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.

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.

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.

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.

A baseline model for the co-evolution of hosts and pathogens.

The basic reproduction ratio (R (0)) is the expected number of secondary cases per primary in a totally susceptible population. In a baseline model, faced with an individual host strain pathogen virulence evolves to maximise R (0) which yields monomorphism. The basic depression ratio (D (0)) is the amount by which the total population is decreased, per infected individual, due to the presence of infection. Again, in a baseline model, faced with an individual pathogen strain host resistance evolves to minimise D (0) which yields monomorphism. With this in mind we analyse the community dynamics of the interaction between R (0) and D (0) and show that multi-strain co-existence (polymorphism) is possible and we discuss the possibility of stable cycles occurring within the co-existence states. We show for co-existence, the number of host and pathogen strains present need to be identical in order to achieve stable equilibria. For polymorphic states we observe contingencies (outcome dependent on initial conditions) between both point equilibrium and sustained oscillations. Invasion criteria for host and pathogen strains are identified.

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.

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.

Modelling the stability of Stx lysogens.

Shiga-toxin-converting bacteriophages (Stx phages) are temperate phages of Escherichia coli, and can cause severe human disease. The spread of shiga toxins by Stx phages is directly linked to lysogen stability because toxins are only synthesized and released once the lytic cycle is initiated. Lysogens of Stx phages are known to be less stable than those of the related lambda phage; this is often described in terms of a 'hair-trigger' molecular switch from lysogeny to lysis. We have developed a mathematical model to examine whether known differences in operator regions and binding affinities between Stx phages and lambda phage can account for the lower stability of Stx lysogens. The Stx phage 933W has only two binding sites in its left operator region (compared to three in phage lambda), but this has a minimal effect on 933W lysogen stability. However, the relatively weak binding affinity between repressor molecules and the second binding site in the right operator is found to significantly reduce the stability of its lysogens, and may account for the hair-trigger nature of the switch. Reduced lysogen stability can lead to increased frequency of genetic recombination in bacterial genomes. The development of the mathematical model has considerable utility in understanding the behaviour and evolution of the molecular switch, with implications for phage-related diseases.

Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations.

Deterministic and stochastic models motivated by Salmonella transmission in unmanaged/managed populations are studied. The SIRS models incorporate three routes of transmission (direct, vertical and indirect via free-living infectious units in the environment). With deterministic models we are able to understand the effects of different routes of transmission and other epidemiological factors on infection dynamics. In particular, vertical transmission has little influence on this dynamics, whereas the higher the indirect (direct) transmission rate the greater the tendency to persistent oscillation (stable endemic states). We show that the sustained cycles are also prone to demographic effect, i.e., persistent oscillation becomes impossible in the managed case (in the sense of balanced recruitment and death rates) by comparing with results in unmanaged populations (exponential population dynamics). Further, approximations of quasi-stationary distributions are derived for stochastic versions of the proposed models based on a diffusion approximation to the infection process. The effect of transmission parameters on the ratio of mean to standard deviation of the approximating distribution, used to judge the validity of the approximations and the expected time until fade out of infection, is further discussed. We conclude that strengthening any route of transmission may or may not reduce the expected time to fade out of infection, depending on the population dynamics.

Pair approximations and the inclusion of indirect transmission: theory and application to between farm transmission of Salmonella.

The spatial-temporal dynamics of farm animal diseases depend both on disease specific processes and the underlying contact network between farms. Indirect transmission via free-living bacteria in the environment is an important transmission route and contributes significantly to the dynamics. The pair-wise model has been developed to include both direct transmission and indirect transmission via free stages. The model is compared with stochastic simulations of epidemics on contact networks. The network framework is applied to the investigation of the epidemiological dynamics of between-herd transmission of Salmonella spp. The main results help to explain differences in observed epidemiological patterns and to identify possible causes for different strains of Salmonella developing so much variation in their infection dynamics in UK dairy herds. Numerical results show that shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower prevalence of infection and a greater tendency for (damped) oscillation. A possible control strategy is consequently suggested. Furthermore, the effect of network structure on long-term dynamics is examined.

Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks.

The process of infection during an epidemic can be envisaged as being transmitted via a network of routes represented by a contact network. Most differential equation models of epidemics are mean-field models. These contain none of the underlying spatial structure of the contact network. By extending the mean-field models to pair-level, some of the spatial structure can be contained in the model. Some networks of transmission such as river or transportation networks are clearly asymmetric, whereas others such as airborne infection can be regarded as symmetric. Pair-level models have been developed to describe symmetric contact networks. Here we report on work to develop a pair-level model that is also applicable to asymmetric contact networks. The procedure for closing the model at the level of pairs is discussed in detail. The model is compared against stochastic simulations of epidemics on asymmetric contact networks and against the predictions of the symmetric model on the same networks.

A semi-stochastic model of the transmission of Escherichia coli O157 in a typical UK dairy herd: dynamics, sensitivity analysis and intervention/prevention strategies.

When modelling the transmission of infection within small populations, it is necessary to consider the possibility of stochastic fade-out of infection. We present a semi-stochastic model for the transmission of a microparasite, in this case Escherichia coli O157, within a multigroup system, namely a typical UK dairy herd. 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)). We present the results of our simulation study alongside data from empirical studies and also compare simulation results with those for the corresponding deterministic model. We then examine the effects of reducing shedding in the food-producing groups on outbreak size and prevalence of infection. A sensitivity analysis of herd prevalence reveals that, for both the deterministic and the semi-stochastic model, the prevalence within the herd is most sensitive to two parameters relating to the weaned group. This supports our previously reported conclusions for the deterministic model, which were based on an analysis of the next-generation matrix. The sensitivity analysis also indicates that herd prevalence is greatly affected by two other parameters relating to the lactating group. We conclude by discussing the possible efficacy of suggested intervention strategies.

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.

The geometric theory of adaptive evolution: trade-off and invasion plots.

The purpose of this paper is to take an entirely geometrical path to determine the evolutionary properties of ecological systems subject to trade-offs. In particular we classify evolutionary singularities in a geometrical fashion. To achieve this, we study trade-off and invasion plots (TIPs) which show graphically the outcome of evolution from the relationship between three curves. The first invasion boundary (curve) has one strain as resident and the other strain as putative invader and the second has the roles of the strains reversed. The parameter values for one strain are used as the origin with those of the second strain varying. The third curve represents the trade-off. All three curves pass through the origin or tip of the TIP. We show that at this point the invasion boundaries are tangential. At a singular TIP, in which the origin is an evolutionary singularity, the invasion boundaries and trade-off curve are all tangential. The curvature of the trade-off curve determines the region in which it enters the singular TIP. Each of these regions has particular evolutionary properties (EUS, CS, SPR and MI). Thus we determine by direct geometric argument conditions for each of these properties in terms of the relative curvatures of the trade-off curve and invasion boundaries. We show that these conditions are equivalent to the standard partial derivative conditions of adaptive dynamics. The significance of our results is that we can determine whether the singular strategy is an attractor, branching point, repellor, etc. simply by observing in which region the trade-off curve enters the singular TIP. In particular we find that, if and only if the TIP has a region of mutual invadability, is it possible for the singular strategy to be a branching point. We illustrate the theory with an example and point the way forward.

Understanding the dynamics of Salmonella infections in dairy herds: a modelling approach.

There is evidence of variation in the infection dynamics of different Salmonella serotypes in cattle--ranging from transient epidemics to long term persistence and recurrence. We seek to identify possible causes of these differences. In this study we present mathematical models which describe both managed population dynamics and epidemiology and use these to investigate the effects of demographic and epidemiological factors on epidemic behaviour and threshold for invasion. In particular, when the system is perturbed by higher culling or pathogen-induced mortality we incorporate mechanisms to constrain the lactating herd size to remain constant in the absence of pathogen or to lie within a fairly small interval in the presence of pathogen. A combination of numerical and analytical techniques is used to analyse the models. We find that the epidemiologically dominating management group can change from the dry/lactating cycle to the weaned group with increasing culling rate. Pseudovertical transmission is found to have little effect on the invasion criteria, while indirect transmission has significant influence. Pathogen-induced mortality, recovery, immune response and pathogen removal are found to be factors inducing damped oscillations; variation in these factors between Salmonella serotypes may be responsible for some of the observed differences in within herd dynamics. Specifically, higher pathogen-induced mortality, shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower number of infectives and smaller epidemics but a greater tendency for damped oscillations.