Vaccine-driven virulence evolution: consequences of unbalanced reductions in mortality and transmission and implications for pertussis vaccines

Abstract

Many vaccines have heterogeneous effects across individuals. Additionally, some vaccines do not prevent infection, but reduce disease-associated mortality and transmission. Both of these factors will alter selection pressures on pathogens and thus shape the evolution of pathogen virulence. We use a mathematical modelling framework to show that (i) the balance of how vaccines reduce transmission versus mortality and (ii) individual variability in protection conferred both shape the evolution of pathogen virulence. Epidemiological (burden of disease) and evolutionary (pathogen virulence) outcomes are both worse when vaccines confer smaller reductions in transmission than in mortality. Furthermore, outcomes are modulated by variability in vaccine effects, with increased variability limiting the extent of virulence evolution but in some cases preventing eradication. These findings are pertinent to current concerns about the global resurgence of pertussis and the efficacy of pertussis vaccines, as the two classes of these vaccines may reduce disease symptoms without preventing infection and differ in their ability to reduce transmission. Furthermore, these findings point to the importance of generating precise predictions for virulence evolution in Bordetella pertussis (and other similar pathogens) by incorporating empirical characterizations of vaccine effects into models capturing the epidemiological details of this system.

1. Introduction

Vaccination is one of the most powerful public health tools available [1]. When vaccines block within and/or between host propagation of pathogens, they provide both individual benefits through their direct protective effects and population benefits ranging from herd immunity to eradication. However, vaccination can fundamentally change the ecology of infectious disease systems and introduce novel selective pressures that drive pathogen evolution, potentially reducing or negating these benefits. Pathogen life-history traits can evolve in response to vaccination when pathogens retain the ability to infect some or all of the host population. One such pathogen life-history trait is virulence, or the rate of disease-associated host death. Virulence evolution has been a general focus in the fields of epidemiology and evolutionary biology [2–4] and has received particular attention in the context of vaccination.

Virulence determines a pathogen's effects on morbidity and mortality, as it describes the aggressiveness of the pathogen's host manipulation strategy [5]. When host death truncates transmission, virulence is inversely proportional to transmission time. Thus, virulence in itself is never adaptive, but it is often maintained as an unavoidable pleiotropic byproduct of transmission. Theoretical and empirical evidence points towards a saturating relationship between virulence and transmission, creating a trade-off that shapes pathogens' life-history strategies [5–8]. This transmission–virulence trade-off has been shown to emerge from within-host processes [9] and is often reflected phenomenologically in population-level evolutionary epidemiological models [10]. In most scenarios, intermediate virulence strategies confer the highest fitness because increased transmission rate comes at the price of increased host mortality and decreased transmission time ([6], equation (2.1)). Here, we define virulence as the rate of disease-associated mortality in hosts with no mortality-blocking immunity, noting that the operational definition of virulence varies between studies and that the term ‘virulence’ itself is sometimes used to describe infectivity rather than aggressiveness in plant systems [5].

Vaccination can drive virulence evolution by inducing forms of immunity that alter the relationship between transmission rate and duration. Some vaccines are thought to have an ‘anti-growth’ effect (‘r2’ parameter in the models developed by Gandon et al. [11,12]) by slowing the within-host replication rate of a pathogen, leading to coupled reductions in both transmission and mortality. In a population-level model, Gandon et al. [11] (see also [12]) found that vaccines that moderately reduce pathogen replication rate select for increased pathogen virulence, but lead to reductions in prevalence. Analogous within-host models that consider transmission and mortality to be functions of within-host pathogen density also predict that pathogens should evolve increased virulence when vaccine-induced immunity impedes the replication of a pathogen [13,14], and that virulence should increase with increasing immune efficacy [15].

Other vaccines are thought to only block mortality effects without conferring any reduction in transmission. Gandon et al. [11] showed theoretically that the use of such mortality-blocking (‘r4’ in their model specification) vaccines that do not reduce transmission can drive the evolution of very high pathogen virulence. Normally, highly virulent strategies confer low fitness because rapid host mortality truncates the transmission period. But when vaccines remove or reduce this cost of mortality, pathogens are able to increase their total fitness by increasing their transmission rate. Non-immunized individuals are disproportionately affected by the evolution of increased virulence—they experience drastically increased mortality while vaccinated individuals are at least partially protected. Several empirical examples support the predictions of Gandon et al. [11]. Read et al. [16] showed convincingly that when vaccines blocked the mortality effects of Marek's disease virus in commercial poultry the virus evolved higher virulence to the point where vaccinated hosts lost all benefits of vaccine-induced immunity and non-vaccinated individuals experienced increased mortality. Recently, Fleming-Davies et al. [17] showed that incomplete acquired immunity (analogous to vaccination) to the bacterial pathogen Mycoplasma gallisepticum in finches selects for increased virulence, as low-virulence strains are unable to infect previously infected hosts while high-virulence strains retain this ability.

Despite the development of a rich theoretical framework for investigating vaccine-driven virulence evolution, this approach has not yet been applied to many important vaccine types. In particular, there is a clear knowledge gap about vaccines with unbalanced effects on transmission and mortality, whose effects are not fully or accurately captured by either the anti-growth or mortality-blocking models of vaccine action. These unbalanced effects may be characteristic of vaccines targeted towards pathogen proteins like toxins that cause damage to the host, since toxins generally have a dual effect, reducing host survival (as considered in Gandon et al. [11]) and also increasing pathogen growth, often via immunosuppressive effects. As a result, toxin-targeting vaccines can slightly reduce transmission and also have significant mortality-reducing effects. Investigating how such vaccines might drive virulence evolution requires a model that can consider both effects independently in the same framework. Previous model formulations are unsuitable for investigating this range of vaccine effects, because they either consider reductions in transmission and mortality to be necessarily coupled via a reduction in pathogen replication rate (e.g. [18]) or assume that transmission is necessarily reduced to a greater degree than mortality by layering anti-growth and transmission-blocking effects. Here, we present a modification to previous modelling frameworks that allows us to investigate and compare between the effects of various modes of vaccine action on the evolution of pathogen virulence (figure 1).

Figure 1.

A model schematic encompassing multiple modes of vaccine action. In the model framework that we present, vaccine-induced immunity can act in two non-exclusive ways. First, immunity can have a mortality-reducing effect (x-axis). Second, immunity can have a transmission-reducing effect (y-axis) by decreasing the within-host replication rate of the pathogen, which has a nonlinear effect on transmission. Thus, our model allows us to consider not only ‘anti-growth’ and ‘anti-toxin’ effects but also scenarios in which vaccines have unequal effects on transmission and mortality. An important aspect of this model is that immunity does not reduce the rate at which susceptible individuals become infected when challenged—consistent with the action of Bordetella pertussis vaccines. The shaded regions show the parameter spaces that correspond to a greater percentage decrease in a disease-associated mortality rate or an onward transmission rate for an individual treated with a vaccine. The boundaries of the shaded areas were generated from the equations for transmission and disease-associated mortality given below, with c₁ = 1.0, c₂ = 0.33 and i = 1.0. (Online version in colour.)

This model framework might provide particularly important insights into the B. pertussis system. Both whole-cell pertussis (wP) vaccines, which contain killed B. pertussis bacteria, and acellular pertussis (aP) vaccines, which contain a small subset of the repertoire of proteins B. pertussis would normally present to the human immune system [19], are known to significantly reduce disease-associated damage. However, evidence from a primate model system suggests that, unlike naturally acquired immunity, neither form of vaccine-induced immunity is able to prevent infection [20–22]. Epidemiological patterns suggest that both do moderately reduce onward transmission [23,24], but that aP vaccines reduce transmission to a lesser extent [25,26]. Thus, vaccinated individuals challenged by B. pertussis can become asymptomatically infected and can carry out some amount of transmission. The model framework presented in figure 1 is able to represent these effects, which are intermediate between anti-growth and anti-toxin.

Another aspect of vaccine-driven virulence evolution that we seek to investigate relates to the structure of vaccine-induced immunity in host populations. While the importance of characterizing ‘landscapes of immunity’ and predicting their evolutionary and epidemiological effects is increasingly recognized [27], complexity has been conspicuously lacking in current modelling efforts, despite empirical evidence for its existence. To date, only the simplest distributions of immunity have been considered; almost all models (with the exception of [14,28–30]) consider only ‘binary’ distributions of immunity under the assumption that vaccination has a ‘fixed effect’ in all individuals. Yet, it has long been known that vaccines can have a ‘variable effect’, creating continuous distributions of immunity within populations [31,32], and recent efforts have sought to characterize these patterns [33]. Variability in age, sex or environmental factors could create population-level immunological heterogeneity in the absence of vaccination. These factors could also contribute to heterogeneity in combination with vaccination or through interactions with vaccine efficacy.

Incorporating immunological heterogeneity into models could potentially reveal safe implementation strategies for mortality-blocking vaccines, which have previously been found to have dangerous long-term effects as they drive the evolution of hypervirulent pathogens at any level of coverage. Gandon et al. [11] found that mortality-blocking vaccines drove the evolution of hypervirulent pathogens while assuming that vaccination had a fixed effect, creating a binary distribution of immunity in the host population (figure 2). The experimental work of Read et al. [16] that produced similar results used host populations that exhibited such homogeneity; all vaccinated individuals were age matched and kept in identical conditions to reduce the variance in vaccine effect. In both cases, the optimum pathogen virulence strategies in the two host categories were extremely different, especially when vaccination completely eliminated mortality effects. Highly virulent strategies emerged as pathogens gained most of their fitness through the vaccinated host group. Because there were no hosts with intermediate immunity, the costs of a hypervirulent strategy were only incurred in unvaccinated individuals. The few inquiries into the consequences of immunological heterogeneity for virulence evolution have indeed produced intriguing results. For example, Ganusov et al. [28] found that, in a within-host model, heterogeneity in the lethal pathogen density slightly decreased the evolutionarily stable degree of virulence.

Figure 2.

Immunity distributions. Vaccines that have a variable effect create continuous distributions of immunity (modelled here as pseudo-continuous distributions), while vaccines that have a fixed effect create a binary distribution of immunity. In all plots, θ = 0.5.

Here, we apply adaptive dynamics to a non-system-specific epidemiological model to quantify the extent to which patterns of vaccine-driven virulence evolution and their associated epidemiological outcomes are dependent upon (i) the relative strength of mortality and transmission-reducing vaccine effects and (ii) variation in the effects of vaccination between individuals. We find that drastically different evolutionary and epidemiological outcomes can result from the use of vaccines that differ only slightly in their effects on mortality and transmission and/or in their variability in effect among individuals, and that immunological variability can buffer against virulence evolution.

2. Methods

To investigate the evolutionary and epidemiological consequences of the use of vaccines that block mortality, transmission or a combination of the two, we constructed an evolutionary epidemiological model that follows a susceptible–infected–recovered (SIR) framework in which susceptible hosts become infected in a density-dependent fashion and then permanently recover with no chance of reinfection. We include two classes of infected hosts in order to explore competition between two pathogen strains of differing virulence and assume no co-infection or superinfection. The susceptible and infected host populations are structured by immunity. The relative frequencies of individuals born into each immunity class in the susceptible population are fixed, consistent with a constant vaccination rate and outcome and no host evolution. We vary how immunity affects transmission and mortality, and characterize the dynamics of virulence evolution as pathogens adapt to immunity distributions of various shapes using an adaptive dynamics approach. We then explore the epidemiological outcomes associated with evolutionarily stable pathogen virulence, as well as the potential for pathogen eradication. We note that these methods are not intended to make specific predictions about virulence evolution in B. pertussis or any other disease system. Rather, they are designed to explore how certain mechanistic details of vaccine action (that do differ between the two classes of pertussis vaccines) might generally shape pathogen evolution. Definitions of the parameters and variables used throughout the paper are given in table 1.

Table 1.

Parameter and variable definitions.

parameter	definition	parameter	definition
x	mortality-reducing effect of immunity	γ	recovery rate
y	transmission-reducing effect of immunity	K	density-dependent growth-scaling factor
i	immunity	Si	number of susceptible individuals of immune class i
α	virulence	$I_{i,1}$	number of individuals of immune class i infected by the resident pathogen
β	transmission rate	$I_{i,2}$	number of individuals of immune class i infected by the invader pathogen
$c_1,c_2$	shape parameters for virulence–transmission trade-off	R	number of recovered individuals
v	disease-associated mortality rate	${\alpha }_1,{\alpha }_2$	virulence of the resident and invader pathogens
Di	density of immune class i in immunity distribution	$S_{0_i},I_{0_{i,1}},I_{0_{i,2}},R_0$	initial number of individuals in each state class
θ	vaccination rate	${\alpha }_{\mathrm{ES},\mathrm{NV}}$	evolutionarily stable virulence in the absence of vaccination
λ	shape parameter for immunity distribution	αES	evolutionarily stable virulence in the presence of vaccination
M	set of immune categories	αR	virulence associated with the repeller point
b	birth rate	χi	case-fatality ratio of individuals of immune class i
μ	mortality rate	${\chi }_{\mathrm{all},}\hspace{0.17em}{\chi }_{\mathrm{vacc}},{\chi }_{\mathrm{non}-\mathrm{vacc}}$	case-fatality ratio of all individuals, vaccinated individuals and non-vaccinated individuals

2.1. Virulence and immunity

We assume a saturating relationship between virulence (α) and transmission (β),

$$\beta (\alpha ,i)=c_1\hspace{0.17em}{((1-i\hspace{0.17em}y)\hspace{0.17em}\alpha )}^{c_2}.$$ 2.1

The parameter y determines the relative effect of immunity on transmission. We assume that immunity (i) reduces transmission by scaling virulence rather than by scaling total transmission, consistent with immunity limiting the within-host proliferation of pathogens. For all results presented in the main text, we set c₁ = 1 and c₂ = 0.333. We repeated our analyses with an alternate shape of the virulence–transmission trade-off curve (c₁ = 1.2, c₂ = 0.45) and found that our results were qualitatively similar (see electronic supplementary material, figures S1 and S2). We model disease-associated mortality (ν) as a function of virulence and immunity, which takes values in the interval [0,1] and scales the disease-associated host mortality rate,

$$v(\alpha ,i)=\alpha \hspace{0.17em}(1-i\hspace{0.17em}x).$$ 2.2

The parameter x determines the relative effect of immunity on disease-associated mortality.

Both x and y are constrained to take values in the interval [0,1]. When x > 0 and y = 0, immunity reduces disease-associated mortality but has no effect on transmission. When x > 0 and y > 0, immunity reduces both transmission and disease-associated mortality. When x = 1 immunity completely blocks disease-associated mortality, and when y = 1 immunity completely blocks transmission. When x = 0 and y > 0, immunity reduces transmission but has no effect on disease-associated mortality. Here, we consider values of x and y in {0, 0.05, 0.5, 0.95, 1.0}. Note that the relationship between the parameters x and y, and the relative percentage decrease in transmission/mortality experienced in a population, does not necessarily map to the shaded regions in figure 1, which represent individual-level effects, as variable vaccine effects and vaccination rates less than 100% create immunological heterogeneity.

2.2. Host population immunity structures

We explore several forms of vaccine-induced immunity distributions (figure 2). We defined the immunity distributions created by vaccines with variable effects according to the following function, which gives the density of individuals with immunity i in the host population:

$$D_i(i,\theta ,\lambda )=\hspace{0.17em}\{\begin{array}{c}\frac{(1-\theta )+\theta \hspace{0.17em}\lambda \hspace{0.17em}{\mathrm{e}}^{-\lambda \hspace{0.17em}(1-i)}}{{\sum }_{i\hspace{0.17em}\in M}{D}_i}\mathrm{for}\hspace{0.17em}i=0\\ \frac{\theta \hspace{0.17em}\lambda \hspace{0.17em}{\mathrm{e}}^{-\lambda \hspace{0.17em}(1-i)}}{{\sum }_{i\hspace{0.17em}\in M}{D}_i}\hspace{0.17em}\mathrm{for}\hspace{0.17em}i>0.\end{array}$$ 2.3

M is the set of possible immunity categories, θ is the proportion of the population that is vaccinated and λ is the shape parameter of the immunity distribution. For comparison, we also explored ‘binary’ immunity distributions generated by vaccines with a fixed effect. We define such immunity distributions according to the following function, which gives the density of individuals with immunity i in the host population:

$$D_i(i,\theta )=\hspace{0.17em}\{\begin{array}{c}\theta -1\hspace{0.17em}\mathrm{for}\hspace{0.17em}i=0\\ \theta \hspace{0.17em}\mathrm{for}\hspace{0.17em}i=\hspace{0.17em}1\\ 0\hspace{0.17em}\mathrm{for}\hspace{0.17em}i\hspace{0.17em}\notin \{0,1\}.\end{array}$$ 2.4

For all forms of vaccine-induced immunity distributions considered, we considered vaccine coverages θ of {0, 0.05, 0.10, … ,1}. In this model formulation, incomplete vaccine efficacy (i.e. ‘leakiness’) can be considered by setting x or y to a value in the interval (0,1) rather than by constraining the maximum value of i to be less than 1.

2.3. Epidemiological model

We conducted an evolutionary epidemiological investigation of the effects of vaccination on the evolution of virulence using compartmental epidemiological models [34]. The host population is partitioned into 21 immunity categories in the set $M=\{0,0.05,0.10,\dots ,0.95,1.00\}$.

The dynamics of the host population are described by the following set of differential equations, in which the subscript i $(i\in M)$ indicates immunity class. N is the total population size, S_i is the number of susceptible individuals and R is the number of recovered individuals. I_i,1 is the number of individuals infected with the resident pathogen strain (with virulence α₁), and I_i,2 is the number of individuals infected with the invader pathogen (with virulence α₂). In all analyses, we set the background mortality rate, μ, to 0.05, the birthrate, b, to 0.07 and the recovery rate, γ, to 0.1. These parameters are not intended to represent any particular disease system. We set K equal to $b/(b-\mu )$, so that the population size would be 1 at disease-free equilibrium.

$$\begin{array}{rl}\frac{\mathrm{d}{S}_i}{\mathrm{d}t}& =b\hspace{0.17em}N\left(1-\frac{N}{K}\right)\hspace{0.17em}\frac{S_{0_i}+I_{0_{i,1}}+\hspace{0.17em}{I}_{0_{i,2}}+\hspace{0.17em}{R}_{0_i}}{N_0}-\mu \hspace{0.17em}{S}_i\\ & -\sum _{\hspace{0.17em}j\hspace{0.17em}\in \hspace{0.17em}M}\beta ({\alpha }_1,j)\hspace{0.17em}{S}_i\hspace{0.17em}{I}_{\hspace{0.17em}j,1}-\sum _{\hspace{0.17em}j\hspace{0.17em}\in \hspace{0.17em}M}\beta ({\alpha }_2,j)\hspace{0.17em}{S}_i\hspace{0.17em}{I}_{\hspace{0.17em}j,2}\hspace{0.17em},\end{array}$$ 2.5

$$\frac{\mathrm{d}{I}_{i,1}}{\mathrm{d}t}=\sum _{\hspace{0.17em}j\hspace{0.17em}\in \hspace{0.17em}M}\beta ({\alpha }_1,j)\hspace{0.17em}{S}_i\hspace{0.17em}{I}_{\hspace{0.17em}j,1}-\mu \hspace{0.17em}{I}_{i,1}-v({\alpha }_1,i)\hspace{0.17em}{I}_{i,1}-\gamma \hspace{0.17em}{I}_{i,1},$$ 2.6

$$\frac{\mathrm{d}{I}_{i,2}}{\mathrm{d}t}=\sum _{\hspace{0.17em}j\hspace{0.17em}\in \hspace{0.17em}M}\beta ({\alpha }_2,j)\hspace{0.17em}{S}_i{I}_{\hspace{0.17em}j,2}-\mu I_{i,2}-v({\alpha }_2,i)\hspace{0.17em}{I}_{i,2}-\gamma \hspace{0.17em}{I}_{i,2}$$ 2.7

$$\mathrm{and}\frac{\mathrm{d}R}{\mathrm{d}t}=\gamma \hspace{0.17em}{I}_{i,1}+\gamma \hspace{0.17em}{I}_{i,2}-\mu \hspace{0.17em}R.$$ 2.8

We assume that the total number of births is proportional to the total size of the population, and that the proportion of births into each category is equal to the initial density of individuals in that category. This assumption is consistent with a constant vaccination rate and efficacy where the vaccination status and efficacy of vaccination in a newly born individual are independent of the same traits of the parent. We assume density-dependent transmission (the number of new infections proportional to the density of susceptible and infected individuals). We model transmission-reducing immunity (y) as reducing the rate at which infected hosts transmit to susceptible hosts, rather than as reducing the rate at which susceptible hosts become infected. This is consistent with our assumption that vaccination does not reduce the rate at which susceptibles become infected when challenged. For all analyses described below, we constructed the starting host population according to the following equations:

$$S_{0_i}=0.99\hspace{0.17em}{D}_i(i,\theta ),$$ 2.9

$$I_{0_{i,1}}=\hspace{0.17em}0.01\hspace{0.17em}{D}_i(i,\theta )\hspace{0.17em},$$ 2.10

$$I_{0_{i,2}}=\hspace{0.17em}0$$ 2.11

$$\mathrm{and}{R}_0=\hspace{0.17em}0.$$ 2.12

2.4. Evolutionary dynamics

We take an adaptive dynamics approach (see Brännström et al. [35] for an excellent introduction to this topic) to characterize how various distributions of mortality-blocking immunity in the host population affect evolutionary dynamics and create evolutionarily stable virulence strategies (ESSs; [36]). While studies of virulence evolution often use R₀ as a proxy for pathogen fitness, the evolutionarily stable (ES) virulence is determined by a pathogens' invasion fitness (R_E; equivalent to effective reproductive number), which is dependent upon the composition of the host population in which the pathogen is evolving. Whenever hosts vary in their ability to resist infection, the relative frequency of susceptible hosts in each immunity category will change after the introduction of a pathogen. As a result, the R_E and R₀ values associated with a pathogen strategy are not equivalent, and an adaptive dynamics approach is required [37]. Since hosts do not vary in their ability to resist infection in our model, inferring the evolutionary stability of pathogen virulence strategies from their associated R₀ values is equivalent to an adaptive dynamics approach. However, we proceed with a formal adaptive dynamics approach for ease of interpretation and to lay the foundations for extending our methods to scenarios in which vaccines do affect the susceptibility of hosts to infection.

We begin by introducing a ‘resident’ pathogen with virulence α₁ into a disease-free host population and simulating epidemiological dynamics (details described below) until epidemiological equilibrium is reached. Next, at this equilibrium, we calculate the invasion fitness (e.g. effective reproductive number) R_E for various invading pathogen virulence strategies (α₂) using ‘next-generation’ methods [38]. We assume no superinfection and no co-infection. We conducted all pairwise comparisons of resident and invader virulence strategies in {{0, 0.005, … , 0.100}, {0.11, 0.12, … , 1.0}, {1.1, 1.2, … ,10}, {11,12, … ,100}} for various population immunity distributions to generate pairwise-invasibility plots (PIPs).

We found several recurring motifs in the PIPs generated in our adaptive dynamics analyses (figure 3). Many combinations of vaccine coverage and shape parameter resulted in a convergent stable strategy (CSS; [39]), indicating that virulence will evolve to the same stable value regardless of its initial value (figure 3a). Other combinations resulted in more complex evolutionary dynamics involving both an ESS and a repeller point (figure 3b). We denote the virulence associated with a stable strategy (either CSS or ESS) as α_ES, the virulence associated with a stable strategy in an unvaccinated population as α_ES,NV and the virulence strategy at which a repeller point occurs as α_R. If the initial virulence value is greater than α_R, unbounded selection for progressively more virulent strategies would occur. If the initial virulence value is less than α_R, virulence would evolve towards α_ES. If the assumption that evolution occurs in small steps is relaxed, then the direction of selection could change rapidly around the repeller point; a pathogen population with virulence slightly less than α_R could be invaded by a rare mutant with a significantly more virulent strategy, reversing the direction of selection and resulting in the evolution of hypervirulence. Alternatively, a pathogen population with a virulence strategy only slightly higher than α_R could be invaded by a rare mutant with a significantly less virulent strategy, reversing progressive evolution towards higher virulence. The evolutionary dynamics around a region of unviable virulence strategies adjacent to a repeller point (‘repeller region’ motif; figure 3c) would be expected to be similar to those around a repeller point, except that large evolutionary ‘jumps’ in virulence would be required to start or stop evolution towards hypervirulence. The fourth motif that we observed was selection for hypervirulence without any stable strategy (figure 3d).

Figure 3.

Pairwise-invasibility plot motifs and eradication thresholds. Panels show four PIP motifs identified in the analyses of virulence evolution. A fifth motif (not shown) was found when no virulence strategy could invade, resulting in pathogen eradication. (Online version in colour.)

PIPs also provide a means for assessing the evolutionary durability of a vaccine's effect on pathogen eradication. When a vaccine is introduced into an infected population, ‘evolution-proof’ eradication of the pathogen will occur if no virulence strategy will allow for its persistence. Eradication could also occur if the virulence strategy α_ES,NV is not viable after vaccine introduction, but some other virulence strategy is. We term this scenario ‘non-evolution-proof eradication’. This would occur if α_ES,NV falls below the lower eradication threshold in figure 3a–c, above the upper eradication threshold in figure 3a or between the upper eradication threshold and repeller point in figure 3c. For a pathogen population to escape non-evolution-proof eradication, it must evolve a viable virulence strategy through a single large evolutionary step.

2.5. Epidemiological consequences

We investigated the long-term epidemiological consequences of implementing vaccines that create various distributions of immunity. For each immunity distribution considered, we found the ES pathogen virulence as described above. When an immunity distribution was predicted to drive the evolution of hypervirulence, we used α = 10⁶ to calculate epidemiological consequences. Such an extreme value of α may be biologically unattainable, but we found that epidemiological consequences were qualitatively and quantitatively similar when we repeated our analyses while setting α = 10² for all cases in which hypervirulence evolution occurred (results not shown). For each immunity distribution, we calculated the prevalence of the pathogen at epidemiological equilibrium (see Computation methods), and the case-fatality rate (CFR), χ, experienced by vaccinated and unvaccinated individuals, as well as by the entire host population. We calculate the CFR for a population as the average CFR experienced by a new individual entering the population weighted by the proportion of new individuals entering each immunity class. This differs from the average CFR experienced by an individual in a population weighted by the proportion of individuals in each immunity class, as differences in disease-associated mortality between immunity classes result in the distribution of immunity within the infected class differing from the distribution of immunity among individuals entering the population. CFRs were calculated for each immune class as follows:

$${\chi }_i=\hspace{0.17em}\frac{v({\alpha }_{\mathrm{ES}},i)}{v({\alpha }_{\mathrm{ES}},i)+\mu +\gamma }.$$ 2.13

The CFRs of all vaccinated and non-vaccinated individuals were calculated as average CFRs of the relevant immunity categories weighted by the proportion of relevant individuals in each category:

$${\chi }_{\mathrm{all}}=\hspace{0.17em}\sum _{i\in M}\frac{v({\alpha }_{\mathrm{ES}},i)}{v({\alpha }_{\mathrm{ES}},i)+\mu +\gamma }D(i),$$ 2.14

$$\frac{{\chi }_{\mathrm{vacc}}=\begin{array}{rl}& (v({\alpha }_{\mathrm{ES}},0)/v({\alpha }_{\mathrm{ES}},0)+\mu +\gamma )(D_0-(1-\theta ))\\ & +{\sum }_{i\hspace{0.17em}\in (M>0)}(v({\alpha }_{\mathrm{ES}},i)/(v({\alpha }_{\mathrm{ES}},i)+\mu +\gamma ))D_i\end{array}}{\theta }$$ 2.15

$$\mathrm{and}{\chi }_{\mathrm{non}-\mathrm{vacc}}=\hspace{0.17em}\frac{v({\alpha }_{\mathrm{ES}},0)}{v({\alpha }_{\mathrm{ES}},0)+\mu +\gamma }.$$ 2.16

2.6. Computational methods

We implemented the epidemiological model and carried out all analyses in the R statistical software environment [40]. We used the ‘deSolve’ package [41] to simulate the epidemiological dynamics of the system. For all epidemiological simulations, we ran the model for 2000 time units (with a step size of 0.1) and checked that the system had reached epidemiological equilibrium (constant numbers of infected, susceptible and recovered individuals). The code associated with the analyses is available from the Dryad Digital Repository [42].

3. Results

3.1. Evolutionary outcomes

We first investigated the evolution of virulence using pairwise-invasibility analyses while varying vaccine coverage, the strength of mortality-blocking effects (x) and the strength of transmission-blocking effects (y) for both variable and fixed-effect vaccines. We also varied the shape of the immunity distribution (λ) for variable-effect vaccines. For all combinations of mortality- and transmission-reducing effects, we found that evolutionary outcomes for variable-effect vaccines approached those for fixed-effect vaccines as the variation in vaccine effect dropped (i.e. as λ increased). When vaccine-induced immunity did not completely block disease-associated mortality (x < 1), we found that virulence always evolved towards a CSS, or that the pathogen was eradicated when immunity also greatly reduced transmission (y ≥ 0.95, figure 4). We found that evolution-proof eradication occurred when vaccine coverage was high, and when the variability of the vaccine effect was low. However, when vaccine coverage was high but eradication was not achieved, vaccines with less variable effects led to the evolution of more virulent pathogens (figure 5). Similar results were obtained when vaccines completely blocked transmission and mortality effects (x, y = 1). This demonstrates how the outcome of pathogen life-history evolution can be extremely sensitive to the shape of a vaccine-induced immunity distribution.

Figure 4.

Adaptive dynamics results. The evolutionary motifs found for various combinations of vaccine coverage, variability in effects and mortality- and transmission-reducing efficacy are shown. Green regions indicate scenarios in which evolution-proof eradication is achieved and virulence evolution cannot lead to pathogen persistence. Scenarios in which non-evolution-proof eradication is possible were found on the boundaries of the evolution-proof eradication regions (electronic supplementary material, figure S3). Lower and upper eradication thresholds are shown in electronic supplementary material, figures S3 and S4. Qualitatively similar results were found when assuming an alternative shape of the virulence–transmission trade-off curve (electronic supplementary material, figure S1). (Online version in colour.)

Figure 5.

Evolutionarily stable virulence (α_ES). Colours show the ESS (α_ES). No virulence strategy was evolutionarily stable when either eradication or selection for hypervirulence occurred. Qualitatively similar results were found when assuming an alternative shape of the virulence–transmission trade-off curve (electronic supplementary material, figure S2). (Online version in colour.)

When vaccines completely blocked mortality but not transmission (x = 1, y < 1), the evolutionary outcome was highly dependent upon the shape of the immunity distribution and vaccine coverage. As coverage increased, and the effect of the vaccine became less variable, repeller points and regions emerged. The potential for evolutionary ‘jumps’ in virulence to result in the evolution of hypervirulence also increased as vaccine effects became less variable and coverage rose, as the repeller points moved closer to the ES virulence strategies (figure 6). For very high vaccine coverage and low vaccine effect variability, ESSs disappeared and evolution towards hypervirulence always occurred.

Figure 6.

Repeller point virulence (α_R). Colours show the level of virulence at which a repeller point or a region was found (α_R). No repeller points or regions were found for other combinations of mortality (x)- and transmission (y)-blocking effects. (Online version in colour.)

These results illustrate how heterogeneity in the strength of mortality-blocking immunity can buffer against the evolution of increased virulence or hypervirulence. However, when immunity also acts to completely block transmission (y = 1), this effect comes at the cost of pathogen eradication.

3.2. Epidemiological outcomes

Next, we investigated the epidemiological outcomes of virulence evolution driven by vaccination. Again, we found that the outcomes of variable-effect vaccines approach those of fixed-effect vaccines as variation in vaccine effect decreased (as one would hope). When vaccines had an anti-growth effect or reduced transmission to an equal or greater degree than mortality effects (x ≤ y), we found that vaccines always reduced the average CFR experienced in a population (figure 7), decreased infection prevalence (figure 8) and decreased the rate of pathogen spread (i.e. decreased R_E; electronic supplementary material, figure S8). The magnitude of these reductions was greater for vaccines that reduced transmission to a greater degree and mortality to a lesser degree (figures 7 and 8). Greater reductions in the CFR were achieved at higher vaccine coverage and lower variation in vaccine effect (figure 7). Unsurprisingly, reductions in the CFR were mostly driven by a reduction in CFR among vaccinated individuals (electronic supplementary material, figures S6 and S7). In fact, CFR increased among unvaccinated individuals whenever the mortality-blocking effect of the vaccine was strong (0.5 < x < y), except when the pathogen was eradicated. However, we did observe that when the mortality-blocking effect was weak in both an absolute and relative sense or non-existent (x < 0.5 and y > x), reductions in the average CFR were achieved without increases in CFR among unvaccinated individuals (figure 7; electronic supplementary material, figure S7).

Figure 7.

Change in the CFR after vaccine-driven virulence evolution. The change in the CFR was calculated as the difference between the average CFR experienced in a vaccinated population with α = α_ES and the average CFR experienced in an unvaccinated population with α = α_ES,NV. (Online version in colour.)

Figure 8.

Change in the equilibrium prevalence after vaccine-driven virulence evolution. The change in prevalence at epidemiological equilibrium was calculated as the difference between the equilibrium prevalence experienced in a vaccinated population with α = α_ES and the equilibrium prevalence experienced in an unvaccinated population with α = α_ES,NV. (Online version in colour.)

When vaccines’ mortality-reducing effects were strong (x ≥ 0.5) and greater than transmission-reducing effects (x > y), we found that epidemiological outcomes were highly dependent upon vaccine coverage and the shape of the vaccine-induced immunity distribution. In this parameter space, we found that vaccines always increased the potential for the pathogen to spread (electronic supplementary material, figure S8) but decreased infection prevalence in the population in which they were implemented unless selection for hypervirulence occurred (figure 8). Surprisingly, we found that the greatest reductions in prevalence occurred just outside the parameter space corresponding to the evolution of hypervirulence and increased prevalence. This further demonstrates how the population-level consequences of vaccination are extremely sensitive to variation in vaccines' effect. When vaccines drove the evolution of hypervirulent pathogens, a drastic increase in the CFR occurred unless vaccine coverage was very high, and the variation in the vaccine effect was low or non-existent (fixed effect), in which case a reduction in the average CFR (figure 7) was obtained via a drastic reduction in the CFR among vaccinated individuals (electronic supplementary material, figure S6) but at the cost of a drastic increase in the CFR among unvaccinated individuals (electronic supplementary material, figure S7). A similar result was obtained at high vaccine coverage for x = 0.95 and y = 0.5, although the evolution of hypervirulence did not occur in this case.

4. Discussion

We investigated how the strength of mortality- and transmission-reducing vaccine effects and the degree of variability in individual outcomes of vaccination affect evolutionary and epidemiological patterns. Our results show that long-term outcomes of vaccination are positive when transmission is reduced to a greater degree than mortality. In these scenarios, vaccination always reduces the overall CFR and infection prevalence. We also found that when mortality-reducing effects were high, but still less than transmission-reducing effects, slight changes in the variability of individual outcomes of vaccination could determine whether the pathogen is eradicated or evolves increased virulence (relative to the evolutionarily stable virulence in the absence of vaccination), resulting in the increased CFR among unvaccinated individuals.

Vaccines which reduce mortality effects to a greater degree than transmission effects result in relatively worse evolutionary and epidemiological outcomes. For these vaccines, reductions in the overall CFR are only achieved when variation in individual outcomes of vaccination is low and vaccine coverage is close to 100%. We found that these vaccines cannot concurrently reduce CFR and prevalence, and that any reduction in the overall CFR comes at the cost of increased CFR among unvaccinated individuals. When vaccination completely blocked mortality but not transmission, evolutionary and epidemiological outcomes were highly sensitive to variation in vaccine outcome, as slight changes in variability could switch evolutionary patterns from selection towards an evolutionarily stable intermediate virulence strategy to selection for hypervirulence. Notably, we found that immunological variability can not only buffer against the evolution of hypervirulence when immunity reduces mortality but also prevent pathogen eradication when immunity also completely blocks transmission. These results highlight the critical role that variation in vaccine outcome plays in driving pathogen evolution.

Beyond this more complete understanding of how relative vaccine effects on transmission versus mortality shape virulence evolution, our findings show that accounting for variation in vaccine effects qualitatively changes the predictions of evolutionary epidemiological models. This suggests that considering immunological heterogeneity within populations will be important in any investigations into unintended evolutionary consequences of vaccination, especially in those considering other mechanisms of vaccine-induced immunity (e.g. transmission blocking and recovery expediting). Data on variation in vaccine effects are extremely limited or non-existent in most cases. Collecting such data, or leveraging existing datasets to quantify this variation, is an important research direction, with potential to improve predictions about both the short-term epidemiological consequences and long-term evolutionary outcomes associated with disease control efforts. For the latter, a broader scope of models will be required, potentially building on the development of new statistical methods for incorporating continuous heterogeneity into epidemiological models [30]. Together, these new datasets and modified methods should be used to both assess the evolutionary consequences of current vaccine use and evaluate the evolutionary impacts of vaccines in development.

We emphasize that, regardless of their potential to drive the evolution of increased virulence, vaccines that reduce mortality effects to any degree always provide a personal benefit to recipients. The magnitude of this benefit is dependent upon the virulence of the pathogen and may vary between individuals if the vaccine has a variable effect. However, the rational choice for an individual in any scenario is to maximize protection against disease-related mortality by being vaccinated.

Our model generates valuable insights for the pertussis system. Both wP and aP vaccines completely block mortality and moderately reduce transmission, but transmission is reduced to a greater degree by wP vaccines [22]. We found that when vaccines completely block mortality effects, the evolutionarily stable degree of virulence increases and the potential for runaway evolution towards hypervirulence rises as transmission-reducing effects become weaker. While our model was not parametrized for pertussis and lacked many important epidemiological details of this system that could modulate patterns of virulence evolution, including waning immunity and differential mixing patterns, these results do suggest that the use of aP vaccines is riskier than the use of wP vaccines, especially when the population immunity traits (vaccine coverage and shape of immunity distribution) are unknown or expected to change.

Definitive evidence for the recent evolution of B. pertussis [43,44] (including virulence evolution [45]), the global resurgence of pertussis incidence [46–48] and the high worldwide coverage of pertussis vaccines [49] all point to clear importance of modifying our general model to specifically address the epidemiology and biology of the B. pertussis system in order to validate and refine our predictions about vaccine-driven virulence evolution. Major challenges in this endeavour include quantifying the relative transmission- and mortality-reducing effects of pertussis vaccines and establishing which, if any, life-history trade-off limits pertussis virulence evolution. Bordetella pertussis-related deaths are very uncommon in the USA and other developed countries [50], and mostly occur in unvaccinated infants who are unlikely to contribute significantly to transmission [19]. In these locations, it is plausible that a trade-off with the metabolic costs of toxin production [18] is the major factor limiting virulence evolution. In developing countries where B. pertussis mortality rates are much higher [50], a virulence–transmission trade-off might be more significantly involved. Future research about vaccine-driven virulence evolution in B. pertussis should also investigate the time scale of virulence evolution, so that vaccines can be compared in terms of both the magnitude and immediacy (or perhaps temporal irrelevancy) of their projected outcomes.

More broadly, our results have implications for natural populations in which transmission- and/or mortality-reducing immunity is not the product of vaccination. Variation among individuals in terms of immunity due to physiological or eco-immunological factors could potentially drive the evolution of pathogen virulence. Quantifying the extent and basis of immune function variation in natural populations will be an essential first step in establishing the degree to which this occurs. If variation in immune function has a genetic basis, then host and pathogen populations will coevolve. The model we present in this paper could be expanded to investigate how coevolutionary patterns affect the evolution of virulence.

In conclusion, we found that small differences in vaccines’ effects on mortality and transmission can correspond to large differences in evolutionary and epidemiological outcomes. Likewise, we found these results to be highly sensitive to vaccine coverage and the shape of the immunity distributions that they create in populations. Immunological heterogeneity can not only buffer against virulence evolution when immunity reduces mortality but also hinder pathogen eradication when immunity completely blocks transmission. These findings point to the importance of measuring characteristics of vaccines' protective efficacy beyond their ability to provide personal protection, especially when they are thought to reduce mortality to a much greater degree than transmission, as is the case for pertussis vaccines.

Data accessibility

The code associated with the analyses presented in this paper is available from the Dryad Digital Repository: https://dx.doi.org/10.5061/dryad.6m905qfw1 [42].

Authors' contributions

I.F.M. conceived of the study and conducted the analyses. I.F.M. and C.J.M. wrote and revised the manuscript.

Competing interests

We declare we have no competing interests.

Funding

I.F.M. acknowledges support from the National Science Foundation Graduate Research Fellowship Program.