Abstract
The impact of infectious disease is often very different in juveniles and adults, but theory has focused on the drivers of stage-dependent defense in hosts rather than the potential for stage-dependent virulence evolution. Stage-structure has the potential to be important to the evolution of pathogens because it exposes parasites to heterogeneous environments in terms of both host characteristics and transmission routes. We develop a stage-structured (juvenile-adult) epidemiological model and examine the evolutionary outcomes of stage-specific virulence under the classic assumption of a transmission-virulence trade-off. We show that selection on virulence against adults remains consistent with the classic theory. However, the evolution of juvenile virulence is sensitive to both demography and contact structure with higher virulence against juveniles being favored either when the contact structure is assortative (juveniles preferentially interact together) and the juvenile stage is short, or in contrast when the contact structure is disassortative and the juvenile stage is long. These results highlight the potentially profound effects of host stage-structure on determining parasite virulence in nature. This new perspective may have broad implications for both understanding and managing disease severity.
Introduction
Understanding how parasites are selected to exploit their hosts remains a central research question in the evolutionary ecology of host-parasite interactions (Smith 1904; Ball 1943; Anderson & May 1982; Read 1994; Ebert & Herre 1996; Frank 1996; Alizon et al. 2009; Schmid-Hempel 2011; Bull & Lauring 2014; Cressler et al. 2016), with important implications for host persistence (Boots & Sasaki 2003; De Castro & Bolker 2005), disease management (Dieckmann 2005), and host-parasite coevolution (Boots et al. 2009). Although most of the theory of evolution of virulence (defined in this literature as the increased death rate due to infection) focuses on homogenous host populations, heterogeneity within host populations is ubiquitous in nature (Anderson & May 1992, Chapter 8-11). One typical form of host heterogeneity is stage-related structure (e.g., juveniles and adults), and a number of recent ecological studies have examined the impacts of host populations’ stage-related heterogeneity on disease epidemiology (e.g., Dwyer 1991; Fleming-Davies et al. 2015; Hite et al. 2016; for theory, Ashby & Bruns 2018). In these studies, the differences in virulence across life stages have been explained as age-related variation in tolerance, resistance, exposure, immunocompetence, and susceptibility and affected by maternal and acquired immunity (Hudson & Dobson 1995; Wilson et al. 2002).
In addition to age-related variation in the hosts, different host age-classes expose parasites to specific environmental heterogeneity (see Ashby & Bruns 2018, for theory). Given this, parasites may adaptively tune conditional exploitation against certain stage classes, e.g., through plasticity or by infecting tissues and/or cells that are differentially expressed at different stage classes. In principle, stage-specific virulence may occur as a result of parasite adaptation in two ways. First, stage-structure can generate different infectious periods, for instance due to the substantial difference in natural mortality between juveniles and adults (Jones et al. 2013). Infectious period is therefore stage-dependent, which, according to the theory (Day 2001; Gandon et al. 2001; Day & Proulx 2004; Gandon 2004; Alizon et al. 2009; Cressler et al. 2016), may induce selection on virulence such that a shorter infectious period in a certain stage of hosts favors higher virulence. Secondly, the hosts’ stage-structure can generate biased transmission-pathways, thereby exposing the parasites to temporal heterogeneity, which may induce additional selective pressures (reviewed in Lion & Metz 2018). For instance, spatio-temporal segregation between juveniles and adults, which is typical of humans (Rohani et al. 2010), amphibians (Kilpatrick et al. 2010), and insects (Briggs & Godfray 1995), can produce assortative contact structure (i.e., juvenile-juvenile and adult-adult contacts might be more likely than juvenile-adult contacts), such that parasites infecting a certain stage of hosts are likely to be transmitted to the same stage of hosts (“assortative” transmission). Despite these potentially important selective forces on stage-specific virulence, the implications of host stage-structure for parasite fitness and evolution of stage-specific virulence have not been examined.
Here, we extend classic models of virulence evolution to include two host stage-classes (pre and post reproductive) where the juveniles mature into adults, the adults reproduce, and transmission between the stage-classes is characterized by a contact matrix. We explore the evolutionary outcomes of stage-specific virulence in light of classic theory of life-history evolution in heterogeneous populations. We use the adaptive dynamics toolbox (Hofbauer & Sigmund 1990; Dieckmann & Law 1996), first showing that the evolutionary outcomes of virulence against adults (“adult-virulence”) are explained by the classic, optimization-principle (Lion & Metz 2018) (i.e., the evolutionarily stable virulence maximizes the number of secondary infection from infected adult host in a fully susceptible population). Second, we show that the evolutionary outcomes of virulence against juveniles (“juvenile-virulence”) are critically impacted by the interplay between assortativity and the duration of the adult-stage. This is in part because juvenile-virulence can attenuate successful maturation of juvenile hosts and consequently loses future secondary infection from adults. We then use Fisher’s reproductive values (Fisher 1958; Taylor 1990; Williams 2011; Williams & Kamel 2018; Lion 2018) (i.e., the long-term relative contributions of parasites infecting juveniles and adults to future populations), in order to gain a deeper biological insight into the selection gradients. We highlight the pivotal importance of stage-structure to the evolution of infectious disease.
Method
We consider a host population structured into juvenile (J) and adult (A) stages, in which juveniles are obviously incapable of reproduction. The density of susceptible or infected juveniles is denoted SJ or IJ respectively, and that of susceptible or infected adults is denoted SA or IA respectively. Combining an epidemiological SI-model with a stage-structured model yields the following ordinary differential equations (ODEs; Appendix A1): where: r represents a fecundity of the adult hosts per capita (assumed to be the same for susceptible and infected adults), which is reduced by a density-dependent factor κ; juveniles mature into adults at a rate u; mX represents the background mortality for a stage-X host; φxy represents the rate at which a susceptible stage-X host gets transmitted from an infected stage-Y host (i.e., force of infection from infectious Y to susceptible X per capita, Fig 2; also see below for formula); υX represents the virulence against a stage-X host (an evolving trait). The reciprocal of the infectious period for juveniles (or adults) is given by μJ = u + mJ + υJ (or by μA = mA + υA respectively). For alternative approaches including physiologically structured population modeling and infection-age modeling with continuous stage structure, see Day et al. (2011), Mideo et al. (2011), and de Roos & Persson (2013). The present methodology allows us to evaluate the relative virulence against juveniles compared to adults.
Maturation and natural mortality can both affect the relative length of a adult-stage of the hosts. To quantify this, let θA be the expected fraction of time a host individual spends as an adult in the entire lifespan in the absence of disease, which reads (Appendix A2): from which we can check that extremely slow (or fast) maturation, u → 0 (or u → +∞ respectively), leads to θA → 0 (or 1 respectively. We will omit the degenerated case in which u + mJ = mA, as this situation turns out to give no reasonable measure for stage-period; Appendix A2). We use θA as a characteristic parameter of the stage-structured host populations.
The force of infection for a stage-X host from a stage-Y host (with X and Y running across J and A) in Eqn (1) involves with three processes: susceptibility αX (the likeliness for which a stage-X host becomes infected, given a reception of disease-propagule), contact structure σXY (which represents the fraction of time a Y-stage host spends with a X-stage host), and infectiousness βY (the propagule-production from a stage-Y host; see Fig 2): with H = SJ + IJ + SA + IA the total density of hosts, such that the transmission is frequency-dependent, as is assumed in previous studies of stage-structured epidemiological dynamics (e.g., Bernhauerova 2016). Also, to link virulence and transmission, we use the trade-off relationship given by , where kX tunes the efficiency of virulence for infectiousness from stage-X hosts, and bX represents the upper bound of infectiousness from stage-X hosts. Note that we assumed that transmission is a concave function of virulence to restrict our primary attention to stable evolutionary outcomes (Otto & Day 2007).
For quantifying the contact structure, we use a single parameter given by σ = 1 ‒ σAJ = σJJ = 1 ‒ σJA = σAA, (Diekmann et al. 2012, Chapter 12). With this symmetry, the measure of assortativity is given by: where ρ varies from ‒1 to 1. If ‒1 ≤ ρ < 0, then within-stage contact is less frequent compared to between-stage contact (such a contact is said to be “disassortative”). Instead, if 0 < ρ ≤ 1, then within-stage contact is more likely than between-stage contact (“assortative” contact). ρ = 0 indicates that contact is unbiased (“random” contact). In the extreme case, ρ = 1 (or ‒1) indicates that contact occurs exclusively within the stages (or between the stages, respectively).
We use the adaptive dynamics toolbox (Hofbauer & Sigmund 1990; Dieckmann & Law 1996) to study the long-term evolutionary dynamics of stage-specific virulence. Throughout the paper we assume that parasites show specific virulence: υJ and υA, with no association or correlation between them (i.e., we study the joint evolutionary dynamics of (υJ, υA)). First, suppose that the system of ODEs in Eqn (1) has reached a steady state: for a given (or wild-type) virulence strategy v := (υJ, υA). We then introduce a rare mutant attempting to invade a monomorphic wild-type virulence v. We assume weak selection (|v′ - v| is very small). For more details, see Appendix A3.
To assess the possibility of mutant invasion, we define the invasion fitness, denoted w, by using the Next-Generation Theorem (van den Driessche & Watmough 2002; Hurford et al. 2010). The “next-generation matrix” (that determines the long term growth of the mutant) can be written as the product of five matrices: (see Appendix), where (the total density of the hosts at the endemic equilibrium), (the reciprocal of the infectious period of juveniles infected by the mutant), and (the mortality rate of adults infected by the mutant). Eqn (5) offers a natural interpretation of the reproductive success of the mutant by partitioning the epidemiological process, in agreement with models of transmission dynamics in heterogeneous host populations (Craft 2015; VanderWaal & Ezenwa 2016; White et al. 2017). The first matrix represents the availability of susceptible hosts, each with a specific susceptibility (the second matrix); the third matrix represents the contact pattern across stages; the fourth matrix represents the infectiousness of infected hosts per capita, and the fifth matrix represents the stage-specific infectious period with the effect of maturation being included.
The invasion fitness is determined by the dominant eigenvalue of G′ (denoted Λ[G′]), which turns out to exhibit a complicated expression. We therefore use a simpler (but equivalent) measure for the invasion fitness: with short-hand notation (the probability of maturation of juveniles infected by the mutant) and (the availability of X-stage hosts to the parasites infecting a Y-stage host per propagule-production). The factor represents the total number of propagule produced by a X-stage host infected by the mutant. We find that the condition for which the mutant outcompetes the wild type ω(v′, v) > 1 holds if and only if Λ[G′] > 1, under weak selection (see Appendix A4).
Direction of selection is determined by selection gradient: with the partial derivatives evaluated at v′ = v (“neutrality”). Virulence evolves in the direction of selection gradient until singular strategy (SS) is reached, gJ(v) = gA(v) = 0 at v = v*. We assess two stability criteria of SS: the first criterion, attainability (Takada & Kigami 1991; Christiansen 1991), concerns whether SS can be reached by recurrent small mutations. The second is referred to as evolutionary stability (Maynard Smith & Price 1973), which assures that SS can resist against any invasion of alternative, mutant strategies. If SS meets both of these criteria, it is then called as Continuously Stable Strategy (CSS; Eshel 1983). Analytical investigation revealed that the SS is always a CSS, and thus we do not detail the stability analyses in the main text (see results).
We use the following default parameter-values: r = 6, k = 0.06, mJ = 1, αJ = αA = 1, kJ = kA = 1, bJ = bA = 10, while varying mA, u (thus θA) and ρ. That is, the parameter values are symmetric for juveniles and adults. We subsequently check the effects of the differences in α (susceptibility), k (efficiency of exploitation for transmission), and b (upper bound in infectiousness). In addition, we check whether recovery or tolerance in the host can affect the results. Finally, we examined the outcomes when we assume density-dependent rather than frequency dependent transmission in the dynamics.
Results
The selection gradient for adult-virulence reads: (where ° represents neutrality, v′ = v; Appendix A5), which is consistent with a number of previous studies: under the transmission-virulence trade-off, higher exploitation is expected to increase the infectiousness (i.e., a marginal benefit) at the immediate (marginal) cost owing to a reduced infectious period (Day 2001; Gandon et al. 2001; Day & Proulx 2004; Gandon 2004; Alizon et al. 2009; Cressler et al. 2016). This is because the reproductive success of parasites infecting adults is, in effect, determined by a single transmission-pathway, from adults to any susceptible hosts in the population, regardless of the contact-structure and the stationary stage-distribution. Therefore, the direction of selection on adult virulence is completely determined by the balance between such marginal benefits and costs, regardless of the characteristics of hosts’ stage-structure.
In contrast, juvenile-virulence in the model is influenced by additional costs associated with hosts’ stage-structure. To make the biological meaning clearer, we present the reproductive-value based form of the selection gradient (Fisher 1958; Taylor 1990; Caswell 2001; Frank 1998; Grafen 2006; Lion 2018). Reproductive values give a proper weightings of fitness effects for age-classes, by taking the contributions of classes to future gene-pool into account (Fisher 1958; Taylor 1990; Caswell 2001; Frank 1998; Grafen 2006; Lion 2018). Using reproductive values, gJ(v) reads:
(Appendix A6-8), where represents the pair of individual reproductive values of the parasites infecting juvenile and adult hosts (or the left eigenvector of G at neutrality; Appendix A6) and the factor for instance represents the reproductive success due to transmission from an adult host to a juvenile per propagule-production. The first term is multiplied by , which represents the reproductive success of a parasite infecting juveniles, who can receive the marginal benefit due to increased infectiousness but pay the marginal cost due to the reduction in infectious periods as in the selection gradient for adult-virulence (Eqn (8)). In addition, juvenile-virulence incurs the additional cost of increased mortality of infected juveniles associated with the loss of expected reproductive success via adult hosts that the parasites could otherwise gain through the maturation of the juvenile hosts, and thus multiplied by (the reproductive success of a parasite infecting adults) and the marginal decrease in the probability of maturation (πI/μJ)°. Hence, Eqn (9) clearly captures the selection forces on juvenile-virulence, including the marginal benefits, marginal costs, and maturation-mediated costs.
We investigated the effects of two stage-structure characteristics: (i) post-maturation span θA and (ii) stage-assortativity ρ on the evolutionary outcomes (i.e., CSS; Appendix A9). Strikingly, the CSS for adult-virulence is necessarily which is independent of any demographic and disease characteristics of juveniles (as expected from Eqn (8)). Therefore, we used as a reference value in comparison with
In contrast, CSS for juvenile-virulence is dramatically affected by hosts’ stage-structure and maturation. This is because the parasites infecting the juveniles can utilize two transmission-pathways: either from the juvenile (to any susceptible hosts), or from the adult who has successfully matured from the juvenile stage. The expression for is analytically intractable, and as such we numerically evaluated by jointly solving the wild-type ODE Eqn (1) and the selection gradients Eqns (8) and (9).
We can immediately see that the evolutionary outcome of adult-virulence increases with adult natural mortality, in agreement with the previous studies (reviewed in Alizon et al. 2009; Cressler et al. 2016). To assess when selection favors higher juvenile-virulence than adult-virulence, we quantified u as a function of the assortatvity (ρ, abscissa) and post-maturation span (θA, ordinate; Fig 3). We found that either disassortative hosts with a long post-maturation span or assortative hosts with a short post-maturation span select for higher virulence against juveniles. This result slightly changes given stage-specific mortality rates (mJ ≠ mA) such that a higher mortality for juveniles can bias the outcomes towards higher virulence for juveniles, but the general trend is robust (Fig 3A-C). Also, the combination of disassortativity and long post-maturation span (one of the conditions favoring higher virulence against juveniles) leads to parasite extinction as a result of overexploitation against juveniles (Fig 3B, C; Appendix A10). Note that under these conditions, IJ is very small (tending to zero; Fig 3B), while SJ is not (Fig 3C), meaning that hosts do persist but parasites do not.
By relaxing the assumptions of the symmetry in disease-related parameters kJ, kA (efficiency of exploitation for transmission), bJ, bA (maximum infectiousness), and αJ, αA (susceptibility) for juveniles and adults, or by incorporating recovery or tolerance, we showed that the results are robust and qualitatively unchanged (Appendix B). Finally, we checked that density-dependent transmission yields quantitatively similar results (Appendix B). Therefore, we conclude that the combined effects of maturation and assortativity are critical to the evolution of virulence.
Discussion
We have shown how parasites may be subject to different selective pressures when they infect adults as opposed to adults. Our key insight is that the combination between maturation rates and contact-structure determines the evolutionary outcomes of juvenile-virulence. Higher virulence against juveniles is favored either if: (i) the adult-stage is relatively long and the contact-structure is disassortative (between age class interactions are high; Fig 3, left-top zone), or (ii) the juvenile-stage is relatively long and the contact structure is assortative (interactions occur preferentially within classes; Fig 3, right-bottom zone). These results can be understood as follows: when the post-maturation span is long and the contact structure is disassortative, adult hosts are abundant in the population and the transmission from juveniles to adults is more likely than between juveniles; in this case, the availability of adult hosts is higher, which selects for higher exploitation against juveniles to exploit more abundant resource of adults. Equivalent reasoning explains higher juvenile-virulence in short maturating and assortative hosts. These considerations mean that host-demography alongside the maturation of juveniles strongly affects the evolutionary outcomes of parasite virulence. Spatial and/or temporal segregation in the niches of juveniles and adults therefore has the potential to drive the evolution of differential virulence. Our novel result is therefore, that virulence is highly sensitive to stage-structured life-history characteristics of hosts such as ontogeny and any associated, spatio-temporal niche-shifts.
The incorporation of the maturation shows that higher parasite exploitation against juveniles incurs an additional cost associated with increased maturation failure. In particular, while the marginal value theorem (Charnov 1976) does correctly predict the evolutionary outcomes of adult virulence it does not predict that of juvenile virulence. Therefore, sources of heterogeneity in hosts can clearly lead to different predictions than classic virulence evolution theory based on the marginal value theorem, as claimed in a recent conceptual review (Lion & Metz 2018). Gandon (2004) and Osnas & Dobson (2011) introduced multiple hosts’ types or species and studied virulence against them. In previous theory, heterogeneity is incorporated on the basis of hosts species (Regoes et al. 2000; Gandon 2004; Osnas & Dobson 2011), vaccination (Gandon et al. 2001; Gandon et al. 2003; Yates et al. 2006; Zurita-Gutiérrez & Lion 2015), or sex (Úbeda & Jansen 2016) (reviewed in Lion & Metz 2018). However, none of these studies incorporated stage-structure with associated stage-specific virulence. Our novel results arise because we explicitly assumed stage-structure with maturation from juveniles to adults and reproduction by adults rather than more generic heterogeneity between different types of hosts.
Finding examples of stage-specific virulence in empirical systems can be difficult due to the intricacies of specific host-pathogen systems. Stage-related trends in virulence can be complicated by age-related trends in maternal immunity, adaptive immunity, and exposure rate, and specific host-parasite system characteristics including maladaptation and immuno-pathogenicity (Hudson & Dobson 1995; Wilson et al. 2002). Additionally, studies looking at age-related virulence or case mortality do not exclusively look at differences between adult and juvenile stages and may focus on old age-mediated declines in immuno-competence. However, despite these issues, we found data on several empirical systems that lend support to our predictions and may offer opportunities for testing our hypotheses (Fig 1). For most of these systems, we were unable to find data on the assortativity of transmission which therefore limited our ability to make conclusions about trends in the data. However, both of the two wildlife systems for which we found data describing all three of our variables (, post-maturation lifespan, and transmission assortativity) matched our model’s predictions. Wanelik et al. (2017) showed that Great Island Virus (GIV) transmission in guillemots (Uria aalge) is assortative across age classes because of the spatial structure of breeding grounds. GIV is transmitted by poorly motile ticks and pre-breeding stages of guillemots do not enter breeding areas of the colony. As a consequence, the virus does not readily transmit between guillemot age-stages (Wanelik et al. 2017). Previous work on guillemot life-history shows that the birds spend more than three quarters of their life-span as mature breeders (Harris & Wanless 1995), and therefore the combination of assortative transmission and fast maturation predicts that GIV should be more virulent in breeders. In line with the predictions of our model, infection associated mortality risk is 1.45 times higher for adults than for juveniles (Nunn et al. 2006).
In the second example, Jones et al. (2008) showed that salmon louse caused morality in juvenile pink salmon (Oncorhynchus gorbuscha), but had no effect on mortality risk for adults. Salmon louse is also assortatively transmitted between age classes, because pink salmon have strict two-year lifespans where they are only ever associated with individuals of their same age class (Heard 1991; Krkošek et al. 2007). The salmon only reproduce once at the very end of their lives (semelparity), and therefore have a short adult period. This short post-maturation stage and assortative transmission correctly predicts the higher salmon louse virulence in juveniles.
Better data on mixing matrices for more disease systems could provide interesting insights into the maintenance of either high juvenile or high adult virulence. One system where these insights could prove especially important is in Bd (Batrachochytrium dendrobatidis, or chytrid fungus) infection in frogs, which has been causing catastrophic worldwide declines in frog populations (Kilpatrick et al. 2010). Bd infection has been shown to have different virulence effects in the different frog life-stages (Medina et al. 2015; Hite et al. 2016) and these effects also vary by frog species (Berger et al. 1998; Blaustein et al. 2005). Recent work has shown that adult virulence in several frog populations has not decreased even after 20 years of Bd presence (Voyles et al. 2018). Already, frog demography has been implicated as an important factor for population persistence in the face of Bd with frog species where adults move away from breeding waters being more resistant to population declines (Lips et al. 2006; McCaffery et al. 2015), and frogs in habitats with multi-year larvae having more severe epidemics because the older stages maintain high levels of infection that then spill over to infect other stages and species (Medina et al. 2015; Hite et al. 2016). Changes in the assortivity of mixing clearly has important implications for disease transmission across stages, and our model suggests that it could also have implications for the maintenance of high virulence in different age stages.
While data on age-related contact patterns are difficult to find in wildlife populations, a wealth of mixing data exists for humans (Mossong et al. 2008; Rohani et al. 2010). These suggest that contacts relevant for the transmission of directly transmitted pathogens are highly assortative by age. While the evolutionary drivers of human pathogens are often complicated, we posit that chickenpox (varicella virus) virulence in humans proves an intriguing case study. Given that humans have a long juvenile period in the context of our model, even when we only consider pre-reproductive and reproductive periods (Bogin & Smith 1996), the higher virulence in adults of chickenpox (23-29 times higher mortality risk in adults (Heininger & Seward 2006) for primary infections in naïve individuals fits the predictions of our model. This higher mortality risk corresponds to increased viral titers with age (Malavige et al. 2008) and, perhaps most interestingly, while varicella virus infects many cell-types, T cell infection is thought to be important for transport and pathogenesis (Zerboni & Arvin 2016). Therefore, age-related trends in T-cell abundance could be implicated in chickenpox pathogenesis, although this relationship is complicated by the fact that VSV-specific T-cell responses are also correlated with decreased viral titer and diminish with age (Erkeller-Yuksel et al. 1992; Nader et al. 1995; Malavige et al. 2008). Still, this example points towards one mechanism that may underlie the mediation of age-specific virulence in pathogens.
Our models have implications for disease management especially in farmed and other managed animal populations. For instance, if the post-maturation span is short (i.e. if u is small), then artificial restriction of the contacts between stages is predicted to select for higher virulence. However, if the post-maturation span is long, restricting the contacts into juvenile-juvenile and adult-adult (by e.g., separating the cohorts) can lead to the parasite extinction as a result of overexploitation against the juveniles. These contrasting outcomes can occur for any given host species, depending on how management modulates host stage-structure. Our models thus predict that, to prevent evolutionary changes towards higher virulence, one needs to carefully take into account the cohort structure.
For simplicity and tractability we chose to use simple two-stage models rather than continuous “infection-age” models (which would entail the formalism based on partial differential equations and dynamic programming approach). Future studies that capture more continuous age structure are an important next step. Also, although we assumed that parasites can express conditional virulence depending on the stage of the hosts they infect with, more data are needed to test this idea. In addition, coevolutionary models and multiple infections are both likely to give further important insights to the determinants of age-dependent disease interactions in nature. Our approach offers the basis for modeling these coevolutionary dynamics between hosts and parasites when there is stage structure.
Footnotes
Conflict of interest: None.
Data accessibility: No data is used for this study; a supplementary material for detailed mathematical analyses has been submitted in a separate file.