key: cord-0427559-2fnjkbpv
authors: McLeod, David V.; Gandon, Sylvain
title: Multilocus adaptation to vaccination
date: 2021-09-15
journal: bioRxiv
DOI: 10.1101/2021.06.01.446592
sha: 0ef235697af1d944e9e175b0f283390b3fb47884
doc_id: 427559
cord_uid: 2fnjkbpv

Pathogen adaptation to public health interventions, such as vaccination, may take tortuous routes and involve multiple mutations at distinct locations in the pathogen genome, acting on distinct phenotypic traits. Despite its importance for public health, how these multilocus adaptations jointly evolve is poorly understood. Here we consider the joint evolution of two adaptations: the pathogen’s ability to escape the vaccine-induced immune response and adjustments to the pathogen’s virulence and transmissi-bility. We elucidate the role played by epistasis and recombination, with an emphasis on the different protective effects of vaccination. We show that vaccines reducing transmission and/or increasing clearance generate positive epistasis between the vaccine-escape and virulence alleles, favouring strains that carry both mutations, whereas vaccines reducing virulence mortality generate negative epistasis, favouring strains that carry either mutation, but not both. High rates of recombination can affect these predictions. If epistasis is positive, frequent recombination can lead to the sequential fixation of the two mutations and prevent the transient build-up of more virulent escape strains. If epistasis is negative, frequent recombination between loci can create an evolutionary bistability, such that whichever adaptation is more accessible tends to be favoured in the long-term. Our work provides a timely alternative to the variant-centered perspective on pathogen adaptation and captures the effect of different types of vaccines on the interference between multiple adaptive mutations.

e) amplifies the protective effect(s) of vaccination against allele N relative to allele E, and so can either increasingly favour (ρ 1 , ρ 2 , ρ 3 , ρ 5 > 0) or disfavour (ρ 4 > 0) allele E. Generally 115 speaking, if the costs of escape are not too high relative to the degree of escape, and vaccine 116 protection and coverage are not too low, allele E is favoured over allele N on the genetic 117 background A (s E > 0). 118 If instead evolution is restricted to the virulence locus (so µ E = 0 and f E = D = 0), 119 system (1) reduces to

Thus the more virulent strain will be favoured by selection if s V > 0. Selection for virulence the other (say s E s V ), then allele E will rapidly increase, favouring whichever virulence 237 allele it is initially associated with (i.e., the virulence allele hitch-hikes [54] ). Consequently, 238 if mutations at each locus are independent, strain EA will transiently dominate as it is more 239 mutationally accessible from strain N A (Fig. 4a ). If instead double mutations (e.g., N A 240 mutates to EV ) occur with comparable frequency to single mutations, or strains EA and 241 EV are initially equally abundant (e.g., due to chance fluctuations), strain EV (and positive 242 LD) will transiently dominate, as it benefits from both the positive epistasis and selection 243 on allele V . One key determinant of the relative magnitudes of s E and s V will be the costs 244 (c β , c γ ), and degree (e), of vaccine escape. If costs are low and vaccine escape is easily 245 accessible, typically s E s V , whereas costly and/or limited escape can lead to s V s E . 246 In the long-term, one (or both) of the alleles will approach quasi-fixation and LD will 247 vanish. Suppose allele E reaches quasi-fixation; then the dynamics of (1) reduce to

Here s V + s EV = r EV − r EA is the selection coefficient for allele V on the genetic background 249 E; thus whether allele A or V is ultimately favoured depends upon its sign. For example, infections, thereby increasing hosts and favouring allele V (Sup. Info. S1.7).

Recombination can affect these dynamics. By breaking up LD, recombination can prevent 256 the transient selection for the EV strain. If instead strain EV is favoured both transiently 257 and in the long-term, frequent recombination will prevent positive LD from building up in 258 the population. Therefore evolution occurs sequentially such that either allele E or V will fix 259 first (depending upon which allele is more strongly selected for), before evolution proceeds 260 at the other loci (Fig. 4b) . concern than the evolution of drug resistance. Proc. Natl. Acad. Sci. 115, 12878-12886. gov.uk/government/uploads/system/uploads/attachment data/file/1007566/ 553 S1335 Long term evolution of SARS-CoV-2.pdf).

[80] Polack, F. P., Thomas, S. J., Kitchin, N., Absalon, J., Gurtman, A., Lockhart, S., 

Pathogen genetics:

Per-capita growth rate of strain ij: In panel a, we highlight how each of the possible vaccine protections affect the epidemiological dynamics. In particular, vaccines may: reduce the risk of infection (ρ 1 ), reduce within-host pathogen growth (ρ 2 ), reduce infectiousness (ρ 3 ), reduce virulence (ρ 4 ), and/or hasten infection clearance (ρ 5 ). In panel b, we highlight the two-locus, diallelic evolutionary model. Each locus undergoes mutation (µ V , µ E ), while recombination occurs between loci (σI T ). Selection occurs through the additive selection coefficients (s E , s V ), and epistasis (s EV ); each of these is defined in terms of the per-capita growth rates of the different strains, r ij [42, 50]. In turn, the per-capita growth rates, r ij , depend upon the epidemiological model (and so are not constant), and are the average growth rate of strain ij across the two selective environments, vaccinated and unvaccinated hosts. Note that δ Ei is the Kronecker delta, and is equal to one if E = i and zero otherwise. If both vaccine-escape and virulence are advantageous (s E > 0 and s V > 0), but epistasis is negative (s EV < 0), alleles E and V are in competition. Starting from the strain N A equilibrium (solid black circle), strains N V and EA will increase in frequency, producing negative LD until the population is in the vicinity of the curve, Γ(f E ) (dashed black curve; see Sup. Info. S1.6). Along Γ(f E ), strains EA and N V are in direct competition, and so whichever allele (E or V ) has the larger selection coefficient will go to fixation (panel a, σ = 0); blue curves correspond to fixation of strain EA, red curves to fixation of strain N V . Recombination can create an evolutionary bistability such that 'faster' growing strains tend to reach fixation, even if they are competitively inferior (panel b, σ = 0.05); here the colours indicate which strain would fix in the absence of recombination. Each simulation starts with a monomorphic pathogen population (only the N A genotype is present initially) at it's endemic equilibrium following vaccination; mutation introduces genetic variation and allows pathogen adaptation to vaccination. Panels a,b show 100 simulations using the parameter set: p = 0.6, e = 1, β[α] = √ α, λ = d = 0.05, γ = 0.1, µ E = 10 −4 , µ V = 10 −4 , ρ 1 = ρ 3 = ρ 5 = 0, and with ρ 2 , ρ 4 , c β , c γ chosen uniformly at random on the intervals [0, 0.95], [0, 0.95], [0, 0.05], [0, 0.25], respectively, subject to the constraints that at the strain N A equilibrium, s E > 0, s V > 0, and α V /α A > 1.25. α A and α V were chosen to be the optimal virulence in an entirely unvaccinated and vaccinated population, respectively (Sup. Info. S1.4). , respectively, subject to the constraints that at the strain N A equilibrium, s E > 0, s V > 0, and α V /α A > 1.25. α A and α V were chosen to be the optimal virulence in an entirely unvaccinated and vaccinated population, respectively (Sup. Info. S1.4). (1) It reduces the probability of infection by a factor 0 ≤ ρ 1 ≤ 1.

(2) It reduces the growth of the pathogen within the host, and so reduces virulence by a 585 factor 0 ≤ ρ 2 ≤ 1. Since transmission is a function of virulence, this will also reduce 586 transmission.

(3) It reduces transmissibility (without affecting virulence) by a factor 0 ≤ ρ 3 ≤ 1.

(4) It reduces pathogen virulence (without affecting transmission) by a factor 0 ≤ ρ 4 ≤ 1.

(5) It hastens the clearance of the pathogen by a factor 0 ≤ ρ 5 .

Vaccine protection is life-long.

The pathogen has two diallelic loci of interest. The first locus controls vaccine escape: Info. S1.4).

There are thus four pathogen strains, ij ∈ {N A, N V, EA, EV }, and we will let I ij and 606Î ij denote the density of ij-infections in unvaccinated and vaccinated hosts, respectively, 607 at time t. Mutations occur at the escape and virulence loci at per-capita rates µ E and 608 µ V , respectively. Recombination (or genetic reassortment) between the two loci may also 609 occur. Specifically, if σ is a rate constant controlling the incidence of recombination, then 610 recombination between strains ij and k occurs at rate 2σI ij I k . Given recombination has 611 occurred, either strain is equally likely to be the 'recipient' or 'donor' of the allele at a loci chosen with equal probability. Thus the change in density of ij infections in unvaccinated 613 and vaccinated hosts, respectively, due to recombination is

where i = k and j = .

Let S andŜ denote the density of uninfected unvaccinated and vaccinated hosts, respec-616 tively, and let h ij denote the force of infection of strain ij, that is,

where δ Ei is the Kronecker delta, equal to one if E = i and zero otherwise. Then the 618 dynamics of the different strains in vaccinated and unvaccinated hosts are

for i = k and j = . Likewise, the dynamics of S andŜ are

From the preceding assumptions, if we let v ij =Î ij /(I ij +Î ij ) denote the fraction of ij-621 infections found in vaccinated hosts, the per-capita growth rate of ij-infections (so ignoring 622 mutation and recombination), denoted r ij , is equal to

denote the average transmission, virulence, and clearance, respectively, of strain ij. The 625 formulation of the r ij , specifically theβ ij ,ᾱ ij , andγ ij terms, emphasizes that the per-capita 

The additive selection coefficients for alleles E and V are defined as s E ≡ r EA − r N A and 639 s V ≡ r N V − r N A , respectively, while the epistasis in per-capita growth for alleles E and V , 640 is defined as s EV ≡ r EV − r EA + r N A − r N V (Fig. 2) . Using this notation, the evolutionary 641

variables are (f E , f V , D), and the evolutionary dynamics are described by

In addition to the evolutionary dynamics we also have the set of epidemiological variables, 

and

and

are the change in v ij infections due to mutation and recombination, respectively. 652 S1.2.1 Relationships between the v ij (t)

In general, following any brief transient dynamics which may be under the influence of the 654 initial conditions, we expect that v N V (t) ≥ v N A (t). The logic is that strain N A is fitter 655 in unvaccinated hosts, while strain N V is fitter in vaccinated hosts (see Sup. Info. S1.4).

Numerical results suggest the relation v N V (t) ≥ v N A (t) holds (Fig. S1) ; but to provide a 657 rough mathematical argument for why, observe that N j infections in unvaccinated hosts are 658 produced at rate h N j S, and on average will last (d + α j + γ) −1 time units. Similarly, N j 659 infections in vaccinated hosts are produced at rate (1 − ρ 1 )h N jŜ , and on average will last 660 (d + α j (1 − ρ 2 )(1 − ρ 4 ) + γ(1 + ρ 5 )) −1 time units. Thus we may be motivated to approximate

.

(S13)

Over long timescales (t → ∞), if strain N j is maintained by selection and mutation is small

; if strain N j is instead maintained by mutation, then although selection 664 (and the approximation (S13)) roughly apply, v N A will be closer to v N V than predicted by 665 (S13) (due to mutation). Over short-to-medium timescales, although v * N j (t) does not exactly 666 match v N j (t) (since this would require v N j (t) to be in quasi-equilibrium), numerical results

.

(S15)

Thus (S14) is positive for all t, i.e., v * N V (t) > v * N A (t), whenever α V > α A , vaccines imperfectly 671 prevent infection (ρ 1 < 1), and at least one of ρ 2 , ρ 4 , or ρ 5 are positive. It is easy to see 672 that v * EV (t) − v * EA (t) will assume a similar form to (S14), with each of the ρ i replaced by 673 (1 − e)ρ i , γ replaced with γ(1 + c γ ), and v * N j (t) replaced with v * Ej (t); thus we should also 674 expect v * EV (t) > v * EA (t) whenever α V > α A , at least one of ρ 2 , ρ 4 , or ρ 5 are positive, and 675 e < 1 (Fig. S1 ).

In Figure S1 we provide numerical support that the relation v * iV (t) ≥ v * iA (t) holds over 677 a wide range of parameter space using three different initial conditions; in all conditions we 678 start from a situation in which there is only strain N A present and we allow mutation to 679 introduce variation.

(1) For the first set of initial conditions (Fig. S1a-c) , we assume that we start from a vacci-681 nated population at the strain N A endemic equilibrium, that is, for a given parameter 

(2) For the second set of initial conditions (Fig. S1d-f ) , we assume the population is at the 685 strain N A endemic equilibrium before vaccination, that is,

(3) Finally, in the third set of initial conditions (Fig. S1g-i) we assume at the strain N A endemic equilibrium in the unvaccinated population (equation (S16)), we vaccinate a 689 fraction p of the uninfected hosts; therefore we have I T =Ī T , S =S(1 − p) andŜ =Sp, 690 and v N A = 0, whereĪ T andS are given in (S16).

For all three initial conditions, we see the prediction that v * iV (t) ≥ v * iA (t) holds (Fig. S1 ).

If evolution is restricted to the escape locus (so µ V = 0 and f V = D = 0), system (1) reduces

where the selection coefficient for vaccine escape, s E , is

By inspection of s E , decreasing the costs of escape (c β , c γ ) increases transmissibility and 

where the selection coefficient for virulence, s V , is

Since theβ N j ,ᾱ N j , andγ N j depend upon the v N j , selection for virulence on the genetic 

For example, if we assume that allele A is optimal in unvaccinated hosts (or in an entirely 722 unvaccinated population) and allele V is optimal in vaccinated hosts (or in an entirely 723 vaccinated population), and take β[α] = α b , then

In this circumstance, increasing coverage (p) and/or decreasing the vaccines ability to block simulations, we will assume β[α] = α b for b ∈ (0, 1) and that α A and α V are given in (S21).

Epistasis is generated whenever the fitness of an allele depends upon its genetic background, 741 and for continuous time models is defined as affecting the population structure (v ij (t)), and/or availability of susceptible hosts (S(t) and 748Ŝ (t)). Note that for this section we will explicitly write out the dependence of the variables 749 upon t to indicate that epistasis can temporally vary.

In the interests of clarity, we first will assume that vaccine escape is complete (i.e., 751 e = 1), before considering how partial vaccine escape affects epistasis. We will assume that The logic here is that transmission reducing vaccines lower the fitness of allele N relative to allele E. On the genetic background N , this disproportionately affects allele V relative to allele A because: (i) allele V is more transmissible (and the parameters ρ 1 , ρ 2 , ρ 3 multiplicatively interact with the quantity β[α]), and (ii) a greater proportion of strain N V infections are in vaccinated hosts

. By disproportionately reducing the fitness (equivalently, percapita growth) of strain N V , from (S22) this creates positive epistasis.

(2) Vaccines that reduce virulence mortality (ρ 2 , ρ 4 ) produce negative epistasis.

From (S22), the epistatic contribution of vaccine protection against virulence mortality

, equation (S24) is negative and so vaccines reducing 767 virulence mortality contributes negative epistasis.

The logic here is that, ignoring any concomitant effects to transmission (by ρ 2 ), vaccines which reduce virulence mortality increase the fitness of allele N relative to allele E by increasing infection duration. On the genetic background N , this increase in fitness disproportionately affects allele V relative to allele A because: (i) allele V is more virulent (and the parameters ρ 2 , ρ 4 multiplicatively interact with α j ), and (ii) a greater proportion of strain N V infections are in vaccinated hosts (v N V (t) ≥ v N A (t)). By disproportionately increasing the fitness (per-capita growth) of strain N V , from (S22), this creates positive epistasis.

(3) Vaccines that increase clearance (ρ 5 ) produce positive epistasis. The epistatic 770 contribution of vaccines hastening clearance is

Since v N V (t) ≥ v N A (t), equation (S25) is negative, and so vaccines increasing clearance contribute negative epistasis.

The logic here is that vaccines which increase clearance reduce the fitness of allele N relative to allele E by decreasing infection duration. On the genetic background N , this decrease in fitness disproportionately affects allele V relative to allele A because a greater proportion of strain N V infections are in vaccinated hosts (v N V (t) ≥ v N A (t)). By disproportionately decreasing the fitness (per-capita growth) of strain N V , from (S22), this creates negative epistasis.

In addition to the three categories of epistatic contributions of the vaccine protections, 775 which were also highlighted in the main text, the costs of escape can also produce epistasis. 

is negative, and so costs of escape to transmission 780 contributes negative epistasis.

The logic here is that costs of escape reduce the fitness of allele E relative to allele N by blocking transmission. On the genetic background E, this decrease in fitness disproportionately affects allele V relative to allele A because allele V is more transmissible and the costs of transmission interact multiplicatively with β[α]. By disproportionately decreasing the fitness (per-capita growth) of strain EV , from (S22) this creates negative epistasis.

The costs of escape to duration of carriage, c γ , do not directly contribute epistasis because 783 when vaccine escape is complete, they neither interact directly with the virulence locus 784 (i.e., through multiplicative interactions with α j and/or β[α]) nor are they influenced by 785 population structure (i.e., the difference between v iA (t) and v iV (t)). Therefore when e = 1, sume that the costs of escape are negligible. We also will assume throughout that β

As we will see, allowing for incomplete escape (e < 1) yields the same predictions for the 797 sign of the epistatic contribution as when escape is complete (e = 1) provided ψ V (t) > ψ A (t) 798 and θ V (t) > θ A (t). Notice that

Info. S1.2.1), and given that for should generally expect that the difference v iV (t) − v iA (t) will be larger when i = N rather 807 than when i = E, which would imply θ V (t) > θ A (t). Indeed, numerical results show that 808 θ V (t) > θ A (t) generally holds, even for small e (Fig. S2) . However, although rare, there 809 are exceptions (Fig. S2) ; thus all that can be said is that generally speaking θ V (t) > θ A (t)

(regardless of e), and that there is always a level of escape beyond which θ V (t) > θ A (t) will 811

hold. In what follows we will assume θ V (t) > θ A (t) holds.

With this in mind, the epistatic contribution of each of the ρ i under incomplete vaccine 813 escape (e < 1) are as follows:

(1) Vaccines reduce infection, ρ 1 > 0. Here

(2) Vaccines reduce pathogen growth, ρ 2 > 0. In this case,

Observe that the effect on transmission (first term) is positive since

(3) Vaccines reduce transmission, ρ 3 > 0. Here

(4) Vaccines reduce virulence, ρ 4 > 0. Here

which is negative because α V > α A and θ V (t) > θ A (t).

which is positive because θ V (t) > θ A (t). 826 Thus we see that the inclusion of partial vaccine escape will not change the sign of the 

If e = 1, this is negative (see equation (S26)). If e < 1, however, the sign of (S35) is 841 less clear since 

which is negative provided e < 1 (and zero otherwise).

As we typically lack knowledge about the precise type, or magnitude, of the costs of vaccine 

which is negative provided c β > 0. So in the unvaccinated population, epistasis can only be 865 negative.

At the other extreme, when the entire population is vaccinated, p = 1, then S(t) = 0 867 and v ij (t) = 1, and so equation (S22) reduces to 

In the vicinity of curve Γ(f E ), and assuming mutation and recombination rates are sufficiently 895 small (i.e., µ E , µ V , and σ small), system (S8) reduces to

Therefore whichever allele has the larger additive selection coefficient (i.e., s E or s V ), will 897 outcompete the other.

When recombination is frequent, then LD will be continuously removed from the popu- follows similarly). In this case, system (S8) can be approximated by

Thus from (S43), the escape allele will rapidly increase in frequency in the population due 934 to directional selection (s E ), irrespective of its genetic background (allele A or V ), while the 935 rate of increase in allele V (relative to allele E) will depend primarily upon the amount of 936 LD present. Specifically, whichever genetic background allele E is most commonly found on 937 initially will dominate; thus hitch-hiking occurs at the virulence locus [54] . As a consequence, 938 whichever of strain EA or EV has the initial numerical advantage will tend to dominant and N V are present, but rare); in this circumstance, strain EA (although rare) must 946 be initially much more abundant than strain EV (to ensure D(0) = 0 since f N A ≈ 1) 947 (Fig. S3a ).

• If instead strain EA and EV are initially equally abundant (but rare) or double-949 mutations (e.g., strain N A mutating to strain EV ) occur at a comparable rate to 950 single mutations, then because epistasis is positive, while (weak) directional selection 951 favours allele V , strain EV will transiently increase and substantial positive LD will 952 build up in the population (Fig. S3c) .

Thus if s E s V , even though (positive) epistasis produces positive LD, since s E s V , 954 whichever of strains EA or EV has the initial numerical advantage will dominate the tran-955 sient dynamics. Alternatively, selection favouring the escape allele is of equal strength to 956 selection favouring the virulence allele (i.e. s E ≈ s V ). In this case, both allele E and V will 957 increase at a similar rate in the population and so owing to the positive epistasis, substantial 958 positive LD can transiently build up (Fig. S3c ).

Although a variety of factors impact the strength of selection (the relative magnitude 960 of s E to s V ; see Sup. Info. S1.3 and Sup. Info. S1.4), one key mechanism is the costs of 961 escape (c γ , c β ) and the degree of escape (e). For example, if escape is near complete (e → 1), Figure S1 : A greater fraction of iV infections are in vaccinated hosts than iA infections, v iV (t) ≥ v iA (t). Each panel shows the difference between v iV (t) and v iA (t) for 250 randomly chosen parameter sets with the fraction vaccinated (p) given above each column. Note that in all cases, as expected (see Sup. Info. S1.2.1), v iV (t) ≥ v iA (t). Each row corresponds to a different set of initial conditions (ICs) described in Sup. Info. S1.2.1: the first row corresponds to ICs (1), the second to ICs (2), and the third to ICs (3). For each simulation, β[α] = √ α, with α A and α V as in equation (S21), while λ = d = 0.05, γ = 0.1, µ E = 10 −4 , µ V = 10 −4 , σ = 0, and the remaining parameters are chosen uniformly at random over the given intervals: (1 − e), ρ 1 , ρ 2 , ρ 3 , ρ 4 , c β ∈ [0, 0.95], ρ 5 ∈ [0, 8], c γ ∈ [0, 2]. Hence, each simulation start with a monomorphic pathogen population (only the N A genotype is present initially) and mutation introduces genetic variation and allows pathogen adaptation to vaccination. All parameter sets satisfied the constraints that s E > 0 and s V > 0 at the strain N A endemic equilibrium in a vaccinated population (ICs (1), see Sup. Info. S1.2.1), and that α V /α A > 1.2. Figure S2 : Relationship between θ V (t) and θ A (t) when vaccine-escape is incomplete. For almost all simulations, θ V (t) ≥ θ A (t); the few simulations that do not satisfy this condition are associated with limited vaccine-escape (small e; see Sup. Info. S1.5). Each panel shows the simulation results of 500 randomly chosen parameter sets for a different set of initial conditions (ICs) described in Sup. Info. S1.2.1: panel a corresponds to ICs (1); panel b to ICs (2); and panel c to ICs (3). For all simulations, β[α] = √ α, α A and α V are given by equation (S21), with p = 0.6, λ = d = 0.05, γ = 0.1, µ E = 10 −4 , µ V = 10 −4 , σ = 0, and the remaining parameters chosen uniformly at random over the intervals: (1 − e), ρ 1 , ρ 2 , ρ 3 , ρ 4 , c β ∈ [0, 0.95], ρ 5 ∈ [0, 8], c γ ∈ [0, 2]. Finally, all parameter sets satisfied the constraints that s E > 0 and s V > 0 at the strain N A endemic equilibrium in a vaccinated population (ICs (1), see Sup. Info. S1.2.1), and that α V /α A > 1.2. Figure S3 : When epistasis is positive, the evolutionary dynamics are sensitive to initial conditions. Panels a, c are when epistasis is positive, while panels b, d are when epistasis is negative. In panels a, b, double-mutations are not possible (e.g., strain N A cannot mutate to strain EV and vice-versa), whereas in panels a, d double mutations occur at the same per-capita rate as single mutations. If epistasis is positive (panel d), this leads to a substantial, transient build-up of LD, whereas if epistasis is negative (panel c), this has negligible effect. Each panel shows 100 simulations starting from a monomorphic pathogen population (only the N A genotype is present initially) at it's endemic equilibrium following vaccination; mutation introduces genetic variation and allows pathogen adaptation to vaccination. Parameters used were β[α] = √ α p = 0.6, e = 1, λ = d = 0.05, γ = 0.1, µ E = 10 −4 , µ V = 10 −4 , σ = 0. For panels a, c, ρ 2 = ρ 4 = c β = 0, while ρ 1 , ρ 3 , ρ 5 , c γ were chosen uniformly at random from the intervals [0, 0.95], [0, 0.95], [0, 15], [0, 5], respectively. For panels b, d, ρ 1 = ρ 3 = ρ 5 = 0, while ρ 2 , ρ 4 , c β , c γ were chosen uniformly at random from the intervals [0, 0.95], [0, 0.95], [0, 0.1], [0, 0.2], respectively. All parameter sets satisfied the constraints that s E > 0 and s V > 0 at the strain N A equilibrium and α V /α A > 1.25. α A and α V were chosen to be the optimal virulence in an entirely unvaccinated and vaccinated population, respectively (Sup. Info. S1.4).

Vaccines and their impact on the control of disease

Vaccines increase clearance, ρ 5 > 0. In this case, Assume both traits are under directional selection (s E > 0, s V > 0) and epistasis is