Five challenges in evolution and infectious diseases.

Evolution is a key aspect of the biology of many pathogens, driving processes ranging from immune escape to changes in virulence. Because evolution is inherently subject to feedbacks, and because pathogen evolution plays out at scales ranging from within-host to between-host and beyond, evolutionary questions provide special challenges to the modelling community. In this article, we provide an overview of five challenges in modelling the evolution of pathogens and their hosts, and point to areas for development, focussing in particular on the issue of linking theory and data.

A generalized functional response for predators that switch between multiple prey species.

We develop a theory for the food intake of a predator that can switch between multiple prey species. The theory addresses empirical observations of prey switching and is based on the behavioural assumption that a predator tends to continue feeding on prey that are similar to the prey it has consumed last, in terms of, e.g., their morphology, defences, location, habitat choice, or behaviour. From a predator's dietary history and the assumed similarity relationship among prey species, we derive a general closed-form multi-species functional response for describing predators switching between multiple prey species. Our theory includes the Holling type II functional response as a special case and makes consistent predictions when populations of equivalent prey are aggregated or split. An analysis of the derived functional response enables us to highlight the following five main findings. (1) Prey switching leads to an approximate power-law relationship between ratios of prey abundance and prey intake, consistent with experimental data. (2) In agreement with empirical observations, the theory predicts an upper limit of 2 for the exponent of such power laws. (3) Our theory predicts deviations from power-law switching at very low and very high prey-abundance ratios. (4) The theory can predict the diet composition of a predator feeding on multiple prey species from diet observations for predators feeding only on pairs of prey species. (5) Predators foraging on more prey species will show less pronounced prey switching than predators foraging on fewer prey species, thus providing a natural explanation for the known difficulties of observing prey switching in the field.

Endemic disease, awareness, and local behavioural response.

The spread of a contagious disease is often accompanied by a rise in awareness of those in the social vicinity of infected individuals, and a subsequent change in behaviour. Such reactions can manifest themselves in lower susceptibility as people try to prevent themselves from catching the disease, but also in lower infectivity because of self-imposed quarantine or better hygiene, shorter durations of infectiousness or longer immunity. We here focus on the scenario of an endemic disease of which members of the population can be either aware or unaware, and consider a broad set of possible reactions. We quantify the impact on the endemicity of a disease in a well-mixed population under the variation of different disease parameters as a consequence of growing awareness in the population. Applying a pair-closure scheme allows us to analyse the effect of local correlations if aware individuals tend to occur near infected cases, and to link this to the amount of overlap between the networks underlying the spread of awareness and disease, respectively. Lastly, we study the consequences on the dynamics when the pathogen and awareness spread at different velocities.

Evolutionary consequences of a search image.

Many predators are able to become better at spotting cryptic prey by recognising specific clues, but by concentrating on one prey type they will become worse at spotting other prey types. This phenomenon is known as the formation of a search image for a certain prey by a predator and is related to apostatic selection. Here, we study the evolution of a search image in the predator by formulating and analysing a mathematical model. The predator forages for two prey types and is able to form an independent search image for both prey. The results show that the evolutionary dynamics can be divided into two parts: a fast and a slow part. At first selection pressure will be strong towards a stable ratio of prey, which is the same as the ratio found for the unbeatable prey choice for predators with a Holling type II functional response. Following this, the slow dynamics will keep this ratio constant independent of the trait values, but the predator will slowly evolve towards a stronger search image and ultimately become a specialist predator or slowly evolve towards generalist with a weak search image. In conclusion, the formation of a search image causes the predator to control the prey densities such that the ratio of available prey is kept constant by the predator.

How population dynamics shape the functional response in a one-predator-two-prey system.

The type III functional response has historically been associated with switching predators; when there is a choice of prey the predator favors the more abundant prey type. Although this functional response has been found in experiments where both prey densities are manipulated, in real world studies the type II functional response is more commonly found. In modeling, the type III functional response is often used in systems where the second prey type is, implicitly, assumed to be constant. Here we define a functional response that takes into account both prey densities. This causes the functional response to show both type II and type III behavior, dependent on the interaction between the two prey densities. If we take into account population dynamics, we find a type II functional response in most cases, because predation regulates the relative prey densities. This explains why type III functional responses are found in experiments where both prey densities are manipulated, but type II functional responses occur when the feedback of population dynamics on the functional response is important. Furthermore, the results show that switching can have a stabilizing or destabilizing effect and can even lead to predator extinction.

The dual role of CD4 T helper cells in the infection dynamics of HIV and their importance for vaccination.

Given the role of the CD4 T helper cells in the development of memory CTL precursors, it seems beneficial to boost the CD4 T helper response in the context of vaccination against the human immunodeficiency virus (HIV). However, CD4 T cells are also the preferred targets of infection by HIV. Here, we address the question as to whether it is advantageous to stimulate the CD4 T helper cell response, as this will increase the pool of potential target cells of infection. To do so we formulated a mathematical model describing the interactions between virus-infected cells, susceptible cells, HIV-specific CD4 helper T cells, and CTL precursor (CTLp) and effector cells (CTLe). The effect of increased initial CD4 helper and CTLp numbers on the outcome of infection, as well as the effect on viral set point of increased CD4 T helper growth rate, CTL responsiveness and the rate at which CTLp and CTLe are produced were studied. We found that only when the virus has a low basic reproductive number does the number of CTLp and CD4 T helper cells at the moment of infection influence the outcome of infection. In this situation, high initial T helper and CTL numbers can switch the outcome from full-blown infection to virus control. However, this holds for virus with infectivity in a limited range, and current estimates of virus infectivity suggest that it is higher. In that case, only a vaccination protocol that increases CTL responsiveness, ideally in combination with the rate of production of CD4 T helper cells, may offer a solution as it can reduce the viral set point considerably. If brought under a certain level, the viral population might be unable to replicate any further. However, changing these parameters of the immune response is only beneficial when infection is controlled by CTL in the long term. When a CD4 lymphoproliferative response is mounted but the CTL response is not maintained, increasing the CD4 T helper growth rate is deleterious.

Effector cytotoxic T lymphocyte numbers induced by vaccination should exceed levels in chronic infection for protection from HIV.

Recent technological advances have revolutionised our capacity to induce cytotoxic T lymphocyte (CTL) responses with a variety of vaccine formulations and delivery systems. However, the conditions required for a CTL-inducing vaccine to provide protection from infection or disease are poorly understood, and the results of challenge experiments have not been consistent. Here we use a mathematical model to examine the requirements necessary for successful vaccination against human immunodeficiency virus (HIV) through cellular immunity. We describe the interaction between cytotoxic T cells and infected lymphocytes, capturing the essence of a persistent infection of immune cells. We conclude that to protect from infection, the cellular immune response should be boosted to levels exceeding those in chronic infection. This requires either that effector CTL exceed this threshold before infection, or that a memory CTL population is established that can yield this level of effector CTL very quickly upon infection.

The role of T cell help for anti-viral CTL responses.

Cytotoxic T lymphocyte (CTL or CD8) responses are a major branch of the immune system involved in controlling viral infections. Murine models have shown that the development of effective and sustained CD8 cell responses requires CD4 T cell help. However, the precise mechanism in which CD4 cells provide help for CD8 cell responses is still controversial. In the literature, mainly two mechanisms are discussed. According to the "classical" pathway, CD4 cells secrete cytokines, such as IL-2, which promote the responsiveness of the CD8 cells. According to the "CD4-APC-CD8" pathway, CD4 cells specifically activate antigen presenting cells (APCs), and APCs specifically interact with CD8 cells, thereby delivering help. Here, we derive kinetic models in order to describe and compare both pathways of help. We find that the two pathways might have different roles in different situations. The classical pathway is more efficient at inducing CD8 cell expansion at high virus loads, while the CD4-APC-CD8 pathway is more efficient at inducing CD8 cell proliferation at low virus loads. From this, it follows that the classical pathway might be needed in order to kick-start a CD8 cell response in the acute phase of the infection, while the CD4-APC-CD8 pathway is needed in order to ensure virus clearance when virus load is reduced by the immune system. These findings have implications for the interpretation of experimental data from virus infection in helper-deficient hosts. In particular, the models offer further suggestions for the development of treatment regimes aimed at achieving immunological control of HIV infection which has been shown to crucially depend on the availability of helper cell responses.

Host life history and the evolution of parasite virulence.

We present a general epidemiological model of host-parasite interactions that includes various forms of superinfection. We use this model to study the effects of different host life-history traits on the evolution of parasite virulence. In particular, we analyze the effects of natural host death rate on the evolutionarily stable parasite virulence. We show that, contrary to classical predictions, an increase in the natural host death rate may select for lower parasite virulence if some form of superinfection occurs. This result is in agreement with the experimental results and the verbal argument presented by Ebert and Mangin (1997). This experiment is discussed in the light of the present model. We also point out the importance of superinfections for the effect of nonspecific immunity on the evolution of virulence. In a broader perspective, this model demonstrates that the occurrence of multiple infections may qualitatively alter classical predictions concerning the effects of various host life-history traits on the evolution of parasite virulence.

The measured level of prion infectivity varies in a predictable way according to the aggregation state of the infectious agent.

Transmissible spongiform encephalopathies are believed to be caused by an infectious form of the prion protein, designated PrP(Sc). The concentration of PrP(Sc) is often poorly correlated to the level of infectivity. Infectivity can be measured in two ways, namely endpoint titration and the incubation time assay, but patterns of infectivity vary depending on which method is used. These discrepancies can be explained by variation in the aggregation state of PrP(Sc). Both methods of measuring infectivity are modelled mathematically, and the theoretical results are in agreement with published data. It was found to be theoretically impossible to characterise prion infectivity by a multiple of a single quantity representing 'one prion', no matter how it is measured. Infectivity is instead characterised by both the number and sizes of the PrP(Sc) aggregates. Apparent discrepancies arise when these complexities are reduced to a single number.

The dynamics of two diffusively coupled predator-prey populations.

I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator-prey metapopulation or as a spatially explicit predator-prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator-prey populations can be regulated through the interplay of local dynamics and migration.

Understanding bacteriophage therapy as a density-dependent kinetic process.

Studies of bacteriophage as therapeutic agents have had mixed and unpredictable outcomes. We argue that interpretation of these apparently paradoxical results requires appreciation of various density-dependent threshold effects. We use a mathematical model to delineate different categories of outcome, including therapy by simple inundation, by active biocontrol, and by delayed active biocontrol. Counter-intuitively, there are situations in which earlier inoculation can be less efficacious, and simultaneous inoculation with antibiotics can be detrimental. Predictions of therapeutic responses are made using formulae dependent on biologically meaningful parameters; experimental measurement of the parameters will be a prerequisite of application of the model to particular study systems. Such modelling can point to which aspects of phage biology might most fruitfully be engineered so as to enhance the viability of bacteriophage therapy.

Local stability analysis of spatially homogeneous solutions of multi-patch systems.

Multi-patch systems, in which several species interact in patches connected by dispersal, offer a general framework for the description and analysis of spatial ecological systems. This paper describes how to analyse the local stability of spatially homogeneous solutions in such systems. The spatial arrangement of the patches and their coupling is described by a matrix. For a local stability analysis of spatially homogeneous solutions it turns out to be sufficient to know the eigenvalues of this matrix. This is shown for both continuous and discrete time systems. A bookkeeping scheme is presented that facilitates stability analyses by reducing the analysis of a k-species, n-patch system to that of n uncoupled k-dimensional single-patch systems. This is demonstrated in a worked example for a chain of patches. In two applications the method is then used to analyse the stability of the equilibrium of a predator-prey system with a pool of dispersers and of the periodic solutions of the spatial Lotka-Volterra model.

Phage therapy: the peculiar kinetics of self-replicating pharmaceuticals.

The specter of antibiotic-resistant bacteria has provoked renewed interest in the possible use of bacteriophages to control bacterial infections. We argue that clinical application of phage therapy has been held back by a failure to appreciate the extent to which the pharmacokinetics of self-replicating agents differ from those of normal drugs. For self-replicating pharmaceutical agents, treatment outcome depends critically on various density-dependent thresholds, often with apparently paradoxical consequences. An ability to predict these thresholds and associated critical time points is a necessity if phage therapy is to become clinically practicable.

The evolution of syntactic communication.

Animal communication is typically non-syntactic, which means that signals refer to whole situations. Human language is syntactic, and signals consist of discrete components that have their own meanings. Syntax is a prerequisite for taking advantage of combinatorics, that is, "making infinite use of finite means. The vast expressive power of human language would be impossible without syntax, and the transition from non-syntactic to syntactic communication was an essential step in the evolution of human language. We aim to understand the evolutionary dynamics of this transition and to analyse how natural selection can guide it. Here we present a model for the population dynamics of language evolution, define the basic reproductive ratio of words and calculate the maximum size of a lexicon. Syntax allows larger repertoires and the possibility to formulate messages that have not been learned beforehand. Nevertheless, according to our model natural selection can only favour the emergence of syntax if the number of required signals exceeds a threshold value. This result might explain why only humans evolved syntactic communication and hence complex language.

Intra-host competition between nef-defective escape mutants and wild-type human immunodeficiency virus type 1.

Various forms of nef genes with deletions at conserved positions along the sequence have been reported to persist in human immunodeficiency virus type 1 infected patients. We investigate the forces maintaining such variants in the proviral population. The main selection pressures are preservation of function and host immune response. The crippled Nef protein might have fewer epitopes, and as such be less visible to the specific immune response, but it will lose some function. Does a trade-off between avoidance of the immune response and loss of function explain the dynamics of the crippled virus found in the patients? To answer this question, we formulated a deterministic model of the virus-host interactions. We found that when the crippled protein presents few epitopes and suffers little loss of function, the two viral types can coexist. Otherwise, the wild-type comes to prevail. The mutant form might initially dominate, but as the selective pressure by the CD84+ T cells decreases over the course of infection, the advantage for the crippled form of losing epitopes disappears. Hence, we go from a situation of coexistence of wild-type and mutant, to a situation of only full-length nef. The results are discussed in the context of the suggested use of live attenuated vaccines having deletions in nef.

Designing drugs to stop the formation of prion aggregates and other amyloids.

Amyloid protein aggregates are implicated in many neurodegenerative diseases, including Alzheimer's disease and the prion diseases. Therapeutics to block amyloid formation are often tested in vitro, but it is not clear how to extrapolate from these experiments to a clinical setting, where the effective drug dose may be much lower. Here we address this question using a theoretical kinetic model to calculate the growth rate of protein aggregates as a function of the dose of each of three categories of drug. We find that therapeutics which block the growing ends of amyloids are the most promising, as alternative strategies may be ineffective or even accelerate amyloid formation at low drug concentrations. Our mathematical model can be used to identify and optimise an end-blocking drug in vitro. Our model also suggests an alternative explanation for data previously thought to prove the existence of an entity known as protein X.

The kinetics of proteinase K digestion of linear prion polymers.

Transmissible spongiform encephalopathies such as scrapie are caused by a protein-only infectious agent, known as a prion. It is not clear how a protein can be capable of replicating itself, and the mechanism remains controversial. One influential model hypothesizes that prions are nucleated, macroscopically linear polymers. We investigated the theoretical kinetics of this model and derived predictions which could be used to test the model. In the model, the polymerization and depolymerization rates are independent polymer size. This leads to an exponential size distribution at equilibrium. In agreement with a prediction stemming from this size distribution, the average size of PrP-res polymers was proportional to the square root of the concentration of PrP-res in a published study of in vitro conversion. Prion digestion by proteinase K (PK) is predicted to be biphasic. The second phase of digestion should be virtually independent of the PK concentration and should depend on the initial size distribution of prion polymers. For initially equilibrated polymers with an exponential size distribution, phase two digestion is exponential at a predicted rate. This rate varies in a defined way with the concentration used for equilibration and with other parameters which affect the average polymer size.