemic. In 887 many parts of the world, relatively weak NPIs are periodically supplanted by more stringent measures 888 when case numbers grow too large; after a period of sustained decline, the more severe control 889 measures are turned off again. We use an alternative model configuration to simulate this strategy, 890

which we refer to as "rolling lockdowns." 891

In the alternative configuration, NPIs switch between two different intensities, which reduce 892 transmission by 30% and 70%, respectively. The high-intensity control measures are triggered when the 893 number of current infections -lagged by 14 days to simulate various delays between infection and 894

reporting -exceeds 1% of the total population. Control measures revert to the lower intensity when the 895 number of infections (lagged by 14 days) drops below the threshold again. Shifts between low and high 896 intensity of NPIs cannot occur less than 14 days apart (the length of the reporting lag), to ensure that 897

shift is precipitated only by events following shift − 1.

All-or-nothing immunity 900 901

The default model configuration assumes that partial cross-protection from infection or partial 902 immunity from vaccination is imperfect at the individual level, meaning the probability of infection given 903 exposure is reduced but not eliminated, and the degree of protection is the same for all individuals. This 904 is sometimes called "leaky" immunity; an alternative formulation, termed "all-or-nothing" immunity, 905

does not accommodate partial protection for individuals. Upon infection or vaccination, some 906 individuals acquire complete, sterilizing protection, and others acquire no protection at all. 907 We use an alternative version of the ODE model described above to re-run the same scenarios 908 with all-or-nothing immunity instead of leaky immunity. Although the names of the state variables are 909 largely unchanged, many are defined differently (see below Previously vaccinated, not immune to either strain, currently infected with strain 916

The model parameters are generally the same, except that parameters relating to partial 917 immunity ( , , 1 , and 2 ) are defined in terms of probabilities of protection, rather than degrees of 918 protection (see below). Rates of movement between compartments are very different from the default 919 model (Fig S7B) , as are the differential equations (Eq 25-44). 920 921

Probability of cross-protection given infection with either strain 0 or 0. 

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Changes in symptomatology, reinfection, and transmissibility 1017 associated with the SARS-CoV-2 variant B.1.1.7: an ecological study

Impact and effectiveness of mRNA BNT162b2 vaccine against SARS-CoV-2 1020 infections and COVID-19 cases, hospitalisations, and deaths following a nationwide vaccination 1021 campaign in Israel: an observational study using national surveillance data

SARS-CoV-2 variant B.1.617 is resistant to 1024 bamlanivimab and evades antibodies induced by infection and vaccination

Why does drug resistance readily evolve but vaccine resistance 1026 does not?

Evidence for increased breakthrough rates of SARS-CoV-2 variants of concern in 1029 BNT162b2-mRNA-vaccinated individuals

Reduced neutralization of SARS-CoV-2 B.1.617 by vaccine and 1032 convalescent serum

Effectiveness of Covid-19 Vaccines against the B.1.617.2 (Delta)

Neutralising capacity against Delta (B.1.617.2) and other variants of 1038 concern following Comirnaty (BNT162b2, BioNTech/Pfizer) vaccination in health care workers

Efficacy of the ChAdOx1 nCoV-19 Covid-19 Vaccine against the 1042 B.1.351 Variant

A global database of COVID-19 vaccinations

BNT162b2 vaccine effectiveness was marginally affected by 1047 the SARS-CoV-2 beta variant in fully vaccinated individuals

Neutralization of SARS-CoV-2 lineage B.1.1.7 pseudovirus by BNT162b2 vaccine-1051 elicited human sera

Clinical and virological features of SARS-CoV-2 variants of concern: a retrospective cohort 1054 study comparing B.1.1.7 (Alpha), B.1.315 (Beta), and B.1.617.2 (Delta)

Sensitivity of infectious SARS-CoV-2 B.1.1.7 and B.1.351 1058 variants to neutralizing antibodies

Reduced sensitivity of SARS-CoV-2 variant Delta to antibody 1061 neutralization

Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine

Safety and Efficacy of Single-Dose Ad26.COV2.S Vaccine against 1067 Covid-19

SARS-CoV-2 Delta VOC in Scotland: 1069 demographics, risk of hospital admission, and vaccine effectiveness

Reduced neutralization of SARS-CoV-2 B.1.1.7 variant by 1072 convalescent and vaccine sera

Impact of SARS-CoV-2 variants on the total CD4(+) and CD8(+) T cell reactivity in 1075 infected or vaccinated individuals

Detection of a SARS-CoV-2 variant of concern in South Africa

Assessing transmissibility of SARS-CoV-2 lineage B.1.1.7 in 1081 England

Safety and efficacy of the ChAdOx1 nCoV-19 vaccine 1084 (AZD1222) against SARS-CoV-2: an interim analysis of four randomised controlled trials in Brazil, South 1085 Africa, and the UK

Increased resistance of SARS-CoV-2 variant P.1 to antibody neutralization

Antibody resistance of SARS-CoV-2 variants B.1.351 and B.1.1.7

SARS-CoV-2 501Y.V2 escapes neutralization by 1093 South African COVID-19 donor plasma

Weekly epidemiological update on COVID-19 -10

Evidence of escape of SARS-CoV-2 variant B.1.351 from 1097 natural and vaccine-induced sera

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