Evolution of antigen drift/switching: continuously evading pathogens

Evolutionary stability of antigenically escaping viruses

Antigenic variation is the main immune escape mechanism for RNA viruses like influenza or SARS-CoV-2. While high mutation rates promote antigenic escape, they also induce large mutational loads and reduced fitness. It remains unclear how this cost-benefit trade-off selects the mutation rate of viruses. Using a traveling wave model for the coevolution of viruses and host immune systems in a finite population, we investigate how immunity affects the evolution of the mutation rate and other nonantigenic traits, such as virulence. We first show that the nature of the wave depends on how cross-reactive immune systems are, reconciling previous approaches. The immune-virus system behaves like a Fisher wave at low cross-reactivities, and like a fitness wave at high cross-reactivities. These regimes predict different outcomes for the evolution of nonantigenic traits. At low cross-reactivities, the evolutionarily stable strategy is to maximize the speed of the wave, implying a higher mutation rate and increased virulence. At large cross-reactivities, where our estimates place H3N2 influenza, the stable strategy is to increase the basic reproductive number, keeping the mutation rate to a minimum and virulence low.

The influence of immune cells on the existence of virus quasi-species

This article investigate a nonlocal reaction-diffusion system of equations modeling virus distribution with respect to their genotypes in the interaction with the immune response. This study demonstrates the existence of pulse solutions corresponding to virus quasi-species. The proof is based on the Leray-Schauder method, which relies on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions. Furthermore, linear stability analysis of a spatially homogeneous stationary solution identifies the critical conditions for the emergence of spatial and spatiotemporal structures. Finally, numerical simulations are used to illustrate nonlinear dynamics and pattern formation in the nonlocal model.

Antigenic escape is accelerated by the presence of immunocompromised hosts

The repeated emergence of SARS-CoV-2 escape mutants from host immunity has obstructed the containment of the current pandemic and poses a serious threat to humanity. Prolonged infection in immunocompromised patients has received increasing attention as a driver of immune escape, and accumulating evidence suggests that viral genomic diversity and emergence of immune-escape mutants are promoted in immunocompromised patients. However, because immunocompromised patients comprise a small proportion of the host population, whether they have a significant impact on antigenic evolution at the population level is unknown. We consider an evolutionary epidemiological model that combines antigenic evolution and epidemiological dynamics. Applying this model to a heterogeneous host population, we study the impact of immunocompromised hosts on the evolutionary dynamics of pathogen antigenic escape from host immunity. We derived analytical formulae of the speed of antigenic evolution in heterogeneous host populations and found that even a small number of immunocompromised hosts in the population significantly accelerates antigenic evolution. Our results demonstrate that immunocompromised hosts play a key role in viral adaptation at the population level and emphasize the importance of critical care and surveillance of immunocompromised hosts.

Antigenic escape selects for the evolution of higher pathogen transmission and virulence

Despite the propensity for complex and non-equilibrium dynamics in nature, eco-evolutionary analytical theory typically assumes that populations are at equilibria. In particular, pathogens often show antigenic escape from host immune defences, leading to repeated epidemics, fluctuating selection and diversification, but we do not understand how this impacts the evolution of virulence. We model the impact of antigenic drift and escape on the evolution of virulence in a generalized pathogen and apply a recently introduced oligomorphic methodology that captures the dynamics of the mean and variance of traits, to show analytically that these non-equilibrium dynamics select for the long-term persistence of more acute pathogens with higher virulence. Our analysis predicts both the timings and outcomes of antigenic shifts leading to repeated epidemics and predicts the increase in variation in both antigenicity and virulence before antigenic escape. There is considerable variation in the degree of antigenic escape that occurs across pathogens and our results may help to explain the difference in virulence between related pathogens including, potentially, human influenzas. Furthermore, it follows that these pathogens will have a lower R₀, with clear implications for epidemic behaviour, endemic behaviour and control. More generally, our results show the importance of examining the evolutionary consequences of non-equilibrium dynamics.

Redundancy-selection trade-off in phenotype-structured populations

Realistic fitness landscapes generally display a redundancy-fitness trade-off: highly fit trait configurations are inevitably rare, while less fit trait configurations are expected to be more redundant. The resulting sub-optimal patterns in the fitness distribution are typically described by means of effective formulations, where redundancy provided by the presence of neutral contributions is modelled implicitly, e.g. with a bias of the mutation process. However, the extent to which effective formulations are compatible with explicitly redundant landscapes is yet to be understood, as well as the consequences of a potential miss-match. Here we investigate the effects of such trade-off on the evolution of phenotype-structured populations, characterised by continuous quantitative traits. We consider a typical replication-mutation dynamics, and we model redundancy by means of two dimensional landscapes displaying both selective and neutral traits. We show that asymmetries of the landscapes will generate neutral contributions to the marginalised fitness-level description, that cannot be described by effective formulations, nor disentangled by the full trait distribution. Rather, they appear as effective sources, whose magnitude depends on the geometry of the landscape. Our results highlight new important aspects on the nature of sub-optimality. We discuss practical implications for rapidly mutant populations such as pathogens and cancer cells, where the qualitative knowledge of their trait and fitness distributions can drive disease management and intervention policies.

The dynamics of evolutionary rescue from a novel pathogen threat in a host metapopulation

When a novel disease strikes a naïve host population, there is evidence that the most immediate response can involve host evolution while the pathogen remains relatively unchanged. When hosts also live in metapopulations, there may be critical differences in the dynamics that emerge from the synergy among evolutionary, ecological, and epidemiological factors. Here we used a Susceptible-Infected-Recovery model to explore how spatial and temporal ecological factors may drive the epidemiological and rapid-evolutionary dynamics of host metapopulations. For simplicity, we assumed two host genotypes: wild type, which has a positive intrinsic growth rate in the absence of disease, and robust type, which is less likely to catch the infection given exposure but has a lower intrinsic growth rate in the absence of infection. We found that the robust-type host would be strongly selected for in the presence of disease when transmission differences between the two types is large. The growth rate of the wild type had dual but opposite effects on host composition: a smaller increase in wild-type growth increased wild-type competition and lead to periodical disease outbreaks over the first generations after pathogen introduction, while larger growth increased disease by providing more susceptibles, which increased robust host density but decreased periodical outbreaks. Increased migration had a similar impact as the increased differential susceptibility, both of which led to an increase in robust hosts and a decrease in periodical outbreaks. Our study provided a comprehensive understanding of the combined effects among migration, disease epidemiology, and host demography on host evolution with an unchanging pathogen. The findings have important implications for wildlife conservation and zoonotic disease control.

Immune response and within-host viral evolution: Immune response can accelerate evolution

The objectives of this paper are to explore the impact of immune response on within-host viral evolution towards higher Darwinian fitness and, in particular, to verify a hypothesis that immune response, which is insufficient to annihilate a viral infection, can accelerate this evolution. To address this issue, a model of within-host viral evolution with immune response is formulated. This model is an extension of a continuous phenotype space model of viral evolution that was earlier suggested by A. Korobeinikov and C. Dempsey, which incorporates strain-specific immune response with cross-immunity. The model is based upon Nowak-May and Wodarz models of within-host HIV dynamics and is mechanistic (based upon first principles); this allows straightforward interpretation of the model's parameters and simulation results, as well as its further developments. In order to make the simulation results and conclusions robust and reliable and to ensure that they do not depend on the particularities of an immune response model, four different mathematical models of cell-mediated immune response are considered with the proposed model. Simulations confirmed that immune response, when it is unable to eliminate viruses, accelerates viral evolution.

A mathematical model of marine bacteriophage evolution

To explore how particularities of a host cell-virus system, and in particular host cell replication, affect viral evolution, in this paper we formulate a mathematical model of marine bacteriophage evolution. The intrinsic simplicity of real-life phage-bacteria systems, and in particular aquatic systems, for which the assumption of homogeneous mixing is well justified, allows for a reasonably simple model. The model constructed in this paper is based upon the Beretta-Kuang model of bacteria-phage interaction in an aquatic environment (Beretta & Kuang 1998 Math. Biosci.149, 57-76. (doi:10.1016/S0025-5564(97)10015-3)). Compared to the original Beretta-Kuang model, the model assumes the existence of a multitude of viral variants which correspond to continuously distributed phenotypes. It is noteworthy that the model is mechanistic (at least as far as the Beretta-Kuang model is mechanistic). Moreover, this model does not include any explicit law or mechanism of evolution; instead it is assumed, in agreement with the principles of Darwinian evolution, that evolution in this system can occur as a result of random mutations and natural selection. Simulations with a simplistic linear fitness landscape (which is chosen for the convenience of demonstration only and is not related to any real-life system) show that a pulse-type travelling wave moving towards increasing Darwinian fitness appears in the phenotype space. This implies that the overall fitness of a viral quasi-species steadily increases with time. That is, the simulations demonstrate that for an uneven fitness landscape random mutations combined with a mechanism of natural selection (for this particular system this is given by the conspecific competition for the resource) lead to the Darwinian evolution. It is noteworthy that in this system the speed of propagation of this wave (and hence the rate of evolution) is not constant but varies, depending on the current viral fitness and the abundance of susceptible bacteria. A specific feature of the original Beretta-Kuang model is that this model exhibits a supercritical Hopf bifurcation, leading to the loss of stability and the rise of self-sustained oscillations in the system. This phenomenon corresponds to the paradox of enrichment in the system. It is remarkable that under the conditions that ensure the bifurcation in the Beretta-Kuang model, the viral evolution model formulated in this paper also exhibits a rise in self-sustained oscillations of the abundance of all interacting populations. The propagation of the travelling wave, however, remains stable under these conditions. The only visible impact of the oscillations on viral evolution is a lower speed of the evolution.

Multi-scale problem in the model of RNA virus evolution

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

Order reduction for an RNA virus evolution model

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

Within-host stochastic emergence dynamics of immune-escape mutants

Predicting the emergence of new pathogenic strains is a key goal of evolutionary epidemiology. However, the majority of existing studies have focussed on emergence at the population level, and not within a host. In particular, the coexistence of pre-existing and mutated strains triggers a heightened immune response due to the larger total pathogen population; this feedback can smother mutated strains before they reach an ample size and establish. Here, we extend previous work for measuring emergence probabilities in non-equilibrium populations, to within-host models of acute infections. We create a mathematical model to investigate the emergence probability of a fitter strain if it mutates from a self-limiting strain that is guaranteed to go extinct in the long-term. We show that ongoing immune cell proliferation during the initial stages of infection causes a drastic reduction in the probability of emergence of mutated strains; we further outline how this effect can be accurately measured. Further analysis of the model shows that, in the short-term, mutant strains that enlarge their replication rate due to evolving an increased growth rate are more favoured than strains that suffer a lower immune-mediated death rate (‘immune tolerance’), as the latter does not completely evade ongoing immune proliferation due to inter-parasitic competition. We end by discussing the model in relation to within-host evolution of human pathogens (including HIV, hepatitis C virus, and cancer), and how ongoing immune growth can affect their evolutionary dynamics.

The ongoing evolution of infectious diseases provides a constant health threat. This evolution can either result in the production of new pathogens, or new strains of existing pathogens that escape prevailing drug treatments or immune responses. The latter process, also known as immune escape, is a predominant reason for the persistence of several viruses, including HIV and hepatitis C virus (HCV), in their human host. As a consequence, the within-host emergence of new strains has been the intense focus of modelling studies. However, existing models have neglected important feedbacks that affects this emergence probability. Specifically, once a mutated pathogen arises that spreads more quickly than the initial (resident) strain, it potentially triggers a heightened immune response that can eliminate the mutated strain before it spreads. Our study outlines novel mathematical modelling techniques that accurately quantify how ongoing immune growth reduces the emergence probability of mutated pathogenic strains over the course of an infection. Analysis of this model suggests that, in order to enlarge its emergence probability, it is evolutionary beneficial for a mutated strain to increase its growth rate rather than tolerate immunity by having a lower immune-mediated death-rate. Our model can be readily applied to existing within-host data, as demonstrated with application to HIV, HCV, and cancer dynamics.

Coevolutionary immune system dynamics driving pathogen speciation

We introduce and analyze a within-host dynamical model of the coevolution between rapidly mutating pathogens and the adaptive immune response. Pathogen mutation and a homeostatic constraint on lymphocytes both play a role in allowing the development of chronic infection, rather than quick pathogen clearance. The dynamics of these chronic infections display emergent structure, including branching patterns corresponding to asexual pathogen speciation, which is fundamentally driven by the coevolutionary interaction. Over time, continued branching creates an increasingly fragile immune system, and leads to the eventual catastrophic loss of immune control.

Cross-reactive immune responses as primary drivers of malaria chronicity

The within-host dynamics of an infection with the malaria parasite Plasmodium falciparum are the result of a complex interplay between the host immune system and parasite. Continual variation of the P. falciparum erythrocyte membrane protein (PfEMP1) antigens displayed on the surface of infected red blood cells enables the parasite to evade the immune system and prolong infection. Despite the importance of antigenic variation in generating the dynamics of infection, our understanding of the mechanisms by which antigenic variation generates long-term chronic infections is still limited. We developed a model to examine the role of cross-reactivity in generating infection dynamics that are comparable to those of experimental infections. The hybrid computational model we developed is attuned to the biology of malaria by mixing discrete replication events, which mimics the synchrony of parasite replication and invasion, with continuous interaction with the immune system. Using simulations, we evaluated the dynamics of a single malaria infection over time. We then examined three major mechanisms by which the dynamics of a malaria infection can be structured: cross-reactivity of the immune response to PfEMP1, differences in parasite clearance rates, and heterogeneity in the rate at which antigens switch. The results of our simulations demonstrate that cross-reactive immune responses play a primary role in generating the dynamics observed in experimentally untreated infections and in lengthening the period of infection. Importantly, we also find that it is the primary response to the initially expressed PfEMP1, or small subset thereof, that structures the cascading cross-immune dynamics and allows for elongation of the infection.

The evolution of mutation rate in an antagonistic coevolutionary model with maternal transmission of parasites

By constantly selecting for novel genotypes, coevolution between hosts and parasites can favour elevated mutation rates. Models of this process typically assume random encounters. However, offspring are often more likely to encounter their mother's parasites. Because parents and offspring are genetically similar, they may be susceptible to the same parasite strains and thus, in hosts, maternal transmission should select for mechanisms that decrease intergenerational genetic similarity. In parasites, however, maternal transmission should select for genetic similarity. We develop and analyse a model of host and parasite mutation rate evolution when parasites are maternally inherited. In hosts, we find that maternal transmission has two opposing effects. First, it eliminates coevolutionary cycles that previous work shows select for higher mutation. Second, it independently selects for higher mutation rates, because offspring that differ from their mothers are more likely to avoid infection. In parasites, however, the two effects of maternal transmission act in the same direction. As for hosts, maternal transmission eliminates coevolutionary cycles, thereby reducing selection for increased mutation. Unlike for hosts, however, maternal transmission additionally selects against higher mutation by favouring parasite offspring that are the same as their mothers.

Ploidy and the evolution of parasitism

Levels of parasitism are continuously distributed in nature. Models of host-parasite coevolution, however, typically assume that species can be easily characterized as either parasitic or non-parasitic. Consequently, it is poorly understood which factors influence the evolution of parasitism itself. We investigate how ploidy level and the genetic mechanisms underlying infection influence evolution along the continuum of parasitism levels. In order for parasitism to evolve, selective benefits to the successful invasion of hosts must outweigh the losses when encountering resistant hosts. However, we find that exactly where this threshold occurs depends not only on the strength of selection, but also on the genetic model of interaction, the ploidy level in each species, and the nature of the costs to virulence and resistance. With computer simulations, we are able to incorporate more realistic dynamics at the loci underlying species interactions and to extend our analyses in a number of directions, including finite population sizes, multiple alleles and different generation times.

Critical interplay between parasite differentiation, host immunity, and antigenic variation in trypanosome infections

Increasing availability of pathogen genomic data offers new opportunities to understand the fundamental mechanisms of immune evasion and pathogen population dynamics during chronic infection. Motivated by the growing knowledge on the antigenic variation system of the sleeping sickness parasite, the African trypanosome, we introduce a mechanistic framework for modeling within-host infection dynamics. Our analysis focuses first on a single parasitemia peak and then on the dynamics of multiple peaks that rely on stochastic switching between groups of parasite variants. A major feature of trypanosome infections is the interaction between variant-specific host immunity and density-dependent parasite differentiation to transmission life stages. In this study, we investigate how the interplay between these two types of control depends on the modular structure of the parasite antigenic archive. Our model shows that the degree of synchronization in stochastic variant emergence determines the relative dominance of general over specific control within a single peak. A requirement for multiple-peak dynamics is a critical switch rate between blocks of antigenic variants, which implies constraints on variant surface glycoprotein (VSG) archive genetic diversification. Our study illustrates the importance of quantifying the links between parasite genetics and within-host dynamics and provides insights into the evolution of trypanosomes.

Coexistence conditions for strains of influenza with immune cross-reaction

The accumulation of cross-immunity in the host population is an important factor driving the antigenic evolution of viruses such as influenza A. Mathematical models have shown that the strength of temporary non-specific cross-immunity and the basic reproductive number are both key determinants for evolutionary branching of the antigenic phenotype. Here we develop deterministic and stochastic versions of one such model. We examine how the time of emergence or introduction of a novel strain affects co-existence with existing strains and hence the initial establishment of a new evolutionary branch. We also clarify the roles of cross-immunity and the basic reproductive number in this process. We show that the basic reproductive number is important because it affects the frequency of infection, which influences the long term immune profile of the host population. The time at which a new strain appears relative to the epidemic peak of an existing strain is important because it determines the environment the emergent mutant experiences in terms of the short term immune profile of the host population. Strains are more likely to coexist, and hence to establish a new clade in the viral phylogeny, when there is a significant time overlap between their epidemics. It follows that the majority of antigenic drift in influenza is expected to occur in the earlier part of each transmission season and this is likely to be a key surveillance period for detecting emerging antigenic novelty.

Modifiers of mutation-selection balance: general approach and the evolution of mutation rates

A general approach is developed to estimate secondary selection at a modifier locus that influences some feature of a population under mutation-selection balance. The approach is based on the assumption that the properties of all available genotypes at this locus are similar. Then mutation-selection balance and weak associations between genotype distributions at selectable loci and the modifier locus are established rapidly. In contrast, changes of frequencies of the modifier genotypes are slow, and lead to only slow and small changes of the other features of the population. Thus, while these changes occur, the population remains in a state of quasi-equilibrium, where the mutation-selection balance and the associations between the selectable loci and the modifier locus are almost invariant. Selection at the modifier locus can be estimated by calculating quasiequilibrium values of these associations. This approach is developed for the situation where distributions of the number of mutations per genome within the individuals with a given modifier genotype are close to Gaussian. The results are used to study the evolution of the mutation rate. Because beneficial mutations are ignored, secondary selection at the modifier locus always diminishes the mutation rate. The coefficient of selection against an allele which increases the mutation rate by υ is approximately υδ2/[U(2−ρ)] = υŝ, where υ is the genomic deleterious mutation rate, δ is the selection differential of the number of mutations per individual in units of the standard deviation of the distribution of this number in the population, ρ is the ratio of variances of the number of mutations after and before selection, and ŝ is the selection coefficient against a mutant allele in the quasiequilibrium population. However, the decline of the mutation rate can be counterbalanced by the cost of fidelity, which can lead to an evolutionary equilibrium mutation rate.

Acute or chronic? Within-host models with immune dynamics, infection outcome, and parasite evolution

There is ample theoretical and experimental evidence that virulence evolution depends on the immune response of the host. In this article, we review a number of recent studies that attempt to explicitly incorporate the dynamics of the immune system (instead of merely representing it by a single black box parameter) in models for the evolution of parasite virulence. A striking observation is that the type of infection (acute or chronic) is invariably considered to be a constraint that model assumptions have to satisfy rather than as a potential outcome of the interaction of the parasite with its host's immune system. We argue that avoiding making assumptions about the type of infection will lead to a better understanding of infectious diseases, even though a number of fundamental and technical problems remain. Dynamical modeling of the immune system opens a wide range of perspectives: for understanding how the immune system eradicates a parasite (which it does for most pathogens but not for all, HIV being a notorious example of a virus that is not completely eliminated), for studying multiple infections through concomitant immunity, for understanding the emergence and evolution of the immune system in animals, and for evolutionary epidemiology in general (e.g., predicting evolutionary consequences of new therapies and public health policies). We conclude by discussing new approaches based on embedded (or nested) models and identify future perspectives for the modeling of infectious diseases.

Applying population-genetic models in theoretical evolutionary epidemiology

Much of the existing theory for the evolutionary biology of infectious diseases uses an invasion analysis approach. In this Ideas and Perspectives article, we suggest that techniques from theoretical population genetics can also be profitably used to study the evolutionary epidemiology of infectious diseases. We highlight four ways in which population-genetic models provide benefits beyond those provided by most invasion analyses: (i) they can make predictions about the rate of pathogen evolution; (ii) they explicitly draw out the mechanistic way in which the epidemiological dynamics feed into evolutionary change, and thereby provide new insights into pathogen evolution; (iii) they can make predictions about the evolutionary consequences of non-equilibrium epidemiological dynamics; (iv) they can readily incorporate the effects of multiple host dynamics, and thereby account for phenomena such as immunological history and/or host co-evolution.

The fastest evolutionary trajectory

Given two mutants, A and B, separated by n mutational steps, what is the evolutionary trajectory which allows a homogeneous population of A to reach B in the shortest time? We show that the optimum evolutionary trajectory (fitness landscape) has the property that the relative fitness increase between any two consecutive steps is constant. Hence, the optimum fitness landscape between A and B is given by an exponential function. Our result is precise for small mutation rates and excluding back mutations. We discuss deviations for large mutation rates and including back mutations. For very large mutation rates, the optimum fitness landscape is flat and has a single peak at type B.

Cross-immunity, invasion and coexistence of pathogen strains in epidemiological models with one-dimensional antigenic space

Several epidemic models with many co-circulating strains have shown that partial cross-immunity between otherwise identical strains of a pathogen can lead to exclusion of a subset of the strains. Here we examine the mechanisms behind these solutions by considering a host population in which two strains are endemic and ask when it can be invaded by a third strain. If the function relating antigenic distance to cross-immunity is strictly concave or linear invasion is always possible. If the function is strictly convex and has an initial gradient of zero invasion depends on the degree of antigenic similarity between strains and the basic reproductive number. Examining specific concave and convex functions shows that the shape of the cross-immunity function affects the role of secondary infections in invasion. The basic reproductive number affects the importance of tertiary infections. Thus the form of the relationship between antigenic distance and cross-immunity determines whether the pathogen population will consist of an unstructured cloud of strains or a limited number of strains with strong antigenic structuring. In the latter case the basic reproductive number determines the maximum number of strains that can coexist. Analysis of the evolutionary trajectory shows that attaining the maximum diversity requires large spontaneous changes in antigenic structure and cannot result from a sequence of small point mutations alone.

The cost of replication fidelity in human immunodeficiency virus type 1

Mutation rates should be governed by at least three evolutionary factors: the need for beneficial mutations, the benefit of minimizing the mutational load and the cost of replication fidelity. RNA viruses show high mutation rates compared with DNA micro-organisms, and recent findings suggest that the cost of fidelity might play a role in the evolution of increased mutation rates. Here, by analysing previously published data from HIV-1 reverse transcriptase in vitro assays, we show a trade-off between enzymatic accuracy and the maximum rate of polymerization, thus providing a biochemical basis for the fitness cost of fidelity in HIV-1. This trade-off seems to be related to inefficient extension of mispairs, which increases fidelity at the expense of the polymerization rate. Since in RNA viruses fast replication is critical for survival, this could impose a high cost of fidelity and favour the evolution of high mutation rates.

Epidemic dynamics and antigenic evolution in a single season of influenza A

We use a mathematical model to study the evolution of influenza A during the epidemic dynamics of a single season. Classifying strains by their distance from the epidemic-originating strain, we show that neutral mutation yields a constant rate of antigenic evolution, even in the presence of epidemic dynamics. We introduce host immunity and viral immune escape to construct a non-neutral model. Our population dynamics can then be framed naturally in the context of population genetics, and we show that departure from neutrality is governed by the covariance between a strain's fitness and its distance from the original epidemic strain. We quantify the amount of antigenic evolution that takes place in excess of what is expected under neutrality and find that this excess amount is largest under strong host immunity and long epidemics.

Virus evolution within patients increases pathogenicity

Viruses like the human immunodeficiency virus (HIV), the hepatitis B virus (HBV), the hepatitis C virus (HCV) and many others undergo numerous rounds of inaccurate reproduction within an infected host. The resulting viral quasispecies is heterogeneous and sensitive to any selection pressure. Here we extend earlier work by showing that for a wide class of models describing the interaction between the virus population and the immune system, virus evolution has a well-defined direction toward increased pathogenicity. In particular, we study virus-induced impairment of the immune response and certain cross-reactive stimulation of specific immune responses. For eight different mathematical models, we show that virus evolution reduces the equilibrium abundance of uninfected cells and increases the rate at which uninfected cells are infected. Thus, in general, virus evolution makes things worse. An idea for combating HIV infection, however, is constructing a virus mutant that could outcompete the existing infection without being pathogenic itself.

Some basic properties of immune selection

We analyze models for the evolutionary dynamics of viral or other infectious agents within a host. We study how the invasion of a new strain affects the composition and diversity of the viral population. We show that--under strain-specific immunity--the equilibrium abundance of uninfected cells declines during viral evolution. In addition, for cytotoxic immunity the absolute force of infection, and for non-cytotoxic immunity the absolute cellular virulence increases during viral evolution. We prove global stability by means of Lyapunov functions. These unidirectional trends of virus evolution under immune selection do not hold for general cross-reactive immune responses, which introduce frequency-dependent selection among viral strains. Therefore, appropriate cross-reactive immunity can lead to a viral evolution within a host which limits the extent of the disease.

Traveling waves in a model of influenza A drift

Between major pandemics, the influenza A virus changes its antigenic properties by accumulating point mutations (drift) mainly in the RNA genes that code for the surface proteins hemagglutinin (HA) and neuraminidase (NA). The successful strain (variant) that will cause the next epidemic is selected from a reduced number of progenies that possess relatively high transmissibility and the ability to escape from the immune surveillance of the host. In this paper, we analyse a one-dimensional model of influenza A drift (Z. Angew. Math. Mech. 76 (2) (1996) 421) that generalizes the classical SIR model by including mutation as a diffusion process in a phenotype space of variants. The model exhibits traveling wave solutions with an asymptotic wave speed that matches well those obtained from numerical simulations. As exact solutions for these waves are not available, asymptotic estimates for the amplitudes of infected and recovered classes are provided through an exponential approximation based on the smallness of the diffusion constant. Through this approximation, we find simple scaling properties to several parameters of relevance to the epidemiology of the disease.

Human cytomegalovirus glycoprotein N (gpUL73-gN) genomic variants: identification of a novel subgroup, geographical distribution and evidence of positive selective pressure

Human cytomegalvirus (HCMV) ORF UL73 is a polymorphic locus, encoding the viral glycoprotein gpUL73-gN, a component of the gC-II envelope complex. The previously identified gN genomic variants, denoted gN-1, gN-2, gN-3 and gN-4, were further investigated in this work by analysing a large panel of HCMV clinical isolates collected from all over the world (223 samples). Sequencing and phylogenetic analysis confirmed the existence of the four gN genotypes, but also allowed the identification of a novel subgroup belonging to the gN-3 genotype, which was designated gN-3b. The number of non-synonymous (d(N)) and synonymous (d(S)) nucleotide substitutions and their ratio (d(N)/d(S)) were estimated among the gN genotypes to evaluate the possibility of positive selection. Results showed that the four variants evolved by neutral (random) selection, but that the gN-3 and gN-4 genotypes are maintained by positive selective pressure. The 223 HCMV clinical isolates were subdivided according to their geographical origin, and four main regions of gN prevalence were identified: Europe, China, Australia and Northern America. The gN variants were found to be widespread and represented within the regions analysed without any significant difference, and no new genotype was detected. Finally, for clinical and epidemiological purposes, a rapid and low-cost method for genetic grouping of the HCMV clinical isolates was developed based on the RFLP revealed by SacI, ScaI and SalI digestion of the PCR-amplified UL73 sequence. This technique enabled us to distinguish all four gN genomic variants and also their subtypes.

Dynamics and selection of many-strain pathogens

Strain structure is of fundamental importance in the underlying dynamics of a number of pathogens. However, previous models have been too complex to accommodate many strains. This paper offers a solution to this problem, in the form of a simple model that is capable of capturing the dynamics of a large number of antigenic types that interact via host cross-immunity. We derive the structure of the model, which can manage the complexity of many strains by using a status-based formulation, assuming polarized immunity and cross-immunity act to reduced transmission probability. We then apply the model to address basic questions in strain dynamics, focusing particularly on the interpandemic dynamics of influenza. This model shows that strains have a tendency to "cluster." For a long infectious period, relative to host lifetime, clusters may coexist. By contrast, a short infectious period leads to a single dominant cluster at any given time. We show how the speed of cluster replacement depends on the specificity of cross-immunity and on the underlying pathogen mutation rate.

Coevolution of quasispecies: B-cell mutation rates maximize viral error catastrophes

Coevolution of two coupled quasispecies is studied, motivated by the competition between viral evolution and adapting immune response. In this coadaptive model, besides the classical error catastrophe for high virus mutation rates, a second "adaptation" catastrophe occurs, when virus mutation rates are too small to escape immune attack. Maximizing both regimes of viral error catastrophes is a possible strategy for an optimal immune response, reducing the range of allowed viral mutation rates to a minimum. From this requirement, one obtains constraints on B-cell mutation rates and receptor lengths, yielding an estimate of somatic hypermutation rates in the germinal center in accordance with observation.

Reevaluation of amino acid variability of the human immunodeficiency virus type 1 gp120 envelope glycoprotein and prediction of new discontinuous epitopes

To elucidate the evolutionary mechanisms of the human immunodeficiency virus type 1 gp120 envelope glycoprotein at the single-site level, the degree of amino acid variation and the numbers of synonymous and nonsynonymous substitutions were examined in 186 nucleotide sequences for gp120 (subtype B). Analyses of amino acid variabilities showed that the level of variability was very different from site to site in both conserved (C1 to C5) and variable (V1 to V5) regions previously assigned. To examine the relative importance of positive and negative selection for each amino acid position, the numbers of synonymous and nonsynonymous substitutions that occurred at each codon position were estimated by taking phylogenetic relationships into account. Among the 414 codon positions examined, we identified 33 positions where nonsynonymous substitutions were significantly predominant. These positions where positive selection may be operating, which we call putative positive selection (PS) sites, were found not only in the variable loops but also in the conserved regions (C1 to C4). In particular, we found seven PS sites at the surface positions of the alpha-helix (positions 335 to 347 in the C3 region) in the opposite face for CD4 binding. Furthermore, two PS sites in the C2 region and four PS sites in the C4 region were detected in the same face of the protein. The PS sites found in the C2, C3, and C4 regions were separated in the amino acid sequence but close together in the three-dimensional structure. This observation suggests the existence of discontinuous epitopes in the protein's surface including this alpha-helix, although the antigenicity of this area has not been reported yet.

The evolution of parasite virulence and transmission rate in a spatially structured population

If the transmission occurs through local contact of the individuals in a spatially structured population, the evolutionarily stable (ESS) traits of parasite might be quite different from what the classical theory with complete mixing predicts. In this paper, we theoretically study the ESS virulence and transmission rate of a parasite in a lattice-structured host population, in which the host can send progeny only to its neighboring vacant site, and the transmission occurs only in between the infected and the susceptible in the nearest-neighbor sites. Infected host is assumed to be infertile. The analysis based on the pair approximation and the Monte Carlo simulation reveal that the ESS transmission rate and virulence in a lattice-structured population are greatly reduced from those in completely mixing population. Unlike completely mixing populations, the spread of parasite can drive the host to extinction, because the local density of the susceptible next to the infected can remain high even when the global density of host becomes very low. This demographic viscosity and group selection between self-organized spatial clusters of host individuals then leads to an intermediate ESS transmission rate even if there is no tradeoff between transmission rate and virulence. The ESS transmission rate is below the region of parasite-driven extinction by a finite amount for moderately large reproductive rate of host; whereas, the evolution of transmission rate leads to the fade out of parasite for small reproductive rate, and the extinction of host for very large reproductive rate.

'Small worlds' and the evolution of virulence: infection occurs locally and at a distance

Why are some discases more virulent than others? Vector-borne diseases such as malaria and water-borne diseases such as cholera are generally more virulent than diseases spread by direct contagion. One factor that characterizes both vector- and water-borne diseases is their ability to spread over long distances, thus causing infection of susceptible individuals distant from the infected individual. Here we show that this ability of the pathogen to infect distant individuals in a spatially structured host population leads to the evolution of a more virulent pathogen. We use a lattice model in which reproduction is local but infection can vary between completely local to completely global. With completely global infection the evolutionarily stable strategy (ESS) is the same as in mean-field models while a lower virulence is predicted as infection becomes more local. There is characteristically a period of relatively moderate increase in virulence followed by a more rapid rise with increasing proportions of global infection as we move beyond a 'critical connectivity'. In the light of recent work emphasizing the existence of 'small world' networks in human populations, our results suggests that if the world is getting 'smaller'--as populations become more connected--diseases may evolve higher virulence.

Rates of spontaneous mutation

Rates of spontaneous mutation per genome as measured in the laboratory are remarkably similar within broad groups of organisms but differ strikingly among groups. Mutation rates in RNA viruses, whose genomes contain ca. 10(4) bases, are roughly 1 per genome per replication for lytic viruses and roughly 0.1 per genome per replication for retroviruses and a retrotransposon. Mutation rates in microbes with DNA-based chromosomes are close to 1/300 per genome per replication; in this group, therefore, rates per base pair vary inversely and hugely as genome sizes vary from 6 x 10(3) to 4 x 10(7) bases or base pairs. Mutation rates in higher eukaryotes are roughly 0.1-100 per genome per sexual generation but are currently indistinguishable from 1/300 per cell division per effective genome (which excludes the fraction of the genome in which most mutations are neutral). It is now possible to specify some of the evolutionary forces that shape these diverse mutation rates.

Evolutionary pattern of intra-host pathogen antigenic drift: effect of cross-reactivity in immune response

Several viruses are known to change their surface antigen types after infecting a host, thereby escaping the immune defence and ensuring persistent infection. In this paper, we theoretically study the pattern of intra-host micro-evolution of pathogen antigen variants under the antigen specific immune response. We assume that the antigen types of the pathogen can be indexed in one-dimensional space, and that a mutation can produce a new antigen variant that is one step distant from the parental type. We also assume that antibodies directed to a specific antigen can also neutralize similar antigen types with a decreased efficiency (cross-reactivity). The model reveals that the pattern of intra-host antigen evolution critically depends on the width of cross-reactivity. If the width of cross-reactivity is narrower than a certain threshold, antigen variants gradually evolve in antigen space as a travelling wave with a constant wave speed, and the total pathogen density approaches a constant. In contrast, if the width of cross-reactivity exceeds the threshold, the travelling wave loses stability and the distribution of antigen variants fluctuates both in time and in genotype space. In the latter case, the expected episodes after infection are a series of intermittent outbreaks of pathogen density, caused by distantly separated antigen types. The implication of the model to intra-host evolution of equine infectious anaemia virus and human immunodeficiency virus is discussed.

Modeling and optimization of populations subject to time-dependent mutation

It has become clear that many organisms possess the ability to regulate their mutation rate in response to environmental conditions. So the question of finding an optimal mutation rate must be replaced by that of finding an optimal mutation schedule. We show that this task cannot be accomplished with standard population-dynamic models. We then develop a "hybrid" model for populations experiencing time-dependent mutation that treats population growth as deterministic but the time of first appearance of new variants as stochastic. We show that the hybrid model agrees well with a Monte Carlo simulation. From this model, we derive a deterministic approximation, a "threshold" model, that is similar to standard population dynamic models but differs in the initial rate of generation of new mutants. We use these techniques to model antibody affinity maturation by somatic hypermutation. We had previously shown that the optimal mutation schedule for the deterministic threshold model is phasic, with periods of mutation between intervals of mutation-free growth. To establish the validity of this schedule, we now show that the phasic schedule that optimizes the deterministic threshold model significantly improves upon the best constant-rate schedule for the hybrid and Monte Carlo models.