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

Why is drug resistance common and vaccine resistance rare? Drugs and vaccines both impose substantial pressure on pathogen populations to evolve resistance and indeed, drug resistance typically emerges soon after the introduction of a drug. But vaccine resistance has only rarely emerged. Using well-established principles of population genetics and evolutionary ecology, we argue that two key differences between vaccines and drugs explain why vaccines have so far proved more robust against evolution than drugs. First, vaccines tend to work prophylactically while drugs tend to work therapeutically. Second, vaccines tend to induce immune responses against multiple targets on a pathogen while drugs tend to target very few. Consequently, pathogen populations generate less variation for vaccine resistance than they do for drug resistance, and selection has fewer opportunities to act on that variation. When vaccine resistance has evolved, these generalities have been violated. With careful forethought, it may be possible to identify vaccines at risk of failure even before they are introduced.


Mathematical framework
Here we construct mathematical models that describe resistance emergence against various interventions. We use these models to derive distributions of times until resistance emerges for interventions that have or lack early action and multiple target sites. We find that the combined effect of early action and a high multiplicity of target sites greatly reduces the rate at which resistance emerges, relative to interventions that lack one or both of these features.
To begin, we posit four scenarios (summarized in Table S1): a prophylactic intervention with a single effector (A), a prophylactic intervention with multiple effectors (B), a therapeutic intervention with a single effector (C), and a therapeutic intervention with multiple effectors (D). We ask how these different interventions are expected to differ in the rates at which resistance first appears.
To simplify the problem, we make six assumptions. First, by removing the transmission advantage of partial resistance (see Figure 2), incomplete resistance does not transmit when using prophylactic interventions (A and B), but it does transmit when using therapeutic interventions (C and D).
Second, epistatic interactions do not occur between resistance mutations.
Below we discuss how this assumption could be relaxed. Third, evolution through the neutral force of genetic drift can be ignored. Fourth, selection is extremely strong such that any new resistance capable of transmission instantly becomes fixed in the population. Fifth, the probability of mutation towards resistance is both independent of the current level of resistance and equal for every replicative event, where a replicative event is defined as one pathogen cell or virion becoming two. Sixth, the probability of multiple resistance mutations occurring within a single replicative event is sufficiently small that it can be ignored when other pathways to resistance exist. Note that the qualitative ranking we give below would be unchanged if any of these last four assumptions were violated.
Given these assumptions, we can construct a model that describes the number of replicative events until resistance appears. Consider each replicative event to be a toss of a weighted coin where the two possible outcomes are 1) a mutation occurs that confers some degree of resistance, or 2) no resistance is gained. If we knew that r mutational events were required to achieve complete resistance, and that the probability of one of these events occurring in any particular replicative event was p, then the total number or replicative events at which complete resistance first appears would be r plus a negative binomially distributed random variable with parameters r and p.
It then follows from the mean of a negative binomial distribution that the average number of replicative events until this evolutionary process results in complete resistance to an intervention would be µ = r p . Note that in some cases, there may be two potential pathways to resistance. Complete resistance might emerge during a single replicative event, or it might emerge after an accumulation of a series of mutations that each confer partial resistance.
We use this model formulation to compare the four posited interventions above.
For Intervention A, where transmission of incomplete resistance does not occur, resistance will only emerge through one of the two pathways: a single mutation that confers complete resistance. The probability of resistance occurring in any single replicative event could then be described by where β is the probability that a single replicative event will result in a mu-tation that confers some level of resistance to a particular effector, and is the fraction of those mutations that confer complete resistance to that effector. Because Intervention A uses only a single effector, only a single mutation is required for complete resistance (r A = 1). The number of replicative events until resistance appears would then be the random variable W , where W − 1 ∼ NB(1, β), and thus the average number or replicative events until the emergence of resistance to Intervention A would be µ A = 1 β .
For Intervention B, resistance must again emerge during a single replicative event to spread through the pathogen population. However, in contrast to Intervention A, N resistance mutations must simultaneously occur for resistance to be achieved. The probability of resistance being acquired in a single replicative event would therefore be reduced to p B = ( β) N , while the number of mutational events will remain the same, r B = 1. Thus the number of replicative events until resistance appears would be the random variable X, Note that for any value of N greater than 1 and β less than 1, resistance to Intervention A will appear after fewer events on average than resistance to Intervention B.
For Intervention C, where partial resistance can be transmitted, complete resistance can be acquired through an accumulation of mutations that each confers partial resistance. Using the same parameter definitions as above, the probability of a partially resistant mutation occurring during a single replicative event would be p C = (1 − )β, and multiple mutations would be needed to confer complete resistance, r C = γ. The number of replicative events until complete resistance is acquired through an accumulation of partial resistance mutation would thus be the random variable U , where A key point is that resistance could also be acquired through a single mutation of large effect, identically to that described for Intervention A. Thus the number of replicative events until resistance emerges to Intervention C would be Y = min(W, U ). The dependence of W and U on the parameters and γ implies that when << 1 1+γ , resistance will tend to emerge through the accumulation of mutations each conferring partial resistance, resulting in µ C << µ A . In contrast, when >> 1 1+γ , resistance will tend to emerge through a single mutation that confers complete resistance, resulting in µ C ≈ µ A ( Figure S1). Although we are unable to write a closed form solution for µ C , we are able to give it an upper bound. First, note that µ C must of course be at least as small as µ A . Second, note that regardless of whether a mutation confers partial resistance or complete resistance, Intervention C will have failed by the time γ of them have occurred. We can For Intervention D, where the transmission of partial resistance can occur, and where there are N independent effectors, resistance to any single target would appear after Y replicative events identically to that for Intervention C. However, resistance would need to appear against each of N effectors before full resistance was achieved. The number of replicative events until resistance emerged against all of the effectors, would thus be the maximum of N realizations of Y . It thus follows that on average Intervention C will fail after fewer replicative events than Intervention D. Again, we are unable to write a closed form solution for the average number of replicative events until complete resistance is achieved µ D , but we can generate an upper bound.
Noting that the maximal element of a set of non-negative numbers will be less than or equal to the sum of those numbers, and that the sum of multiple negative binomial random variables with the parameters r i and p will itself have a negative binomial random distribution with the parameters i r i and p, the mean of the maximum of N realizations of Y must be less than or equal to N times the mean of Y . An upper bound of µ D is therefore N µ C .
We can quantify the evolutionary benefit of a prophylactic intervention with multiple effectors by dividing the mean number of replicative events until the failure of each intervention, by that of Intervention B, µ B . This analysis demonstrates that complete resistance against Intervention B would take on the order of 1 ( β) N −1 times as many replicative events to fail than any of the other three interventions ( Table S1). Given that β is presumably much less than one, an intervention that employs both prophylaxis and multiple effectors will drastically slow the evolution of resistance relative to an intervention that lacks either or both of these features ( Figure S1). We can thus conclude that prophylaxis and multiple effectors have a synergistic benefit in slowing the evolution of resistance to an intervention. 8 Note that the ordering given above, µ C ≤ µ A ≤ µ B and µ C ≤ µ D ≤ µ B , is based on the number of replicative events until resistance appears. However, a second consideration is that the number of replicative events within a given host will be much smaller for interventions that keep pathogen population sizes small (Interventions A and B) than those that allow pathogen population sizes to become large (Interventions C and D). This could have a tremendous effect on the rate at which resistance evolves, because the mean replicative events measured above must be divided by the number of replications per infected host to determine the average number of hosts that can be infected before an intervention fails. Pathogen population sizes can vary by orders of magnitude between prophylactic and therapeutic interventions, and this could in turn play out as a potentially enormous difference in the speed of evolution.
A final point is that a resistant strain will only persist if it is able to find hosts to infect. If different hosts respond to the same intervention in different ways and a single effector provides protection, this mosaic of host responses may prevent the spread of infection ( Figure S2). Consider an intervention with N effectors that each confer protection in one of k ways. If hosts have equal probability of responding in any of these k ways, then using the binomial distribution, the probability that a pathogen fully resistant to intervention in one host will also be fully resistant to intervention in the next host is In constructing our model, we made six simplifying assumption. It is possible, however, to construct similar models that relax most of these assumptions. If selection for partial resistance were present but weaker against prophylactic interventions than therapeutic interventions, our qualitative con- an extra pathway to resistance through neutral drift, although the probability of following this pathway to resistance is likely to be very small. The assumption that selection is extremely strong could be relaxed by explicitly modeling the spread of resistance mutants through the pathogen population, or this process could be approximated by adding the expected fixation times to the number of replicative events. It is difficult to imagine an easy way to relax our final two assumptions that the probability of a mutation towards resistance is constant for any replicative event and that the probability of two resistance mutations occurring during the same replicative event is very small. These two assumptions, however, have no effect on the qualitative ordering that µ C ≤ µ A ≤ µ B and µ C ≤ µ D ≤ µ B . They also seem likely to have little influence on the quantitative relationship between the interventions when considering biologically reasonable parameter space.  The bottom row shows the same data as the top row, magnified to highlight the differences between Interventions A, C and D. Note that for all parameter sets the average number of replicative events until resistance appears shows µ C ≤ µ A ≤ µ B and µ C ≤ µ D ≤ µ B . The relationship between A and D depends on the parameters , γ, and N . Where unspecified, β = 7 × 10 −6 , = 0.1, N = 3, γ = 5.

Round of transmission
Non-mosaic Mosaic Figure S2: Schematic showing the effect of treatment mosaics on pathogen resistance. White boxes are untreated hosts. Filled and patterned boxes are treated hosts, with different colors and patterns denoting different mechanisms of actions. Circles denote pathogens, with the different colors and patterns indicating resistance to a particular mechanism of action. Open circles are fully susceptible pathogens. Each line shows a chain of transmission that ends when the pathogen infects a host with an effective mechanism of action. Note that the overall levels of resistance and intervention are the same between the non-mosaic situation (top) and the mosaic situation (bottom). The success of a fully susceptible pathogen is therefore the same in both situations, but the resistant pathogens are less successful in the presence of a mosaic. As a result, mosaics reduce the strength of selection for resistance to a particular mechanism of action, because selection acts on the relative success of the different virus strains.