Immunity-driven evolution of virulence and diversity in respiratory diseases.

The time-honored paradigm in the theory of virulence evolution assumes a positive relation between infectivity and harmfulness. However, the etiology of respiratory diseases yields a negative relation, with diseases of the lower respiratory tract being less infective and more harmful. We explore the evolutionary consequences in a simple model incorporating cross-immunity between disease strains that diminishes with their distance in the respiratory tract, assuming that docking rate follows the match between the local mix of cell surface types and the pathogen's surface and cross-immunity the similarity of the pathogens' surfaces. The assumed relation between fitness components causes virulent strains infecting the lower airways to evolve to milder more transmissible variants. Limited cross-immunity, generally, causes a readiness to diversify that increases with host population density. In respiratory diseases that diversity will be highest in the upper respiratory tract. More tentatively, emerging respiratory diseases are likely to start low and virulent, to evolve up, and become milder. Our results extend to a panoply of realistic generalizations of the disease's ecology to including additional epitope axes. These extensions allow us to apply our results quantitatively to elucidate the differences in diversification between rhino- and coronavirus caused common colds.

Environmental dimensionality determines species coexistence.

According to the competitive-exclusion principle, the number n of regulating variables describing a given community dynamics is an upper bound on the number of species (or types or morphs) that can coexist at equilibrium. On occasion, it is possible to reformulate a model with a lower number of regulating variables than appeared in the initial specification. We call the smallest number of such variables the dimension of the environmental feedback, or environmental dimension for short. For studying which species can invade a community, it is enough to know the sign of each species' long-term growth rate, i.e., invasion fitness. Therefore, different indicators of population growth - so-called fitness proxies, such as the basic reproduction number-are sometimes preferred. However, as we show, different fitness proxies may have different dimensions. Fundamental characteristics such as the environmental dimension should not depend on such arbitrary choices. Here, we resolve this difficulty by introducing a refined definition of environmental dimension that focuses on neutral fitness contours. On this basis, we show that this definition of environmental dimension is not only unambiguous, i.e., independent of the choice of fitness proxy, but also constructive, i.e., applicable without needing to assess an infinite number of possible fitness proxies. We then investigate how to determine environmental dimensions by analysing the two components of the environmental feedback: the impact map describing how a community's resident species affect the regulating variables and the sensitivity map describing how population growth depends on the regulating variables. The dimension of the impact map is lower than n when the set of feasible environments is of lower dimension than n, and the dimension of the sensitivity map is lower than n when not all n regulating variables affect the sign of population growth independently. While the minimum of the dimensions of the impact and sensitivity maps provides an upper bound on the environmental dimension, the combined effect of the two maps can result in an even lower environmental dimension, which happens when the sensitivity map is insensitive to some aspects of the impact map's image. To facilitate the applications of the framework introduced here, we illustrate all key concepts with detailed worked examples. In view of these results, we claim that the environmental dimension is the ultimate generalization of the traditional and widely used notions of the "number of regulating variables" or the "number of limiting factors", and is thus the sharpest generally applicable upper bound on the number of species that can robustly coexist in a community.

Evolution of Reproduction Periods in Seasonal Environments.

AbstractMany species are subject to seasonal cycles in resource availability, affecting the timing of their reproduction. Using a stage-structured consumer-resource model in which juvenile development and maturation are resource dependent, we study how a species' reproductive schedule evolves, dependent on the seasonality of its resource. We find three qualitatively different reproduction modes. First, continuous income breeding (with adults reproducing throughout the year) evolves in the absence of significant seasonality. Second, seasonal income breeding (with adults reproducing unless they are starving) evolves when resource availability is sufficiently seasonal and juveniles are more efficient resource foragers. Third, seasonal capital breeding (with adults reproducing partly through the use of energy reserves) evolves when resource availability is sufficiently seasonal and adults are more efficient resource foragers. Such capital breeders start reproduction already while their offspring are still experiencing starvation. Changes in seasonality lead to continuous transitions between continuous and seasonal income breeding, but the change between income and capital breeding involves a hysteresis pattern, such that a population's evolutionarily stable reproduction pattern depends on its initial one. Taken together, our findings show how adaptation to seasonal environments can result in a rich array of outcomes, exhibiting seasonal or continuous reproduction with or without energy reserves.

Correction to: Finite dimensional state representation of physiologically structured populations.

In the original publication of the article, the Subsection 2.1.2 was published incorrectly.

Lotka-Volterra approximations for evolutionary trait-substitution processes.

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka-Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster's total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka-Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community's Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the 'invasion implies substitution' result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

On models of physiologically structured populations and their reduction to ordinary differential equations.

Considering the environmental condition as a given function of time, we formulate a physiologically structured population model as a linear non-autonomous integral equation for the, in general distributed, population level birth rate. We take this renewal equation as the starting point for addressing the following question: When does a physiologically structured population model allow reduction to an ODE without loss of relevant information? We formulate a precise condition for models in which the state of individuals changes deterministically, that is, according to an ODE. Specialising to a one-dimensional individual state, like size, we present various sufficient conditions in terms of individual growth-, death-, and reproduction rates, giving special attention to cell fission into two equal parts and to the catalogue derived in an other paper of ours (submitted). We also show how to derive an ODE system describing the asymptotic large time behaviour of the population when growth, death and reproduction all depend on the environmental condition through a common factor (so for a very strict form of physiological age).

Finite dimensional state representation of physiologically structured populations.

In a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations' outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input-output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent 'population outputs'. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

Beyond R0 Maximisation: On Pathogen Evolution and Environmental Dimensions.

A widespread tenet is that evolution of pathogens maximises their basic reproduction ratio, R0. The breakdown of this principle is typically discussed as exception. Here, we argue that a radically different stance is needed, based on evolutionarily stable strategy (ESS) arguments that take account of the 'dimension of the environmental feedback loop'. The R0 maximisation paradigm requires this feedback loop to be one-dimensional, which notably excludes pathogen diversification. By contrast, almost all realistic ecological ingredients of host-pathogen interactions (density-dependent mortality, multiple infections, limited cross-immunity, multiple transmission routes, host heterogeneity, and spatial structure) will lead to multidimensional feedbacks.

Twelve fundamental life histories evolving through allocation-dependent fecundity and survival.

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with distinct phases of growth and reproduction (determinate growth). We extend this traditional, binary classification by showing that allocation-dependent fecundity and mortality rates allow for a large diversity of optimal allocation schedules. By analyzing a model of organisms that allocate energy between growth and reproduction, we find twelve types of optimal allocation schedules, differing qualitatively in how reproductive allocation increases with body mass. These twelve optimal allocation schedules include types with different combinations of continuous and discontinuous increase in reproduction allocation, in which phases of continuous increase can be decelerating or accelerating. We furthermore investigate how this variation influences growth curves and the expected maximum life span and body size. Our study thus reveals new links between eco-physiological constraints and life-history evolution and underscores how allocation-dependent fitness components may underlie biological diversity.

Epidemiological, evolutionary, and economic determinants of eradication tails.

Despite modern medical interventions, infectious diseases continue to generate huge socio-economic losses. The benefits of eradicating a disease are therefore high. While successful with smallpox and rinderpest, many other eradication attempts have failed. Eradications require huge and costly efforts, which can be sustained only if sufficient progress can be achieved. While initial successes are usually obtained more easily, progress often becomes harder as a disease becomes rare in the eradication endgame. A long eradication tail of slowly decreasing incidence levels can frustrate eradication efforts, as it becomes unclear whether progress toward eradication is still being made and how much more needs to be invested to push the targeted disease beyond its extinction threshold. Realistic disease dynamics are complicated by evolutionary responses to interventions and by interactions among different temporal and spatial scales. Models accounting for these complexities are required for understanding the shapes of eradication tails. In particular, such models allow predicting how hard or costly eradication will be, and may even inform in which manner progress has to be assessed during the eradication endgame. Here we outline a general procedure by analyzing the eradication tails of generic SIS diseases, taking into account two major ingredients of realistic complexity: a group-structured host population in which host contacts within groups are more likely than host contacts between groups, and virulence evolution subject to a trade-off between host infectivity within groups and host mobility among groups. Disentangling the epidemiological, evolutionary, and economic determinants of eradication tails, we show how tails of different shapes arise depending on salient model parameters and on how the extinction threshold is approached. We find that disease evolution generally extends the eradication tail and show how the cost structure of eradication measures plays a key role in shaping eradication tails.

Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective.

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock-scissor-paper configurations. Our concept differs from that of Heino et al. (TREE 13:367-370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.'s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock-scissor-paper configurations. Since in 'realistic' (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case.

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set D in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to D. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to D, just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.

Fast running restricts evolutionary change of the vertebral column in mammals.

The mammalian vertebral column is highly variable, reflecting adaptations to a wide range of lifestyles, from burrowing in moles to flying in bats. However, in many taxa, the number of trunk vertebrae is surprisingly constant. We argue that this constancy results from strong selection against initial changes of these numbers in fast running and agile mammals, whereas such selection is weak in slower-running, sturdier mammals. The rationale is that changes of the number of trunk vertebrae require homeotic transformations from trunk into sacral vertebrae, or vice versa, and mutations toward such transformations generally produce transitional lumbosacral vertebrae that are incompletely fused to the sacrum. We hypothesize that such incomplete homeotic transformations impair flexibility of the lumbosacral joint and thereby threaten survival in species that depend on axial mobility for speed and agility. Such transformations will only marginally affect performance in slow, sturdy species, so that sufficient individuals with transitional vertebrae survive to allow eventual evolutionary changes of trunk vertebral numbers. We present data on fast and slow carnivores and artiodactyls and on slow afrotherians and monotremes that strongly support this hypothesis. The conclusion is that the selective constraints on the count of trunk vertebrae stem from a combination of developmental and biomechanical constraints.

The canonical equation of adaptive dynamics for Mendelian diploids and haplo-diploids.

One of the powerful tools of adaptive dynamics is its so-called canonical equation (CE), a differential equation describing how the prevailing trait vector changes over evolutionary time. The derivation of the CE is based on two simplifying assumptions, separation of population dynamical and mutational time scales and small mutational steps. (It may appear that these two conditions rarely go together. However, for small step sizes the time-scale separation need not be very strict.) The CE was derived in 1996, with mathematical rigour being added in 2003. Both papers consider only well-mixed clonal populations with the simplest possible life histories. In 2008, the CE's reach was heuristically extended to locally well-mixed populations with general life histories. We, again heuristically, extend it further to Mendelian diploids and haplo-diploids. Away from strict time-scale separation the CE does an even better approximation job in the Mendelian than in the clonal case owing to gene substitutions occurring effectively in parallel, which obviates slowing down by clonal interference.

Ontogenetic symmetry and asymmetry in energetics.

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as "ontogenetic symmetry". Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

Necessary and sufficient conditions for R0 to be a sum of contributions of fertility loops.

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341-1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R0 in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases R0 allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R0 that may sometimes be used to good effect in cases where no simple formula for R0 exists.

A rigorous model study of the adaptive dynamics of Mendelian diploids.

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka-Volterra-like dynamics and single locus genetics (as in Kimura's in Proc Natl Acad Sci USA 54: 731-736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the 'long' time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M 1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as 'canonical equation'.

What life cycle graphs can tell about the evolution of life histories.

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.

Invasion and persistence of infectious agents in fragmented host populations.

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio R(0) plays this role. For metapopulations, defining such an 'invasion indicator' is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number R*. However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator R(m) for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.