Host-parasite fluctuating selection in the absence of specificity.

Fluctuating selection driven by coevolution between hosts and parasites is important for the generation of host and parasite diversity across space and time. Theory has focused primarily on infection genetics, with highly specific 'matching-allele' frameworks more likely to generate fluctuating selection dynamics (FSD) than 'gene-for-gene' (generalist-specialist) frameworks. However, the environment, ecological feedbacks and life-history characteristics may all play a role in determining when FSD occurs. Here, we develop eco-evolutionary models with explicit ecological dynamics to explore the ecological, epidemiological and host life-history drivers of FSD. Our key result is to demonstrate for the first time, to our knowledge, that specificity between hosts and parasites is not required to generate FSD. Furthermore, highly specific host-parasite interactions produce unstable, less robust stochastic fluctuations in contrast to interactions that lack specificity altogether or those that vary from generalist to specialist, which produce predictable limit cycles. Given the ubiquity of ecological feedbacks and the variation in the nature of specificity in host-parasite interactions, our work emphasizes the underestimated potential for host-parasite coevolution to generate fluctuating selection.

The importance of who infects whom: the evolution of diversity in host resistance to infectious disease.

Variation for resistance to infectious disease is ubiquitous and critical to host and parasite evolution and to disease impact, spread and control. However, the processes that generate and maintain this diversity are not understood. We examine how ecological feedbacks generate diversity in host defence focussing on when polymorphism can evolve without co-evolution of the parasite. Our key result is that when there is heritable variation in hosts in both their transmissibility and susceptibility along with costs to resistance, there is the possibility of the evolution of polymorphism. We argue that a wide range of behavioural or physiological mechanisms may lead to relationships between transmissibility and susceptibility that generate diversity. We illustrate this by showing that a tendency for higher contacts between related individuals leads to polymorphism. Only dimorphisms can evolve when infection is determined only by an individuals' susceptibility or when transmissibility and susceptibility are simply positively or negatively correlated.

Evolutionary behaviour, trade-offs and cyclic and chaotic population dynamics.

Many studies of the evolution of life-history traits assume that the underlying population dynamical attractor is stable point equilibrium. However, evolutionary outcomes can change significantly in different circumstances. We present an analysis based on adaptive dynamics of a discrete-time demographic model involving a trade-off whose shape is also an important determinant of evolutionary behaviour. We derive an explicit expression for the fitness in the cyclic region and consequently present an adaptive dynamic analysis which is algebraic. We do this fully in the region of 2-cycles and (using a symbolic package) almost fully for 4-cycles. Simulations illustrate and verify our results. With equilibrium population dynamics, trade-offs with accelerating costs produce a continuously stable strategy (CSS) whereas trade-offs with decelerating costs produce a non-ES repellor. The transition to 2-cycles produces a discontinuous change: the appearance of an intermediate region in which branching points occur. The size of this region decreases as we move through the region of 2-cycles. There is a further discontinuous fall in the size of the branching region during the transition to 4-cycles. We extend our results numerically and with simulations to higher-period cycles and chaos. Simulations show that chaotic population dynamics can evolve from equilibrium and vice-versa.

On the determination of evolutionary outcomes directly from the population dynamics of the resident.

In order to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary in an ab initio calculation to find the fitness and its derivatives at a singular point. It has been suggested that the possible evolutionary behaviour can be predicted directly from the resident population dynamics, without the need for calculation, by applying three criteria-one based on the form of the density dependent rates and two on the role played by the evolving parameters. The existing arguments for these criteria are rather limited: they apply to systems in which individuals enter an initial class and can then move through any number of other population classes sequentially. (Extensions are included but only apply for systems of two and three classes.) Additionally, many of the arguments depend on the use of a phenomenologically motivated fitness (shown equivalent to the standard form but in a rather long and indirect manner). The present paper removes all these flaws-the criteria are established directly from the standard definition of fitness and individuals can enter any class and move through the classes non-sequentially without restriction on their number. The criteria thus established underlie a geometric description of the singular behaviour in adaptive dynamics which allows direct inferences to be made from population dynamics to the possible singular behaviour depending on which of the criteria apply and on the nature of the trade-off between evolving parameters. The method has the great advantage of leaving the trade-off explicit but unspecified.

The effect of delayed host self-regulation on host-pathogen population cycles in forest insects.

Delayed host self-regulation using a Beverton-Holt function and delayed logistic self-regulation are included in a host-pathogen model with free-living infective stages (Anderson and May's model G) with the purpose of investigating whether adding the relatively complex self-regulations decrease the likelihood of population cycles. The main results indicate that adding delayed self-regulation to the baseline model increases the likelihood of population cycles. The dynamics display some of the key features seen in the field, such as cycle peak density exceeding the carrying capacity and a locally stable equilibrium coexisting with a stable cycle (bistability). Numerical studies show that the model with more complex forms of self-regulation can generate cycles which match most aspects of the cycles observed in nature.

Epidemiological consequences of an incursion of highly pathogenic H5N1 avian influenza into the British poultry flock.

Highly pathogenic avian influenza and in particular the H5N1 strain has resulted in the culling of millions of birds and continues to pose a threat to poultry industries worldwide. The recent outbreak of H5N1 in the UK highlights the need for detailed assessment of the consequences of an incursion and of the efficacy of control strategies. Here, we present results from a model of H5N1 propagation within the British poultry industry. We find that although the majority of randomly seeded incursions do not spread beyond the initial infected premises, there is significant potential for widespread infection. The efficacy of the European Union strategy for disease control is evaluated and our simulations emphasize the pivotal role of duck farms in spreading H5N1.

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria - one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.

The influence of trade-off shape on evolutionary behaviour in classical ecological scenarios.

Trade-off shapes are crucial to evolutionary outcomes. However, due to different ecological feedbacks their implications may depend not only on the trade-off being considered but also the ecological scenario. Here, we apply a novel geometric technique, trade-off and invasion plots (TIPs), to examine in detail how the shape of trade-off relationships affect evolutionary outcomes under a range of classic ecological scenarios including Lotka-Volterra type and host-parasite interactions. We choose models of increasing complexity in order to gain an insight into the features of ecological systems that determine the evolutionary outcomes. In particular we focus on when evolutionary attractors, repellors and branching points occur and how this depends on whether the costs are accelerating (benefits become 'increasingly' costly), decelerating (benefits become 'decreasingly' costly) or constant. In all cases strongly accelerating costs lead to attractors while strongly decelerating ones lead to repellors, but with weaker relationships, this no longer holds. For some systems weakly accelerating costs may lead to repellors and decelerating costs may lead to attractors. In many scenarios it is weakly decelerating costs that lead to branching points, but weakly accelerating and linear costs may also lead to disruptive selection in particular ecological scenarios. Using our models we suggest a classification of ecological interactions, based on three distinct criteria, that can produce one of four fundamental TIPs which allow for different evolutionary behaviour. This provides a baseline theory which may inform the prediction of evolutionary outcomes in similar yet unexplored ecological scenarios. In addition we discuss the implications of our results to a number of specific life-history trade-offs in the classic ecological scenarios represented by our models.

The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites.

We use adaptive dynamics and pairwise invadability plots to examine the evolutionary dynamics of host resistance to microparasitic infection transmitted indirectly via free stages. We investigate trade-offs between pathogen transmission rate and intrinsic growth rate. Adaptive dynamics distinguishes various evolutionary outcomes associated with repellors, attractors or branching points. We find criteria corresponding to these and demonstrate that a major factor deciding the evolutionary outcome is whether trade-offs are acceleratingly or deceleratingly costly. We compare and contrast two models and show how the differences between them lead to different evolutionary outcomes.

When is evolutionary branching in predator-prey systems possible with an explicit carrying capacity?

In this study we use the theory of adaptive dynamics firstly to explore the differences in evolutionary behaviour of a generalist predator (or more specifically an omnivorous or intraguild predator) in a predator-prey model, with a Holling Type II functional response, when two distinct forms for the carrying capacity are used. The first of these involves the carrying capacity as an emergent property, whilst in the second it appears explicitly in the dynamics. The resultant effect this has on the intraspecific competition in each case is compared. Taking an identical trade-off in each case, we find that only with an emergent carrying capacity is evolutionary branching possible. Our study then concentrates solely on the case where the carrying capacity appears explicitly. Using the same model as above, but choosing alternate trade-offs, we find branching can occur with an explicit carrying capacity. Our investigation finishes by taking a more general functional response in an attempt to derive a condition for when branching can or cannot occur. For a predator-prey model, branching cannot occur if the functional response can be separated into two components, one a function of the population densities, X and Z, and the other a function of the evolving parameter z (traded off against the intrinsic growth rate), i.e. if F(z,X,Z) = F(1)(z)F(2)(X,Z). This search for evolutionary branching is motivated by its possible role in speciation.

A model to assess the efficacy of vaccines for control of liver fluke infection.

Fasciola hepatica, common liver fluke, infects cattle and sheep causing disease and production losses costing approximately $3 billion annually. Current control relies on drugs designed to kill the parasite. However, resistance is evident worldwide and widespread in some areas. Work towards a vaccine has identified several antigens of F. hepatica that show partial efficacy in terms of reducing worm burden and egg output. A critical question is what level of efficacy is required for such a vaccine to be useful? We have created the first mathematical model to assess the effectiveness of liver fluke vaccines under simulated field conditions. The model describes development of fluke within a group of animals and includes heterogeneity in host susceptibility, seasonal exposure to metacercariae and seasonal changes in temperature affecting metacercarial survival. Our analysis suggests that the potential vaccine candidates could reduce total fluke burden and egg output by up to 43% and 99%, respectively, on average under field conditions. It also suggests that for a vaccine to be effective, it must protect at least 90% of animals for the whole season. In conclusion, novel, partial, vaccines could contribute substantially towards fasciolosis control, reducing usage of anthelmintics and thus delaying the spread of anthelmintic resistance.

Adaptive dynamics of Lotka-Volterra systems with trade-offs: the role of interspecific parameter dependence in branching.

We determine the adaptive dynamics of a general Lotka-Volterra system containing an intraspecific parameter dependency--in the form of an explicit functional trade-off between evolving parameters--and interspecific parameter dependencies--arising from modelling species interactions. We develop expressions for the fitness of a mutant strategy in a multi-species resident environment, the position of the singular strategy in such systems and the non-mixed second-order partial derivatives of the mutant fitness. These expressions can be used to determine the evolutionary behaviour of the system. The type of behaviour expected depends on the curvature of the trade-off function and can be interpreted in a biologically intuitive manner using the rate of acceleration/deceleration of the costs implicit in the trade-off function. We show that for evolutionary branching to occur we require that one (or both) of the traded-off parameters includes an interspecific parameter dependency and that the trade-off function has weakly accelerating costs. This could have important implications for understanding the type of mechanisms that cause speciation. The general theory is motivated by using adaptive dynamics to examine evolution in a predator-prey system. The applicability of the general theory as a tool for examining specific systems is highlighted by calculating the evolutionary behaviour in a three species (prey-predator-predator) system.

Evolution, the loss of diversity and the role of trade-offs.

We investigate how the loss of previously evolved diversity in host resistance to disease is dependent on the complexity of the underlying evolutionary trade-off. Working within the adaptive dynamics framework, using graphical tools (pairwise invasion plots, PIPs; trait evolution plots, TEPs) and algebraic analysis we consider polynomial trade-offs of increasing degree. Our focus is on the evolutionary trajectory of the dimorphic population after it has been attracted to an evolutionary branching point. We show that for sufficiently complex trade-offs (here, polynomials of degree three or higher) the resulting invasion boundaries can form closed 'oval' areas of invadability and strategy coexistence. If no attracting singular strategies exist within this region, then the population is destined to evolve outside of the region of coexistence, resulting in the loss of one strain. In particular, the loss of diversity in this model always occurs in such a way that the remaining strain is not attracted back to the branching point but to an extreme of the trade-off, meaning the diversity is lost forever. We also show similar results for a non-polynomial but complex trade-off, and for a different eco-evolutionary model. Our work further highlights the importance of trade-offs to evolutionary behaviour.

The evolution of resistance through costly acquired immunity.

We examine the evolutionary dynamics of resistance to parasites through acquired immunity. Resistance can be achieved through the innate mechanisms of avoidance of infection and reduced pathogenicity once infected, through recovery from infection and through remaining immune to infection: acquired immunity. We assume that each of these mechanisms is costly to the host and find that the evolutionary dynamics of innate immunity in hosts that also have acquired immunity are quantitatively the same as in hosts that possess only innate immunity. However, compared with resistance through avoidance or recovery, there is less likely to be polymorphism in the length of acquired immunity within populations. Long-lived organisms that can recover at intermediate rates faced with fast-transmitting pathogens that cause intermediate pathogenicity (mortality of infected individuals) are most likely to evolve long-lived acquired immunity. Our work emphasizes that because whether or not acquired immunity is beneficial depends on the characteristics of the disease, organisms may be selected to only develop acquired immunity to some of the diseases that they encounter.

Modelling pathogen transmission: the interrelationship between local and global approaches.

We describe two spatial (cellular automaton) host-pathogen models with contrasting types of transmission, where the biologically realistic transmission mechanisms are based entirely on 'local' interactions. The two models, fixed contact area (FCA) and fixed contact number (FCN), may be viewed as local 'equivalents' of commonly used global density- (and frequency-) dependent models. Their outputs are compared with each other and with the patterns generated by these global terms. In the FCN model, unoccupied cells are bypassed, but in the FCA model these impede pathogen spread, extending the period of the epidemic and reducing the prevalence of infection when the pathogen persists. Crucially, generalized linear modelling reveals that the global transmission terms betaSI and beta'SI/N are equally good at describing transmission in both the FCA and FCN models when infected individuals are homogeneously distributed and N is approximately constant, as at the quasi-equilibrium. However, when N varies, the global frequency-dependent term beta'SI/N is better than the density-dependent one, betaSI, at describing transmission in both the FCA and FCN models. Our approach may be used more generally to compare different local contact structures and select the most appropriate global transmission term.

The adaptive dynamics of Lotka-Volterra systems with trade-offs.

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.

A model appropriate to the transmission of a human food-borne pathogen in a multigroup managed herd.

We describe a model of microparasite transmission within a multigroup managed farming system. The model was formulated to represent transmission of Escherichia coli O157 within a typical UK dairy herd and was used to suggest possible on-farm control strategies. The model includes birth, death, maturation, the dry/lactating cycle and various types of transmission (i.e. direct, pseudovertical (representing direct faecal-oral transmission between dam and calf within the first 48 h) and indirect (via free-living infectious units in the environment)). A combination of numerical and analytical techniques was used to analyse the model. We found that pseudovertical transmission and indirect transmission via infectious units in the 'general' environment can lead to more groups being affected, but otherwise have relatively little effect on the invasion criteria. To reduce infection within the herd, we suggest that efforts be directed at reducing the opportunity for group-specific indirect transmission-particularly within the weaned group.

How specificity and epidemiology drive the coevolution of static trait diversity in hosts and parasites.

There is typically considerable variation in the level of infectivity of parasites and the degree of resistance of hosts within populations. This trait variation is critical not only to the evolutionary dynamics but also to the epidemiology, and potentially the control of infectious disease. However, we lack an understanding of the processes that generate and maintain this trait diversity. We examine theoretically how epidemiological feedbacks and the characteristics of the interaction between host types and parasites strains determine the coevolution of host-parasite diversity. The interactions include continuous characterizations of the key phenotypic features of classic gene-for-gene and matching allele models. We show that when there are costs to resistance in the hosts and infectivity in the parasite, epidemiological feedbacks may generate diversity but this is limited to dimorphism, often of extreme types, in a broad range of realistic infection scenarios. For trait polymorphism, there needs to be both specificity of infection between host types and parasite strains as well as incompatibility between particular strains and types. We emphasize that although the high specificity is well known to promote temporal "Red Queen" diversity, it is costs and combinations of hosts and parasites that cannot infect that will promote static trait diversity.

Two-host, two-vector basic reproduction ratio (R(0)) for bluetongue.

Mathematical formulations for the basic reproduction ratio (R(0)) exist for several vector-borne diseases. Generally, these are based on models of one-host, one-vector systems or two-host, one-vector systems. For many vector borne diseases, however, two or more vector species often co-occur and, therefore, there is a need for more complex formulations. Here we derive a two-host, two-vector formulation for the R(0) of bluetongue, a vector-borne infection of ruminants that can have serious economic consequences; since 1998 for example, it has led to the deaths of well over 1 million sheep in Europe alone. We illustrate our results by considering the situation in South Africa, where there are two major hosts (sheep, cattle) and two vector species with differing ecologies and competencies as vectors, for which good data exist. We investigate the effects on R(0) of differences in vector abundance, vector competence and vector host preference between vector species. Our results indicate that R(0) can be underestimated if we assume that there is only one vector transmitting the infection (when there are in fact two or more) and/or vector host preferences are overlooked (unless the preferred host is less beneficial or more abundant). The two-host, one-vector formula provides a good approximation when the level of cross-infection between vector species is very small. As this approaches the level of intraspecies infection, a combination of the two-host, one-vector R(0) for each vector species becomes a better estimate. Otherwise, particularly when the level of cross-infection is high, the two-host, two-vector formula is required for accurate estimation of R(0). Our results are equally relevant to Europe, where at least two vector species, which co-occur in parts of the south, have been implicated in the recent epizootic of bluetongue.

Modelling bluetongue virus transmission between farms using animal and vector movements.

Bluetongue is a notifiable disease of ruminants which, in 2007, occurred for the first time in England. We present the first model for bluetongue that explicitly incorporates farm to farm movements of the two main hosts, as well as vector dispersal. The model also includes a seasonal vector to host ratio and dynamic restriction zones that evolve as infection is detected. Batch movements of sheep were included by modelling degree of mixing at markets. We investigate the transmission of bluetongue virus between farms in eastern England (the focus of the outbreak). Results indicate that most parameters affecting outbreak size relate to vectors and that the infection generally cannot be maintained without between-herd vector transmission. Movement restrictions are effective at reducing outbreak size, and a targeted approach would be as effective as a total movement ban. The model framework is flexible and can be adapted to other vector-borne diseases of livestock.