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.     

Integrating life history and cross-immunity into the evolutionary dynamics of pathogens

Models for the diversity and evolution of pathogens have branched into two main directions: the adaptive dynamics of quantitative life-history traits (notably virulence) and the maintenance and invasion of multiple, antigenically diverse strains that interact with the host's immune memory. In a first attempt to reconcile these two approaches, we developed a simple modelling framework where two strains of pathogens, defined by a pair of life-history traits (infectious period and infectivity), interfere through a given level of cross-immunity. We used whooping cough as a potential example, but the framework proposed here could be applied to other acute infectious diseases. Specifically, we analysed the effects of these parameters on the invasion dynamics of one strain into a population, where the second strain is endemic. Whereas the deterministic version of the model converges towards stable coexistence of the two strains in most cases, stochastic simulations showed that transient epidemic dynamics can cause the extinction of either strain. Thus ecological dynamics, modulated by the immune parameters, eventually determine the adaptive value of different pathogen genotypes. We advocate an integrative view of pathogen dynamics at the crossroads of immunology, epidemiology and evolution, as a way towards efficient control of infectious diseases.     

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.     

On the role of cross-immunity and vaccines on the survival of less fit flu-strains

A pathogen's route to survival involves various mechanisms including its ability to invade (host's susceptibility) and its reproductive success within an invaded host ("infectiousness"). The immunological history of an individual often plays an important role in reducing host susceptibility or it helps the host mount a faster immunological response de facto reducing infectiousness. The cross-immunity generated by prior infections to influenza A strains from the same subtype provide a significant example. The results of this paper are based on the analytical study of a two-strain epidemic model that incorporates host isolation (during primary infection) and cross-immunity to study the role of invasion mediated cross-immunity in a population where a precursor related strain (within the same subtype, i.e. H3N2, H1N1) has already become established. An uncertainty and sensitivity analysis is carried out on the ability of the invading strain to survive for given cross-immunity levels. Our findings indicate that it is possible to support coexistence even in the case when invading strains are "unfit", that is, when the basic reproduction number of the invading strain is less than one. However, such scenarios are possible only in the presence of isolation. That is, appropriate increments in isolation rates and weak cross-immunity can facilitate the survival of less fit strains. The development of "flu" vaccines that minimally enhance herd cross-immunity levels may, by increasing genotype diversity, help facilitate the generation and survival of novel strains.     

Theoretical considerations of cross-immunity, recombination and the evolution of new parasitic strains.

We explore the dynamics of multiple strains of a parasite in order to assess the conditions under which a novel strain, perhaps a mutant or migrant, may invade a population that already carries an endemic strain. Multiple strain dynamics can be modeled through coinfection or complete cross-immunity. We examine these three modes to discuss the relationships among cross-immunity, the basic reproductive rates of each strain, and the invasion of the new strain. Superinfection is more restrictive than coinfection in the proportion of parameters that allows invasion. The coinfection model is extended to allow haploid strains to undergo recombination within the host. We investigate the effects of recombination and cross-immunity on the invasion of new strains. Interestingly, although recombination is understood to generate diversity, it is not always advantageous.     

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.     

Cross-Immunity and Community Structure of a Multiple-Strain Pathogen in the Tick Vector

Many vector-borne pathogens consist of multiple strains that circulate in both the vertebrate host and the arthropod vector. Characterization of the community of pathogen strains in the arthropod vector is therefore important for understanding the epidemiology of mixed vector-borne infections. Borrelia afzelii and B. garinii are two species of tick-borne bacteria that cause Lyme disease in humans. These two sympatric pathogens use the same tick, Ixodes ricinus, but are adapted to different classes of vertebrate hosts. Both Borrelia species consist of multiple strains that are classified using the highly polymorphic ospC gene. Vertebrate cross-immunity against the OspC antigen is predicted to structure the community of multiple-strain Borrelia pathogens. Borrelia isolates were cultured from field-collected I. ricinus ticks over a period spanning 11 years. The Borrelia species of each isolate was identified using a reverse line blot (RLB) assay. Deep sequencing was used to characterize the ospC communities of 190 B. afzelii isolates and 193 B. garinii isolates. Infections with multiple ospC strains were common in ticks, but vertebrate cross-immunity did not influence the strain structure in the tick vector. The pattern of genetic variation at the ospC locus suggested that vertebrate cross-immunity exerts strong selection against intermediately divergent ospC alleles. Deep sequencing found that more than 50% of our isolates contained exotic ospC alleles derived from other Borrelia species. Two alternative explanations for these exotic ospC alleles are cryptic coinfections that were not detected by the RLB assay or horizontal transfer of the ospC gene between Borrelia species.     

Subthreshold and superthreshold coexistence of pathogen variants: the impact of host age-structure

It is well known that in the most general epidemic models with multiple pathogen variants a competitive exclusion principle is valid, such that the variant with the highest reproduction number eliminates the rest. Mechanisms such as super-infection, coinfection, and cross-immunity can lead to pathogen polymorphism where multiple strains coexist. It is also known that variability of infectivity with host age can destabilize the endemic equilibrium and cause oscillations. In this article we show that the hosts' chronological age can itself lead to coexistence of microparasites in the most basic model where competitive exclusion will occur without the age structure. Moreover, the host age-structure leads to multiple subthreshold dominance equilibria, and both weakly and strongly subthreshold coexistence. We find that the two pathogens cannot cooperate to persist subthreshold if neither one of them can persist subthreshold by itself. If, however, one of them can persist subthreshold by itself, it can cause the two pathogens to coexist in a strongly subthreshold equilibrium. The second strain that persists subthreshold through the mediation of the first always has a lower virulence. Our results show that age structure in infectivity can permit the coexistence of competing pathogens when the incidence is of proportionate mixing type (frequency-dependent transmission) and at least one of the strains is virulent.     

Influenza emergence in the face of evolutionary constraints

Different influenza subtypes can evolve at very different rates, but the causes are not well understood. In this paper, we explore whether differences in transmissibility between subtypes can play a role if there are fitness constraints on antigenic evolution. We investigate the problem using a mathematical model that separates the interaction of strains through cross-immunity from the process of emergence for new antigenic variants. Evolutionary constraints are also included with antigenic mutation incurring a fitness cost. We show that the transmissibility of a strain can become disproportionately important in dictating the rate of antigenic drift: strains that spread only slightly more easily can have a much higher rate of emergence. Further, we see that the effect continues when vaccination is considered; a small increase in the rate of transmission can make it much harder to control the frequency at which new strains emerge. Our results not only highlight the importance of considering both transmission and fitness constraints when modelling influenza evolution, but may also help in understanding the differences between the emergence of H1N1 and H3N2 subtypes.     

Estimation of Cross-Immunity Between Drifted Strains of Influenza A/H3N2

To determine the cross-immunity between influenza strains, we design a novel statistical method, which uses a theoretical model and clinical data on attack rates and vaccine efficacy among school children for two seasons after the 1968 A/H3N2 influenza pandemic. This model incorporates the distribution of susceptibility and the dependence of cross-immunity on the antigenic distance of drifted strains. We find that the cross-immunity between an influenza strain and the mutant that causes the next epidemic is 88%. Our method also gives estimates of the vaccine protection against the vaccinating strain, and the basic reproduction number of the 1968 pandemic influenza.     

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity

We investigate the population dynamics of host-pathogen systems in which the pathogen has a potentially arbitrary number of antigenically distinct strains interacting via cross-immunity. The interior equilibrium configuration of the symmetric multiple strain SIR model with cross-immunity is characterized. We develop an efficient iterative method for numerically solving the equilibrium equation together with a number of informative analytical approximations to the full solution. Equilibrium properties are studied as a function of the number of strains, reproduction number, infectious period, and cross immunity profile. We establish that the prevalence in the system increases monotonically with the number of strains and the reduction in cross immunity. Moreover, we demonstrate the existence of a phase transition separating high prevalence and low prevalence parameter regions, with the critical point being defined by R₀1, where is the level of cross-immunity and R₀ is the reproduction number. Above the threshold, prevalence saturates with increasing numbers of strains as a result of the inclusion of prohibition of co-infection in the model. Below the threshold, prevalence saturates much more rapidly as the number of strains increases - indicating that when cross-protection is sufficiently intense, the selective advantage for a pathogen to increase its diversity is substantially less than in the threshold region. Similarly, there is limited benefit to increased transmissibility (or decreased cross-immunity) both for the high and low diversity pathogen systems compared with systems at the threshold R₀1 where small increase in transmissibility can result in significant increase in prevalence.     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996     

Host resistance and pathogen aggressiveness are key determinants of coinfection in the wild

Coinfection, whereby the same host is infected by more than one pathogen strain, may favor faster host exploitation rates as strains compete for the same limited resources. Hence, coinfection is expected to have major consequences for pathogen evolution, virulence, and epidemiology. Theory predicts genetic variation in host resistance and pathogen infectivity to play a key role in how coinfections are formed. The limited number of studies available has demonstrated coinfection to be a common phenomenon, but little is known about how coinfection varies in space, and what its determinants are. Our aim is to understand how variation in host resistance and pathogen infectivity and aggressiveness contribute to how coinfections are formed in the interaction between fungal pathogen Podosphaera plantaginis and Plantago lanceolata. Our phenotyping study reveals that more aggressive strains are more likely to form coinfections than less aggressive strains in the natural populations. In the natural populations most of the variation in coinfection is found at the individual plant level, and results from a common garden study confirm the prevalence of coinfection to vary significantly among host genotypes. These results show that genetic variation in both the host and pathogen populations are key determinants of coinfection in the wild.     

Evolutionary implications for interactions between multiple strains of host and parasite

The interaction between multiple parasite strains within different host types may influence the evolutionary trajectories of parasites. In this article, we formulate a deterministic model with two strains of parasites and two host types in order to investigate how heterogeneities in parasite virulence and host life-history may affect the persistence and spread of diseases in natural systems. We compute the reproductive number of strain i (R(i)) independently, as well as the (conditional) "invasion" reproductive number for strains i (R(i)(j), j not equal i) when strain j is at a positive equilibrium. We show that the disease-free equilibrium is locally asymptotically stable if R(i)<1 for both strains and is unstable if R(i)>1 for one stain. We establish the criterion R(i)(j)>1 for strain i to invade strain j. Subthreshold coexistence driven by coinfection is possible even when R(i) of one strain is below 1. We identify conditions that determine the evolution of parasite specialism or generalism based on the life-history strategies employed by hosts, and investigate how host strains may influence parasite persistence.     

The immune response of cattle to Babesia bovis (syn. B. argentina). Studies on the nature and specificity of protection.

Studies of the immune response to Babesia bovis (syn. B. argentina) in Bos taurus cattle, using the passive transfer of serum from immune animals, indicated that an effector mechanism was mediated by antibodies which reacted with the parasitized erythrocytes. During removal from the peripheral blood, the parasites did not show reduced viability on subinoculation into other non-infected animals, and thus were not dead or irreversibly damaged at this time. It was concluded that opsonization of infected erythrocytes was probably the basis of protection by the system. There was some evidence that minor variation of the protective antigen(s) occurred within strains of the parasite but this had little effect on the efficiency of the host's immune response. However, there was no cross-protection between the antibodies against different strains. These interstrain differences in antibody specificity were reconciled with earlier observations that cross-immunity commonly occurs between different strains in infected animals. It was concluded that the mechanism of cross-immunity relied on priming of the host's immune system by the protective antigen(s) of the strain so that a secondary response against the heterologous strain occurred soon after challenge.     

Invasion and Exclusion among Coexisting Pseudomonas syringae Strains on Leaves

The invasion and exclusion abilities of coexisting Pseudomonas syringae strains were quantified on leaves. Twenty-nine P. syringae strains were inoculated onto plants in 107 pairwise combinations. All pairs were duplicated so that each strain was inoculated both first as an antagonist strain (day 0) and second as a challenge strain (day 3). The population size of each strain in a mixture was quantified on day 6 following incubation under moist conditions. For P. syringae strains, the presence of an established population often significantly reduced the growth of subsequently arriving challenge strains on the leaf surface. Invasion and exclusion abilities, quantified by contrasting population sizes of challenge strains in the presence and in the absence of another strain, varied significantly among P. syringae strains and were partly a function of the particular strain pair. The population size of a strain when present alone on a leaf was not predictive of invasion or exclusion ability. Successful invaders were significantly less likely to exclude challenge populations than were nonsuccessful invaders. Population sizes of successful excluders were negatively correlated with population sizes of coexisting challenge strains, while population sizes of successful invaders were positively correlated with those of coexisting antagonist strains. The patterns of interaction among coexisting strains suggest mechanisms for successful invasion and exclusion among P. syringae strains on leaves.     

The impact of cross-immunity, mutation and stochastic extinction on pathogen diversity

We examine the dynamics of antigenically diverse infectious agents using a mathematical model describing the transmission dynamics of arbitrary numbers of pathogen strains, interacting via cross-immunity, and in the presence of mutations generating new strains and stochastic extinctions of existing ones. Equilibrium dynamics fall into three classes depending on cross-immunity, transmissibility and host population size: systems where global extinction is likely, stable single-strain persistence, and multiple-strain persistence with stable diversity. Where multi-strain dynamics are stable, a diversity threshold region separates a low-prevalence, low-diversity region of parameter space from a high-diversity, high-prevalence region. The location of the threshold region is determined by the reproduction number of the pathogen and the intensity of cross-immunity, with the sharpness of the transition being determined by the manner in which immunity accrues with repeated infections. Host population size and cross-immunity are found to be the most decisive factors in determining pathogen diversity. While the model framework developed is simplified, we show that it can capture essential aspects of the complex evolutionary dynamics of pathogens such as influenza.     

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.