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Chaos, Persistence, and Evolution of Strain Structure in Antigenically Diverse Infectious Agents

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Science  08 May 1998:
Vol. 280, Issue 5365, pp. 912-915
DOI: 10.1126/science.280.5365.912

Abstract

The effects of selection by host immune responses on transmission dynamics was analyzed in a broad class of antigenically diverse pathogens. Strong selection can cause pathogen populations to stably segregate into discrete strains with nonoverlapping antigenic repertoires. However, over a wide range of intermediate levels of selection, strain structure is unstable, varying in a manner that is either cyclical or chaotic. These results have implications for the interpretation of longitudinal epidemiological data on strain or serotype abundance, design of surveillance strategies, and the assessment of multivalent vaccine trials.

New epidemics of an infectious disease can be triggered by the evolution of a novel antigenic type or strain that evades the acquired immunity within the host population created by its predecessors. The most studied case is the influenza virus, where major shifts in the structure of surface antigens can often trigger worldwide pandemics of the novel variant (1, 2). The antigens that are most likely to exhibit diversity are those under strong selection by host immune responses. These polymorphic antigens may be ranked by the degree to which the associated immune response reduces the reproductive or transmission success of the pathogen. We demonstrated previously that those antigens that elicit the strongest immune response, which in turn have the strongest impact on transmission success, may be organized by immune selection acting within the host population into sets of nonoverlapping variants (3). For example, in the case of two antigens each encoded by a distinct locus with two alleles, namely a andb at one locus and c and d at the other, four genotypes exist (ac, ad,bc, and bd). One set of nonoverlapping variants is ac and bd (a discordant set) and the other isbc and ad. The pathogen population may exhibit a discrete strain structure where one set of nonoverlapping variants exists at much greater frequency than the other. For this pattern to emerge and be stable over time, the intensity of acquired immunity to a specific variant antigen (encoded by a given allele) within the host population must reduce considerably the transmission success or fitness of all subsequent infections by genotypes possessing that allele. Pathogen populations may therefore be categorized into discrete “strains” or serotypes according to the genetic loci that encode antigens eliciting immune responses with the greatest effect on the transmission success of the infectious agent (3).

Here, we show that pathogens that possess antigens that do not elicit an immune response that is strong enough to induce a discrete stable strain structure may exist as a set of strains that exhibit cyclical or chaotic fluctuations in frequency over time. They may still be organized by immune selection into discrete groups of variants, where all the members in a given group have different alleles at every locus, but the dominancy of the group, relative to that of other groups, may fluctuate widely over time, either cyclically or chaotically. Pathogen antigens that elicit immune responses that have little effect on transmission success will not be organized to express a discrete nonoverlapping strain structure. In this case, all possible allele combinations will be maintained at abundances commensurate with their individual transmission success or fitness.

We define the conditions under which each of these three outcomes arises, in terms of the biological characteristics of both the pathogen and the immune response of the host to the various antigens of the infectious agent, using a model in which a pathogen strain is defined by the m alleles that exist at each of n loci. We take no account of other important biological complications such as mutation, seasonality in transmission, time delays between the acquisition of infection and infectiousness to susceptible hosts, or genetic diversity in the host population, influencing the immune responses to particular antigenic variants. These exclusions are deliberate, because we wish to assess the impact of selection imposed by acquired immunity in the host population on temporal trends in the frequencies of different variants and the evolution of the associated strain structure in the pathogen population.

Strains that do not share any alleles (hereafter referred to as a discordant set) are assumed not to interfere with each others' transmission success or fitness as mediated by host immune responses. The degree to which infection with a given strain limits the ability of another strain that shares any alleles to infect the same host is defined by a cross-protection or cross-immunity parameter γ. If γ = 0, then the strains do not induce cross-protective responses, whereas if γ = 1, then there is complete cross-protection. It is also assumed that immunity to a given strain i does not prevent infection by any other strain, but only reduces the probability of transmission of a nondiscordant strain j by a factor (1 – γ). This implies that the process of the acquisition of immunity to any one strain is independent to that of its acquisition to any other strain. These simple but realistic biological assumptions can be expressed by a system of differential equations representing the changes in the abundances of each of the m n strains over time in a genetically homogeneous host population (4).

The model system generated three distinct dynamical behaviors (Fig.1). (i) When the degree of cross-immunity is below a threshold value, γL, all the strains coexist in the host population with stable abundance, and no strain structure (NSS) is apparent. Increasing the degree of cross-protection in this dynamical domain acts to decrease strain abundance because of increased immunological interference between strains. (ii) When the degree of cross-immunity exceeds an upper threshold, γU, one discordant set of strains dominates in terms of their prevalence. This situation represents the presence of stable discrete strain structure (DSS). (iii) For γL < γ < γU, no stable strain structure occurs and the relative proportions of the different strains exhibit very complex and often chaotic dynamics with marked fluctuations over time. This is referred to as cyclical or chaotic strain structure (CSS).

Figure 1

Strain structure type is shown as a function of the degree of cross-protection (γ) and the ratio of host life-span to pathogen infectious period (σ/μ) for pathogens with (A) two antigenically active loci, each with two alleles; (B) three loci, each with three alleles; and (C) four loci, each with four alleles (R 0 = 4 for all strains).

Figure 1 plots the parameter regions for which these different dynamical behaviors and associated strain structures pertain. The upper threshold, γU, is given by γU = 1 – 1/(2R 0) if all strains have identical transmission success as defined by the case reproductive rateR 0 [the average number of secondary infections generated by one primary infection in a susceptible host population (5)]. The lower threshold value γL is determined by the ratio σ/μ [where 1/μ is host life expectancy and 1/σ is the average duration of infection (equal to infectiousness)], R 0, and the number of antigen loci and alleles. For pathogens in developed countries, with human life expectancies of around 70 years and infectious periods of between 4 days to 1 year (1/σ = 0.01 to 1), σ/μ is between 70 to 7000. The large size of the parameter space generating cyclical or chaotic behavior (Fig. 1), for relevant parameter assignments for the duration of infection and host life expectancy, implies that such complex dynamics may be the norm for many antigenically variable infectious agents that induce moderate cross-protective immune responses in the human host.

Figure 2 portrays the wide range of complex dynamical behaviors generated by the model in the CSS parameter domain. Initially the strain prevalences follow simple limit cycles (Fig. 2B), but as the value of γ is increased, chaotic intermittency is seen (Fig. 2C), followed by increasingly large-amplitude chaos (Fig. 2, D and E). Several important trends are apparent from extensive numerical studies. First, as γ increases in value, the amplitude and period of the epidemic cycles rise. Second, when γ is close to the two boundaries of the CSS region, complete coherence is seen between the epidemics of strains within discordant sets (that is, the abundances of all strains within a set are identical). This is a direct result of competitive exclusion between genotypes that share alleles where the competition is created by herd immunity. Third, for a large band of cross-protection values in the center of the CSS region, this coherence begins to break down (Fig. 2D), and the trajectories of the prevalence of individual members of discordant sets can be distinguished. The exact dynamical mechanism driving this symmetry breakdown remains unclear, but the behavior appears to be associated with a crossover effect as the system moves from the low-γ region of short-period cycles (a few years) to the large-γ region where long-period (many years) chaos dominates. In the crossover region, both time scales are apparent, with “generation” cycles (with period ∼1/μ = host life expectancy) being modulated by shorter period oscillations, the periods of which are determined by the average infectious period (1/σ) and the transmission success of the pathogen (R 0). In other words, the long periods are determined by host demography and the short periods by the variables that determine the typical course of infection in the host and the transmission dynamics of the infectious agent. As the duration of the infectious period is decreased, the period of the low-γ regime limit cycles decreases, and chaotic behavior increasingly dominates the entire CSS region. Similarly, discordant set decoherence increases as the number of antigenically active loci and alleles increase. Decoherence also increases markedly as the infectious period decreases.

Figure 2

Long-term population dynamics of pathogens with three loci, each with two alleles. R 0 = 4 for all strains, σ = 10 years−1, and μ = 0.02 years−1. (A) Bifurcation diagram shows the location of local maxima of w amx (the fraction of the population exposed to strain amx or any strain sharing alleles with amx) as a function of the degree of cross-protection. (B through D) Time-series of the prevalences, z i, of each strain in the population: (B) for γ = 0.58 (nonchaotic limit cycle), (C) for γ = 0.62 (intermittent chaotic episodes), (D) for γ = 0.72 (chaotic short-period cycles with decoherence between discordant set members), and (E) γ = 0.85 (large-amplitude chaos).

Other theoretical studies indicate that complex dynamics are likely to be common for any pathogen populations that are organized in such a manner that there is cross-immunity within certain strain subsets and none between these subsets. For instance, Andreasenet al. (2) modeled influenza by assuming that interstrain cross-immunity exists only between nearest neighbors in a one-dimensional phenotypic space (that is, immunity to straini affects the transmission success of strains i– 1 and i + 1) and demonstrated that this form of population structuring can also generate stable limit cycles in systems with four or more strains. Other population-structuring mechanisms may also give interaction matrices that generate complex nonlinear dynamics, but it is particularly relevant that such a fundamental biological property of pathogens as the sharing of antigen variants yields this structure.

A key question arising from these analyses is whether we can expect to see deterministic chaos in epidemiological data for common viral, bacterial, and protozoan antigenically variable pathogens in human communities. Although stochastic effects in small populations may interfere with the persistence of long-period cycles (6), we may expect to see the following broad patterns. For polymorphic antigens that elicit weak immune responses, the abundances of the different strains (as defined by different combinations of alleles) will be determined by their respective transmission successes or fitnesses. In the cyclical or chaotic region, where the degree of cross-protection associated with the antigen is moderate, irregular epidemic cycles will be observed with average periods set by the mean duration of infectiousness in the host (short-period cycles of a few years for infectious periods of a few days to very long periods of many years for infectious periods approaching a month or longer). Concomitantly, there will be a high degree of correlation in the prevalence or incidence time-series of strains within the same discordant set and a low degree of correlation between strains or serotypes in different discordant sets. For antigens that elicit very strongly protective immune responses, the correlation between strains within the same discordant set will be extremely high, manifesting as DSS.

These theoretical predictions argue for considerable caution in the interpretation of observed patterns in longitudinal epidemiological data sets that are stratified by strain type or serotype based on molecular or immunological taxonomic characteristics. Although the limited time-series data currently available show significant fluctuations in strain incidences for a variety of pathogens (Fig.3), better quality, longer term data will be required to accurately assess the precise strain structure of a given antigenically variable pathogen population. Essential to this process is the cloning of the relevant antigen genes; for example, the recent cloning of the Plasmodium falciparum gene encoding PfEMP-1 (7-9) is crucial to the validation of the hypothesis that PfEMP-1 is the antigen that structures P. falciparum into discrete strains (10). The analysis we present suggests that it is also worth monitoring other P. falciparum antigens such as the merozoite surface proteins (MSP), which may not have as significant a role in protective immunity as PfEMP-1, because these may be more likely to exhibit cyclical or chaotic fluctuations. A recent study in Senegal (11) involving the typing of successive clinical malaria isolates during a 4-month period of intense transmission shows a large degree of genetic diversity in MSP1 and MSP2 combinations. However, long-term large-scale studies are required to elucidate the precise dynamics of these allele combinations. For bacteria, such as Neisseria meningitidis, large-scale nucleotide sequencing studies of genes encoding surface antigens are essential to understand the population structure of these organisms (12, 13) and, in particular, to elucidate whether the strong association observed between VR1 and VR2 epitopes of the PorA protein (2) forms the basis for the fluctuations in lineages of N. meningitidis (14). Serotypes of group A streptococci also exhibit rapid fluctuations (Fig. 3) (15). These serotypes are defined by an antiphagocytic cell-surface molecule known as M protein, which appears to occur in nonoverlapping combinations with another variable element known as the serum opacity factor (16), and thus, may exemplify a two-locus multiallele system in a state of CSS.

Figure 3

Examples of dynamical changes in serotype frequencies of two major human pathogens: (A) influenza cases in the United States from 1983–94 (20) and (B) group A streptococcal infections in Minnesota, United States, from 1965–1967 (15).

Such information on pathogen population structure is crucial in many different contexts, including the assessment of vaccine trials where the candidate vaccine contains only a subset of the antigens expressed by the pathogen population (17) such as those currently planned for N. meningitidis (18) andStreptococcus pneumoniae (19). Attributing observed changes in strain structure following mass immunization to the intervention may be problematic, given the complex nonlinear dynamics suggested by our analyses. The intricate behavior of these multistrain systems is a consequence of the selective pressures imposed on the pathogen population by the profile of herd immunity in the host population. This profile is, in turn, conditioned by the prevailing antigenic structure of the pathogen population. It is the subtle interplay between these two factors that leads to the unstable evolutionary dynamics we describe.

  • * To whom correspondence should be addressed. E-mail: sunetra.gupta{at}zoology.ox.ac.uk

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