Influenza drift and epidemic size: the race between generating and escaping immunity

Influenza in humans is characterised by strongly annual dynamics and antigenic evolution leading to partial escape from prior host immunity. The variability of new epidemic strains depends on the amount of virus currently circulating. In this paper, the amount of antigenic variation produced each year is dependent on the epidemic size. Our model reduces to a one-dimensional map and a full mathematical analysis is presented. This simple system suggests some basic principles which may be more generally applicable. In particular, for diseases with antigenic drift, vaccination may be doubly beneficial. Not only does it protect the population through classical herd immunity, but the overall case reduction reduces the chance of new variants being produced; hence, subsequent epidemics may be milder as a result of this positive feedback. Also, a disease with a high innate rate of antigenic variation will always be able to invade a susceptible population, whereas a disease with less potential for variation may require several introduction events to become endemic.     

Antigenic distance and cross-immunity, invasibility and coexistence of pathogen strains in an epidemiological model with discrete antigenic space

In models of pathogen interaction and evolution discrete genotypes in the form of bit strings may be mapped to points in a discrete phenotype space based on similarity in antigenic structure. Cross-immunity between strains, that is the reduction in susceptibility to strain A conferred to a host by infection with strain B, can then be defined for pairs of points in the antigenic space by a specified function. Analysis of an SIR type model shows that, if two strains are at equilibrium, the shape of the cross-immunity function has a strong influence on the invasion and coexistence of a third strain and, consequently, the expected evolutionary pathway. A function that is constant except for discontinuities at the end points is expected to result in the accumulation of diversity until a pair of discordant strains occurs that can, depending on parameter values, exclude all other strains. For a function of the form f(h)=h^q, where h is the antigenic distance between two strains, invasion and coexistence is always possible if q≤1 and little antigenic structure is expected in the pathogen population. However, if q>1 invasion and coexistence may be impossible, depending on parameter values, and the pathogen population is expected to show significant antigenic structuring. In addition to illuminating the role of cross-immunity in pathogen evolution, this analysis indicates that the choice of cross-immunity function, the representation of immunity acquired from multiple previous infections and the number of elements used to characterize the antigenic space must be carefully considered in the development and interpretation of more sophisticated models of pathogen dynamics and evolution.     

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.     

Heterogeneity in antibody range and the antigenic drift of influenza A viruses

In this paper we explore the consequences of a heterogeneous immune response in individuals on the evolution of a rapidly mutating virus. We show that several features of the incidence and phylogenetic patterns typical of influenza A may be understood in this framework. In our model, limited diversity and rapid drift of the circulating viral strains result from the interplay of two interacting subpopulations with different types of immune response, narrow or broad, upon infection. The subpopulation with the narrow immune response acts as a reservoir where consecutive mutations escape immunity and can persist. Strains with a number of accumulated mutations escape immunity in the other subpopulation as well, causing larger epidemic peaks in the whole population, and reducing strain diversity. Overall, our model produces a modulation of epidemic peak heights and patterns of antigenic drift consistent with reported observations, suggesting an underlying mechanism for the evolutionary epidemiology of influenza, in particular, and other infectious diseases, more generally.     

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.     

Phenotypic differences in viral immune escape explained by linking within-host dynamics to host-population immunity

Viruses that do not cause life-long immunity persist by evolving rapidly in response to prevailing host immunity. The immune-escape mutants emerge frequently, displacing or co-circulating with native strains even though mutations conferring immune evasion are often detrimental to viral replication. The epidemiological dynamics of immune-escape in acute-infection viruses with high transmissibility have been interpreted mainly through immunity dynamics at the host population level, despite the fact that immune-escape evolution involves dynamical processes that feedback across the within- and between-host scales. To address this gap, we use a nested model of within- and between-host infection dynamics to examine how the interaction of viral replication rate and cross-immunity imprint host population immunity, which in turn determines viral immune escape. Our explicit consideration of direct and immune-mediated competitive interactions between strains within-hosts revealed three insights pertaining to risk and control of viral immune-escape: (1) replication rate and immune-stimulation deficiencies (i.e., original antigenic sin) act synergistically to increase immune escape, (2) immune-escape mutants with replication deficiencies relative to their wildtype progenitor are most successful under moderate cross-immunity and frequent re-infections, and (3) the immunity profile along short host-transmission chains (local host-network structure) is a key determinant of immune escape.     

Localized contacts between hosts reduce pathogen diversity

We investigate the dynamics of a simple epidemiological model for the invasion by a pathogen strain of a population where another strain circulates. We assume that reinfection by the same strain is possible but occurs at a reduced rate due to acquired immunity. The rate of reinfection by a distinct strain is also reduced due to cross-immunity. Individual based simulations of this model on a 'small-world' network show that the proportion of local contacts in the host contact network structure significantly affects the outcome of such an invasion, and as a consequence will affect the patterns of pathogen evolution. In particular, hosts interacting through a 'small-world' network of contacts support lower prevalence of infection than well-mixed populations, and the region in parameter space for which an invading strain can become endemic and coexist with the circulating strain is smaller, reducing the potential to accommodate pathogen diversity. We discuss the underlying mechanisms for the reported effects, and we propose an effective mean-field model to account for the contact structure of the host population in 'small-world' networks.     

Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity

We study the epidemiology of a viral disease with dose-dependent replication and transmission by nesting a differential-equation model of the within-host viral dynamics inside a between-host epidemiological model. We use two complementary approaches for nesting the models: an agent-based (AB) simulation and a mean-field approximation called the growth-matrix (GM) model. We find that although infection rates and predicted case loads are somewhat different between the AB and GM models, several epidemiological parameters, e.g. mean immunity in the population and mean dose received, behave similarly across the methods. Further, through a comparison of our dose-dependent replication model against two control models that uncouple dose-dependent replication from transmission, we find that host immunity in a population after an epidemic is qualitatively different than when transmission depends on time-varying viral abundances within hosts. These results show that within-host dynamics and viral dose should not be neglected in epidemiological models, and that the simpler GM approach to model nesting provides a reasonable tradeoff between model complexity and accuracy of results.     

On the determinants of population structure in antigenically diverse pathogens

Many pathogens exhibit antigenic diversity and elicit strain-specific immune responses. This potential for cross-immunity structure in the host resource motivates the development of mathematical models, stressing competition for susceptible hosts in driving pathogen population dynamics and genetics. Here we establish that certain model formulations exhibit characteristics of prototype pattern-forming systems, with pathogen population structure emerging as three possible patterns: (i) incidence is steady and homogeneous; (ii) incidence is steady but heterogeneous; and (iii) incidence shows oscillatory dynamics, with travelling waves in strain-space. Results are robust to strain number, but sensitive to the mechanism of cumulative immunity.     

Investigation of the demographic and selective forces shaping the nucleotide diversity of genes involved in nod factor signaling in Medicago truncatula

Symbiotic nitrogen-fixing rhizobia are able to trigger root deformation in their Fabaceae host plants, allowing their intracellular accommodation. They do so by delivering molecules called Nod factors. We analyzed the patterns of nucleotide polymorphism of five genes controlling early Nod factor perception and signaling in the Fabaceae Medicago truncatula to understand the selective forces shaping the evolution of these genes. We used 30 M. truncatula genotypes sampled in a genetically homogeneous region of the species distribution range. We first sequenced 24 independent loci and detected a genomewide departure from the hypothesis of neutrality and demographic equilibrium that suggests a population expansion. These data were used to estimate parameters of a simple demographic model incorporating population expansion. The selective neutrality of genes controlling Nod factor perception was then examined using a combination of two complementary neutrality tests, Tajima's D and Fay and Wu's standardized H. The joint distribution of D and H expected under neutrality was obtained under the fitted population expansion model. Only the gene DMI1, which is expected to regulate the downstream signal, shows a pattern consistent with a putative selective event. In contrast, the receptor-encoding genes NFP and NORK show no significant signatures of selection. Among the genes that we analyzed, only DMI1 should be viewed as a candidate for adaptation in the recent history of M. truncatula.     

The Roles of Competition and Mutation in Shaping Antigenic and Genetic Diversity in Influenza

Influenza A (H3N2) offers a well-studied, yet not fully understood, disease in terms of the interactions between pathogen population dynamics, epidemiology and genetics. A major open question is why the virus population is globally dominated by a single and very recently diverged (2–8 years) lineage. Classically, this has been modeled by limiting the generation of new successful antigenic variants, such that only a small subset of progeny acquire the necessary mutations to evade host immunity. An alternative approach was recently suggested by Recker et al. in which a limited number of antigenic variants are continuously generated, but most of these are suppressed by pre-existing host population immunity. Here we develop a framework spanning the regimes described above to explore the impact of rates of mutation and levels of competition on phylodynamic patterns. We find that the evolutionary dynamics of the subtype H3N2 influenza is most easily generated within this framework when it is mutation limited as well as being under strong immune selection at a number of epitope regions of limited diversity.     

The dynamics of cocirculating influenza strains conferring partial cross-immunity

 We develop a model that describes the dynamics of a finite number of strains that confer partial cross-protection among strains. The immunity structure of the host population is captured by an index-set notation where the index specifies the set of strains to which the host has been exposed. This notation allows us to derive threshold conditions for the invasion of a new strain and to show the existence of an endemic multi-strain equilibrium in a special case. The dynamics of systems consisting of more than two strains can exhibit sustained oscillations caused by an overshoot in the immunity to a specific strain if cross-protection is sufficiently strong. Received 15 January 1996; received in revised form 24 June 1996     

Spatially structured superinfection and the evolution of disease virulence

When pathogen strains differing in virulence compete for hosts, spatial structuring of disease transmission can govern both evolved levels of virulence and patterns in strain coexistence. We develop a spatially detailed model of superinfection, a form of contest competition between pathogen strains; the probability of superinfection depends explicitly on the difference in levels of virulence. We apply methods of adaptive dynamics to address the interplay of spatial dynamics and evolution. The mean-field approximation predicts evolution to criticality; any small increase in virulence capable of dynamical persistence is favored. Both pair approximation and simulation of the detailed model indicate that spatial structure constrains disease virulence. Increased spatial clustering reduces the maximal virulence capable of single-strain persistence and, more importantly, reduces the convergent-stable virulence level under strain competition. The spatially detailed model predicts that increasing the probability of superinfection, for given difference in virulence, increases the likelihood of between-strain coexistence. When strains differing in virulence can coexist ecologically, our results may suggest policies for managing diseases with localized transmission. Comparing equilibrium densities from the pair approximation, we find that introducing a more virulent strain into a host population infected by a less virulent strain can sometimes reduce total host mortality and increase global host density.     

Dynamic Correlation between Intrahost HIV-1 Quasispecies Evolution and Disease Progression

Quantifying the dynamics of intrahost HIV-1 sequence evolution is one means of uncovering information about the interaction between HIV-1 and the host immune system. In the chronic phase of infection, common dynamics of sequence divergence and diversity have been reported. We developed an HIV-1 sequence evolution model that simulated the effects of mutation and fitness of sequence variants. The amount of evolution was described by the distance from the founder strain, and fitness was described by the number of offspring a parent sequence produces. Analysis of the model suggested that the previously observed saturation of divergence and decrease of diversity in later stages of infection can be explained by a decrease in the proportion of offspring that are mutants as the distance from the founder strain increases rather than due to an increase of viral fitness. The prediction of the model was examined by performing phylogenetic analysis to estimate the change in the rate of evolution during infection. In agreement with our modeling, in 13 out of 15 patients (followed for 3–12 years) we found that the rate of intrahost HIV-1 evolution was not constant but rather slowed down at a rate correlated with the rate of CD4+ T-cell decline. The correlation between the dynamics of the evolutionary rate and the rate of CD4+ T-cell decline, coupled with our HIV-1 sequence evolution model, explains previously conflicting observations of the relationships between the rate of HIV-1 quasispecies evolution and disease progression.     

Uncovering epidemiological dynamics in heterogeneous host populations using phylogenetic methods

Host population structure has a major influence on epidemiological dynamics. However, in particular for sexually transmitted diseases, quantitative data on population contact structure are hard to obtain. Here, we introduce a new method that quantifies host population structure based on phylogenetic trees, which are obtained from pathogen genetic sequence data. Our method is based on a maximum-likelihood framework and uses a multi-type branching process, under which each host is assigned to a type (subpopulation). In a simulation study, we show that our method produces accurate parameter estimates for phylogenetic trees in which each tip is assigned to a type, as well for phylogenetic trees in which the type of the tip is unknown. We apply the method to a Latvian HIV-1 dataset, quantifying the impact of the intravenous drug user epidemic on the heterosexual epidemic (known tip states), and identifying superspreader dynamics within the men-having-sex-with-men epidemic (unknown tip states).     

Pathogen adaptation under imperfect vaccination: implications for pertussis

Mass vaccination campaigns have drastically reduced the burden of infectious diseases. Unfortunately, in recent years several infectious diseases have re-emerged. Pertussis poses a well-known example. Inspired by pertussis, we study, by means of an epidemic model, the population and evolutionary dynamics of a pathogen population under the pressure of vaccination. A distinction is made between infection in immunologically naive individuals (primary infection) and infection in individuals whose immune system has been primed by vaccination or infection (secondary infection). The results show that (i) vaccination with an imperfect vaccine may not succeed in reducing the infection pressure if the transmissibility of secondary infections is higher than that of primary infections; (ii) pathogen strains that are able to evade the immunity induced by vaccination can only spread if escape mutants incur no or only a modest fitness cost and (iii) the direction of evolution depends crucially on the distribution of the different types of susceptibles in the population. We discuss the implications of these results for the design and use of vaccines that provide temporary immunity.     

Immune selection suppresses the emergence of drug resistance in malaria parasites but facilitates its spread

Although drug resistance in Plasmodium falciparum typically evolves in regions of low transmission, resistance spreads readily following introduction to regions with a heavier disease burden. This suggests that the origin and the spread of resistance are governed by different processes, and that high transmission intensity specifically impedes the origin. Factors associated with high transmission, such as highly immune hosts and competition within genetically diverse infections, are associated with suppression of resistant lineages within hosts. However, interactions between these factors have rarely been investigated and the specific relationship between adaptive immunity and selection for resistance has not been explored. Here, we developed a multiscale, agent-based model of Plasmodium parasites, hosts, and vectors to examine how host and parasite dynamics shape the evolution of resistance in populations with different transmission intensities. We found that selection for antigenic novelty (“immune selection”) suppressed the evolution of resistance in high transmission settings. We show that high levels of population immunity increased the strength of immune selection relative to selection for resistance. As a result, immune selection delayed the evolution of resistance in high transmission populations by allowing novel, sensitive lineages to remain in circulation at the expense of the spread of a resistant lineage.In contrast, in low transmission settings, we observed that resistant strains were able to sweep to high population prevalence without interference. Additionally, we found that the relationship between immune selection and resistance changed when resistance was widespread. Once resistance was common enough to be found on many antigenic backgrounds, immune selection stably maintained resistant parasites in the population by allowing them to proliferate, even in untreated hosts, when resistance was linked to a novel epitope. Our results suggest that immune selection plays a role in the global pattern of resistance evolution.     

Acquired immunity and stochasticity in epidemic intervals impede the evolution of host disease resistance

Disease can generate intense selection pressure on host populations, but here we show that acquired immunity in a population subject to repeated disease outbreaks can impede the evolution of genetic disease resistance by maintaining susceptible genotypes in the population. Interference between the life-history schedule of a species and periodicity of the disease has unintuitive effects on selection intensity, and stochasticity in outbreak period further reduces the rate of spread of disease-resistance alleles. A general age-structured population genetic model was developed and parameterized using empirical data for phocine distemper virus (PDV) epizootics in harbor seals. Scenarios with acquired immunity had lower levels of epizootic mortality compared with scenarios without acquired immunity for the first PDV outbreaks, but this pattern was reversed after about five disease cycles. Without acquired immunity, evolution of disease resistance was more rapid, and long-term population size variation is efficiently dampened. Acquired immunity has the potential to significantly influence rapid evolutionary dynamics of a host population in response to age-structured disease selection and to alter predicted selection intensities compared with epidemiological models that do not consider such feedback. This may have important implications for evolutionary population dynamics in a range of human, agricultural, and wildlife disease settings.     

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.