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

A speed limit on serial strain replacement from original antigenic sin

Many pathogens evolve to escape immunity, yet it remains difficult to predict whether immune pressure will lead to diversification, serial replacement of one variant by another, or more complex patterns. Pathogen strain dynamics are mediated by cross-protective immunity, whereby exposure to one strain partially protects against infection by antigenically diverged strains. There is growing evidence that this protection is influenced by early exposures, a phenomenon referred to as original antigenic sin (OAS) or imprinting. In this paper, we derive constraints on the emergence of the pattern of successive strain replacements demonstrated by influenza, SARS-CoV-2, seasonal coronaviruses, and other pathogens. We find that OAS implies that the limited diversity found with successive strain replacement can only be maintained if [Formula: see text] is less than a threshold set by the characteristic antigenic distances for cross-protection and for the creation of new immune memory. This bound implies a "speed limit" on the evolution of new strains and a minimum variance of the distribution of infecting strains in antigenic space at any time. To carry out this analysis, we develop a theoretical model of pathogen evolution in antigenic space that implements OAS by decoupling the antigenic distances required for protection from infection and strain-specific memory creation. Our results demonstrate that OAS can play an integral role in the emergence of strain structure from host immune dynamics, preventing highly transmissible pathogens from maintaining serial strain replacement without diversification.

A speed limit on serial strain replacement from original antigenic sin

Many pathogens evolve to escape immunity, yet it remains difficult to predict whether immune pressure will lead to diversification, serial replacement of one variant by another, or more complex patterns. Pathogen strain dynamics are mediated by cross-protective immunity, whereby exposure to one strain partially protects against infection by antigenically diverged strains. There is growing evidence that this protection is influenced by early exposures, a phenomenon referred to as original antigenic sin (OAS) or imprinting. In this paper, we derive new constraints on the emergence of the pattern of successive strain replacements demonstrated by influenza, SARS-CoV-2, seasonal coronaviruses, and other pathogens. We find that OAS implies that the limited diversity found with successive strain replacement can only be maintained if R 0 is less than a threshold set by the characteristic antigenic distances for cross-protection and for the creation of new immune memory. This bound implies a "speed limit" on the evolution of new strains and a minimum variance of the distribution of infecting strains in antigenic space at any time. To carry out this analysis, we develop a theoretical model of pathogen evolution in antigenic space that implements OAS by decoupling the antigenic distances required for protection from infection and strain-specific memory creation. Our results demonstrate that OAS can play an integral role in the emergence of strain structure from host immune dynamics, preventing highly transmissible pathogens from maintaining serial strain replacement without diversification.

Evolution of pathogens with cross-immunity in response to healthcare interventions

Cross-immunity, as an evolutionary driver, can contribute to pathogen evolution, particularly pathogen diversity. Healthcare interventions aimed at reducing disease severity or transmission are commonly used to control diseases and can also induce pathogen evolution. Understanding pathogen evolution in the context of cross-immunity and healthcare interventions is crucial for infection control. This study starts by modelling cross-immunity, the extent of which is determined by strain traits and host characteristics. Given that all hosts have the same characteristics, full cross-immunity between residents and mutants occurs when mutation step sizes are small enough. Cross-immunity can be partial when the step size is large. The presence of partial cross-immunity reduces pathogen load and shortens the infectious period inside hosts, reducing transmission between hosts and improving host population survival and recovery. This study focuses on how pathogens evolve through small and large mutational steps and how healthcare interventions affect pathogen evolution. Using the theory of adaptive dynamics, we found that when mutational steps are small (only full cross-immunity is present), pathogen diversity cannot occur because it maximises the basic reproduction number. This results in intermediate values for both pathogen growth and clearance rates. However, when large mutational steps are allowed (with full and partial cross-immunity present), pathogens can evolve into multiple strains and induce pathogen diversity. The study also shows that different healthcare interventions can have varying effects on pathogen evolution. Generally, low levels of intervention are more likely to induce strain diversity, while high levels are more likely to result in strain reduction.

Pathogen diversity in meta-population networks

The pathogen diversity means that multiple strains coexist, and widely exist in the biology systems. The new mutation of SARS-CoV-2 leading to worldwide pathogen diversity is a typical example. What are the main factors of inducing the pathogen diversity? Previous studies indicated the pathogen mutation is the most important reason for inducing the pathogen diversity. The traffic network and gene network are crucial in shaping the dynamics of pathogen contagion, while their roles for the pathogen diversity still lacking a theoretical study. To this end, we propose a reaction-diffusion process of pathogens with mutations on meta-population networks, which includes population movement and strain mutation. We extend the Microscopic Markov Chain Approach (MMCA) to describe the model. Traffic networks make pathogen diversity more likely to occur in cities with lower infection densities. The likelihood of pathogen diversity is low in cities with short effective distances in the traffic network. Star-type gene network is more likely to lead to pathogen diversity than lattice-type and chain-type gene networks. When pathogen localization is present, infection is localized to strains that are at the endpoints of the gene network. Both the increased probability of movement and mutation promote pathogen diversity. The results also show that the population tends to move to cities with short effective distances, resulting in the infection density is high.

Non-pharmaceutical interventions and the emergence of pathogen variants

Non-pharmaceutical interventions (NPIs), such as social distancing and contact tracing, are important public health measures that can reduce pathogen transmission. In addition to playing a crucial role in suppressing transmission, NPIs influence pathogen evolution by mediating mutation supply, restricting the availability of susceptible hosts, and altering the strength of selection for novel variants. Yet it is unclear how NPIs might affect the emergence of novel variants that are able to escape pre-existing immunity (partially or fully), are more transmissible or cause greater mortality. We analyse a stochastic two-strain epidemiological model to determine how the strength and timing of NPIs affect the emergence of variants with similar or contrasting life-history characteristics to the wild type. We show that, while stronger and timelier NPIs generally reduce the likelihood of variant emergence, it is possible for more transmissible variants with high cross-immunity to have a greater probability of emerging at intermediate levels of NPIs. This is because intermediate levels of NPIs allow an epidemic of the wild type that is neither too small (facilitating high mutation supply), nor too large (leaving a large pool of susceptible hosts), to prevent a novel variant from becoming established in the host population. However, since one cannot predict the characteristics of a variant, the best strategy to prevent emergence is likely to be an implementation of strong, timely NPIs.

Modelling the effect of within-host dynamics on the diversity of a multi-strain pathogen

Multi-strain pathogens such as Group A Streptococcus, Streptococcus pneumoniae, and Staphylococcus aureus cause millions of infections each year with a substantial health burden. Control of multi-strain pathogens can be complicated by the high strain diversity often observed in endemic settings. It is not well understood how high strain diversity is maintained in populations, given that they compete with each other both directly (within an individual host) and indirectly (via host immunity). Previous modelling studies have investigated how indirect competition affects the prevalence and diversity of strains. However, these studies often make simplifying assumptions about the direct competition that occurs within hosts. Currently, little data is available to validate these assumptions, hence there is a need to clarify how sensitive model outputs are to these assumptions. In this study, we compare the dynamics of multi-strain pathogens under different assumptions about direct competition between strains using an agent-based model. We find that the assumptions made about direct competition can affect the epidemiological dynamics, particularly when there is no long-term immunity following infections and a low rate of importation of non-circulating strains. Our results suggest that while direct and indirect competition can each decrease strain diversity when they act in isolation, they may increase strain diversity when they act together. This finding highlights the importance of examining sensitivity to assumptions about strain competition. In particular, omitting consideration of direct competition can lead to inaccurate estimates of the likely effectiveness of control strategies as changes in strain diversity shift the level of direct strain competition.

Negative frequency-dependent selection and asymmetrical transformation stabilise multi-strain bacterial population structures

Streptococcus pneumoniae can be divided into many strains, each a distinct set of isolates sharing similar core and accessory genomes, which co-circulate within the same hosts. Previous analyses suggested the short-term vaccine-associated dynamics of S. pneumoniae strains may be mediated through multi-locus negative frequency-dependent selection (NFDS), which maintains accessory loci at equilibrium frequencies. Long-term simulations demonstrated NFDS stabilised clonally-evolving multi-strain populations through preventing the loss of variation through drift, based on polymorphism frequencies, pairwise genetic distances and phylogenies. However, allowing symmetrical recombination between isolates evolving under multi-locus NFDS generated unstructured populations of diverse genotypes. Replication of the observed data improved when multi-locus NFDS was combined with recombination that was instead asymmetrical, favouring deletion of accessory loci over insertion. This combination separated populations into strains through outbreeding depression, resulting from recombinants with reduced accessory genomes having lower fitness than their parental genotypes. Although simplistic modelling of recombination likely limited these simulations' ability to maintain some properties of genomic data as accurately as those lacking recombination, the combination of asymmetrical recombination and multi-locus NFDS could restore multi-strain population structures from randomised initial populations. As many bacteria inhibit insertions into their chromosomes, this combination may commonly underlie the co-existence of strains within a niche.

Epidemiological consequences of enduring strain-specific immunity requiring repeated episodes of infection

Group A Streptococcus (GAS) skin infections are caused by a diverse array of strain types and are highly prevalent in disadvantaged populations. The role of strain-specific immunity in preventing GAS infections is poorly understood, representing a critical knowledge gap in vaccine development. A recent GAS murine challenge study showed evidence that sterilising strain-specific and enduring immunity required two skin infections by the same GAS strain within three weeks. This mechanism of developing enduring immunity may be a significant impediment to the accumulation of immunity in populations. We used an agent-based mathematical model of GAS transmission to investigate the epidemiological consequences of enduring strain-specific immunity developing only after two infections with the same strain within a specified interval. Accounting for uncertainty when correlating murine timeframes to humans, we varied this maximum inter-infection interval from 3 to 420 weeks to assess its impact on prevalence and strain diversity, and considered additional scenarios where no maximum inter-infection interval was specified. Model outputs were compared with longitudinal GAS surveillance observations from northern Australia, a region with endemic infection. We also assessed the likely impact of a targeted strain-specific multivalent vaccine in this context. Our model produced patterns of transmission consistent with observations when the maximum inter-infection interval for developing enduring immunity was 19 weeks. Our vaccine analysis suggests that the leading multivalent GAS vaccine may have limited impact on the prevalence of GAS in populations in northern Australia if strain-specific immunity requires repeated episodes of infection. Our results suggest that observed GAS epidemiology from disease endemic settings is consistent with enduring strain-specific immunity being dependent on repeated infections with the same strain, and provide additional motivation for relevant human studies to confirm the human immune response to GAS skin infection.

Group A Streptococcus (GAS) is a ubiquitous bacterial pathogen that exists in many distinct strains, and is a major cause of death and disability globally. Vaccines against GAS are under development, but their effective use will require better understanding of how immunity develops following infection. Evidence from an animal model of skin infection suggests that the generation of enduring strain-specific immunity requires two infections by the same strain within a short time frame. It is not clear if this mechanism of immune development operates in humans, nor how it would contribute to the persistence of GAS in populations and affect vaccine impact. We used a mathematical model of GAS transmission, calibrated to data collected in an Indigenous Australian community, to assess whether this mechanism of immune development is consistent with epidemiological observations, and to explore its implications for the impact of a vaccine. We found that it is plausible that repeat infections are required for the development of immunity in humans, and illustrate the difficulties associated with achieving sustained reductions in disease prevalence with a vaccine.

Models of immune selection for multi-locus antigenic diversity of pathogens

It is well accepted that pathogens can evade recognition and elimination by the host immune system by varying their antigenic targets. Thus, it has become a truism that host immunity is a major driver and determinant of the antigenic diversity of pathogens. However, it remains puzzling how host immunity selects for antigenic diversity at the level of the pathogen population, given that hosts have acquired immune responses to multiple antigens of most pathogens - sometimes through multiple effectors of both humoral and cellular immunity. In this Opinion article, we address this puzzle and the related question of why pathogens often have diversity at multiple antigenic loci. Here, we describe five hypotheses to explain the polymorphism of multiple antigens in a single pathogen species and highlight research relevant to our current models of thinking about multi-locus antigenic diversity.

Analytical Exploration of Potential Pathways by which Diabetes Mellitus Impacts Tuberculosis Epidemiology

We aimed to develop a conceptual framework of diabetes mellitus (DM) effects on tuberculosis (TB) natural history and treatment outcomes, and to assess the impact of these effects on TB-transmission dynamics. The model was calibrated using TB data for India. A conceptual framework was developed based on a literature review, and then translated into a mathematical model to assess the impact of the DM-on-TB effects. The impact was analyzed using TB-disease incidence hazard ratio (HR) and population attributable fraction (PAF) measures. Evidence was identified for 10 plausible DM-on-TB effects. Assuming a flat change of 300% (meaning an effect size of 3.0) for each DM-on-TB effect, the HR ranged between 1.0 (Effect 9-Recovery) and 2.7 (Effect 2-Fast progression); most effects did not have an impact on the HR. Meanwhile, TB-disease incidence attributed directly and indirectly to each effect ranged between -4.6% (Effect 7-TB mortality) and 34.5% (Effect 2-Fast progression). The second largest impact was for Effect 6-Disease infectiousness at 29.9%. In conclusion, DM can affect TB-transmission dynamics in multiple ways, most of which are poorly characterized and difficult to assess in epidemiologic studies. The indirect (e.g. onward transmission) impacts of some DM-on-TB effects are comparable in scale to the direct impacts. While the impact of several effects on the HR was limited, the impact on the PAF was substantial suggesting that DM could be impacting TB epidemiology to a larger extent than previously thought.

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis

We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero-Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

Opposite outcomes of coinfection at individual and population scales

Coinfecting parasites and pathogens remain a leading challenge for global public health due to their consequences for individual-level infection risk and disease progression. However, a clear understanding of the population-level consequences of coinfection is lacking. Here, we constructed a model that includes three individual-level effects of coinfection: mortality, fecundity, and transmission. We used the model to investigate how these individual-level consequences of coinfection scale up to produce population-level infection patterns. To parameterize this model, we conducted a 4-y cohort study in African buffalo to estimate the individual-level effects of coinfection with two bacterial pathogens, bovine tuberculosis (bTB) and brucellosis, across a range of demographic and environmental contexts. At the individual level, our empirical results identified bTB as a risk factor for acquiring brucellosis, but we found no association between brucellosis and the risk of acquiring bTB. Both infections were associated with reductions in survival and neither infection was associated with reductions in fecundity. The model reproduced coinfection patterns in the data and predicted opposite impacts of coinfection at individual and population scales: Whereas bTB facilitated brucellosis infection at the individual level, our model predicted the presence of brucellosis to have a strong negative impact on bTB at the population level. In modeled populations where brucellosis was present, the endemic prevalence and basic reproduction number ([Formula: see text]) of bTB were lower than in populations without brucellosis. Therefore, these results provide a data-driven example of competition between coinfecting pathogens that occurs when one pathogen facilitates secondary infections at the individual level.

Modeling two strains of disease via aggregate-level infectivity curves

Well formulated models of disease spread, and efficient methods to fit them to observed data, are powerful tools for aiding the surveillance and control of infectious diseases. Our project considers the problem of the simultaneous spread of two related strains of disease in a context where spatial location is the key driver of disease spread. We start our modeling work with the individual level models (ILMs) of disease transmission, and extend these models to accommodate the competing spread of the pathogens in a two-tier hierarchical population (whose levels we refer to as 'farm' and 'animal'). The postulated interference mechanism between the two strains is a period of cross-immunity following infection. We also present a framework for speeding up the computationally intensive process of fitting the ILM to data, typically done using Markov chain Monte Carlo (MCMC) in a Bayesian framework, by turning the inference into a two-stage process. First, we approximate the number of animals infected on a farm over time by infectivity curves. These curves are fit to data sampled from farms, using maximum likelihood estimation, then, conditional on the fitted curves, Bayesian MCMC inference proceeds for the remaining parameters. Finally, we use posterior predictive distributions of salient epidemic summary statistics, in order to assess the model fitted.

Computational Characterization of Transient Strain-Transcending Immunity against Influenza A

The enigmatic observation that the rapidly evolving influenza A (H3N2) virus exhibits, at any given time, a limited standing genetic diversity has been an impetus for much research. One of the first generative computational models to successfully recapitulate this pattern of consistently constrained diversity posits the existence of a strong and short-lived strain-transcending immunity. Building on that model, we explored a much broader set of scenarios (parameterizations) of a transient strain-transcending immunity, ran long-term simulations of each such scenario, and assessed its plausibility with respect to a set of known or estimated influenza empirical measures. We evaluated simulated outcomes using a variety of measures, both epidemiological (annual attack rate, epidemic duration, reproductive number, and peak weekly incidence), and evolutionary (pairwise antigenic diversity, fixation rate, most recent common ancestor, and kappa, which quantifies the potential for antigenic evolution). Taking cumulative support from all these measures, we show which parameterizations of strain-transcending immunity are plausible with respect to the set of empirically derived target values. We conclude that strain-transcending immunity which is milder and longer lasting than previously suggested is more congruent with the observed short- and long-term behavior of influenza.

Capturing the dynamics of pathogens with many strains

Pathogens that consist of multiple antigenic variants are a serious public health concern. These infections, which include dengue virus, influenza and malaria, generate substantial morbidity and mortality. However, there are considerable theoretical challenges involved in modelling such infections. As well as describing the interaction between strains that occurs as a result cross-immunity and evolution, models must balance biological realism with mathematical and computational tractability. Here we review different modelling approaches, and suggest a number of biological problems that are potential candidates for study with these methods. We provide a comprehensive outline of the benefits and disadvantages of available frameworks, and describe what biological information is preserved and lost under different modelling assumptions. We also consider the emergence of new disease strains, and discuss how models of pathogens with multiple strains could be developed further in future. This includes extending the flexibility and biological realism of current approaches, as well as interface with data.

Characterization of the endemic equilibrium and response to mutant injection in a multi-strain disease model

We explore a model of an antigenically diverse infection whose otherwise identical strains compete through cross-immunity. We assume that individuals may produce upon infection different numbers of antibody types, each of which matches the antigenic configuration of a particular epitope, and that one matching antibody type grants total immunity against a challenging strain. In order to reduce the number of equations involved in the analytic description of the dynamics, we follow the strategy proposed by Kryazhimskiy et al. (2007) and apply a low-order closure reminiscent of a pair approximation. Using this approximation, we go beyond the numerical studies of Kryazhimskiy et al. (2007) and explore the analytic properties of the ensuing model in the absence of mutation. We characterize its endemic equilibrium, comparing with the results of agent based simulations of the full model to assess the performance of the closure assumption. We show that a particular choice of immune response leads to a degenerate endemic equilibrium, where different strain prevalences may exist, breaking the symmetry of the model. Finally we study the behavior of the system under the injection of mutant strains. We find that the build up of diversity from a single founding strain is extremely unlikely for different choices of the population׳s immune response.

The impact of coinfections and their simultaneous transmission on antigenic diversity and epidemic cycling of infectious diseases

Epidemic cycling in human infectious diseases is common; however, its underlying mechanisms have been poorly understood. Much effort has been made to search for external mechanisms. Multiple strains of an infectious agent were usually observed and coinfections were frequent; further, empirical evidence indicates the simultaneous transmission of coinfections. To explore intrinsic mechanisms for epidemic cycling, in this study we consider a multistrain Susceptible-Infected-Recovered-Susceptible epidemic model by including coinfections and simultaneous transmission. We show that coinfections and their simultaneous transmission widen the parameter range for coexistence and coinfections become popular when strains enhance each other and the immunity wanes quickly. However, the total prevalence is nearly independent of these characteristics and approximated by that of one-strain model. With sufficient simultaneous transmission and antigenic diversity, cyclical epidemics can be generated even when strains interfere with each other by reducing infectivity. This indicates that strain interactions within coinfections and cross-immunity during subsequent infection provide a possible intrinsic mechanism for epidemic cycling.

Role of stochastic processes in maintaining discrete strain structure in antigenically diverse pathogen populations

Many highly diverse pathogen populations appear to exist stably as discrete antigenic types despite evidence of genetic exchange. It has been shown that this may arise as a consequence of immune selection on pathogen populations, causing them to segregate permanently into discrete nonoverlapping subsets of antigenic variants to minimize competition for available hosts. However, discrete antigenic strain structure tends to break down under conditions where there are unequal numbers of allelic variants at each locus. Here, we show that the inclusion of stochastic processes can lead to the stable recovery of discrete strain structure through loss of certain alleles. This explains how pathogen populations may continue to behave as independently transmitted strains despite inevitable asymmetries in allelic diversity of major antigens. We present evidence for this type of structuring across global meningococcal isolates in three diverse antigens that are currently being developed as vaccine components.

The curse of the Pharaoh revisited: evolutionary bi-stability in environmentally transmitted pathogens

It is increasingly evident that for a number of high-profile pathogens, transmission involves both direct and environmental pathways. Much of the distinguished evolutionary theory has, however, focused on each of transmission component separately. Herein, we use the framework of adaptive dynamics to study the evolutionary consequences of mixed transmission. We find that environmental transmission can select for increased virulence when direct transmission is low. Increasing the efficiency of direct transmission gives rise to an evolutionary bi-stability, with coexistence of different levels of virulence. We conclude that the overlooked contribution of environmental transmission may explain the curious appearance of high virulence in pathogens that are typically only moderately pathogenic, as observed for avian influenza viruses and cholera.

Tracking the Emergence of Pandemic Influenza A/H1N1/2009 and its Interaction with Seasonal Influenza Viruses in Singapore

Introduction: Since the emergence of the pandemic influenza A/H1N1/2009 virus in April 2009, diagnostic testing in many countries has revealed the rapid displacement and then replacement of circulating seasonal influenza viruses by this novel virus. Materials and Methods: In-house seasonal and pandemic influenza-specific polymerase chain reaction assays were introduced and/or developed at the Molecular Diagnosis Centre (MDC) at the National University Hospital (NUH), Singapore. These assays have been used to test all samples received from in-patients, out-patients, staff and visitors for suspected pandemic influenza A/H1N1/2009 infection. Results: Prior to the arrival of the pandemic A/H1N1/2009 virus in Singapore at the end of May 2009, seasonal influenza A/H3N2 predominated in this population, with very little seasonal influenza A/H1N1 and B viruses detected. Within about 1 month of its arrival in Singapore (mainly during June to July 2009), this pandemic virus rapidly displaced seasonal influenza A/H3N2 to become the predominant strain in the Singaporean population served by MDC/NUH. Conclusions: Real-time molecular techniques have allowed the prompt detection of different influenza subtypes during this current pandemic, which has revealed the displacement/replacement of previously circulating seasonal subtypes with A/H1N1/2009. Although some of this may be explained by immunological cross-reactivity between influenza subtypes, more studies are required. Key words: Diagnostic, H1N1, Polymerase chain reaction

Theoretical aspects of immunity

The immune system recognizes a myriad of invading pathogens and their toxic products. It does so with a finite repertoire of antibodies and T cell receptors. We here describe theories that quantify the dynamics of the immune system. We describe how the immune system recognizes antigens by searching the large space of receptor molecules. We consider in some detail the theories that quantify the immune response to influenza and dengue fever. We review theoretical descriptions of the complementary evolution of pathogens that occurs in response to immune system pressure. Methods including bioinformatics, molecular simulation, random energy models, and quantum field theory contribute to a theoretical understanding of aspects of immunity.

Epidemics with multistrain interactions: the interplay between cross immunity and antibody-dependent enhancement

This paper examines the interplay of the effect of cross immunity and antibody-dependent enhancement (ADE) in multistrain diseases. Motivated by dengue fever, we study a model for the spreading of epidemics in a population with multistrain interactions mediated by both partial temporary cross immunity and ADE. Although ADE models have previously been observed to cause chaotic outbreaks, we show analytically that weak cross immunity has a stabilizing effect on the system. That is, the onset of disease fluctuations requires a larger value of ADE with small cross immunity than without. However, strong cross immunity is shown numerically to cause oscillations and chaotic outbreaks even for low values of ADE.

Quantifying the transmission potential of pandemic influenza

This article reviews quantitative methods to estimate the basic reproduction number of pandemic influenza, a key threshold quantity to help determine the intensity of interventions required to control the disease. Although it is difficult to assess the transmission potential of a probable future pandemic, historical epidemiologic data is readily available from previous pandemics, and as a reference quantity for future pandemic planning, mathematical and statistical analyses of historical data are crucial. In particular, because many historical records tend to document only the temporal distribution of cases or deaths (i.e. epidemic curve), our review focuses on methods to maximize the utility of time-evolution data and to clarify the detailed mechanisms of the spread of influenza.

First, we highlight structured epidemic models and their parameter estimation method which can quantify the detailed disease dynamics including those we cannot observe directly. Duration-structured epidemic systems are subsequently presented, offering firm understanding of the definition of the basic and effective reproduction numbers. When the initial growth phase of an epidemic is investigated, the distribution of the generation time is key statistical information to appropriately estimate the transmission potential using the intrinsic growth rate. Applications of stochastic processes are also highlighted to estimate the transmission potential using similar data. Critically important characteristics of influenza data are subsequently summarized, followed by our conclusions to suggest potential future methodological improvements.

Patterns of antigenic diversity and the mechanisms that maintain them

Many of the remaining challenges in infectious disease control involve pathogens that fail to elicit long-lasting immunity in their hosts. Antigenic variation is a common reason for this failure and a contributor to the complexity of vaccine design. Diversifying selection by the host immune system is commonly, and often correctly, invoked to explain antigenic variability in pathogens. However, there is a wide variety of patterns of antigenic variation across space and time, and within and between hosts, and we do not yet understand the determinants of these different patterns. This review describes five such patterns, taking as examples two bacteria (Streptococcus pneumoniae and Neisseria meningitidis), two viruses (influenza A and HIV-1), as well as the pathogens (taken as a group) for which antigenic variation is negligible. Pathogen-specific explanations for these patterns of diversity are critically evaluated, and the patterns are compared against predictions of theoretical models for antigenic diversity. Major remaining challenges are highlighted, including the identification of key protective antigens in bacteria, the design of vaccines to combat antigenic variability for viruses and the development of more systematic explanations for patterns of antigenic variation.

Stochasticity generates an evolutionary instability for infectious disease

Traditional models of disease evolution are based upon the deterministic competition between strains that confer complete cross-immunity, and predict the selection of strains with higher basic reproductive ratios (R(0)). In contrast, evolution in a stochastic setting is determined by a complex mixture of influences. Here, to isolate the impact of stochasticity, we constrain all competing strains to have an equal basic reproductive ratio - thereby eliminating deterministic selection. The resulting stochastic models predict an evolutionary unstable strategy, which separates a region favouring the evolution of rapid-transmission (acute) strains from one favouring persistent (chronic) strains. We find this to be a generic phenomenon with strain evolution consistently driven towards extremes of epidemiological behaviour. Even in the absence of an equal R(0) constraint, such stochastic selective pressures operate in addition to standard deterministic selection and will therefore influence the evolutionary behaviour of disease in all scenarios.

Dynamics of antigenic and genetic changes in the hemagglutinins of influenza A/H3N2 viruses of three consecutive seasons (2002/2003 to 2004/2005) in Austria

Human influenza viruses are subject to continuous antigenic drift and this phenomenon poses great problems for the annual production of vaccines which should ideally be manufactured from strains closely matching the predominant strains of the coming influenza season. We have investigated the dynamics of antigenic and genetic changes in the hemagglutinins of circulating influenza A/H3N2 strains in three consecutive seasons (2002/2003 to 2004/2005) in Austria by sequence analysis of the HA1 domain and by antigenic characterization using a hemagglutination inhibition assay. Each of the three seasons was dominated by a single and different H3N2 variant, but in all cases sequencing revealed the co-circulation of a drift variant which would have been missed by conventional antigenic analysis. These emerging strains always showed already a close genetic relationship to the dominating strain of the following season. Our results underscore the value of monitoring seasonal influenza strain dynamics by sequence analysis as an instrument that can provide important and timely information on the appearance of strains with epidemiologic significance.

Vaccination and the dynamics of immune evasion

Vaccines exert strong selective pressures on pathogens, favouring the spread of antigenic variants. We propose a simple mathematical model to investigate the dynamics of a novel pathogenic strain that emerges in a population where a previous strain is maintained at low endemic level by a vaccine. We compare three methods to assess the ability of the novel strain to invade and persist: algebraic rate of invasion; deterministic dynamics; and stochastic dynamics. These three techniques provide complementary predictions on the fate of the system. In particular, we emphasize the importance of stochastic simulations, which account for the possibility of extinctions of either strain. More specifically, our model suggests that the probability of persistence of an invasive strain (i) can be minimized for intermediate levels of vaccine cross-protection (i.e. immune protection against the novel strain) and (ii) is lower if cross-immunity acts through a reduced infectious period rather than through reduced susceptibility.

Tracking the dynamics of pathogen interactions: Modeling ecological and immune-mediated processes in a two-pathogen single-host system

Traditionally, epidemiological studies have focused on understanding the dynamics of a single pathogen, assuming no interactions with other pathogens. Recently, a large body of work has begun to explore the effects of immune-mediated interactions, arising from cross-immunity and antibody-dependent enhancement, between related pathogen strains. In addition, ecological processes such as a temporary period of convalescence and pathogen-induced mortality have led to the concept of ecological interference between unrelated diseases. There remains, however, the need for a systematic study of both immunological and ecological processes within a single framework. In this paper, we develop a general two-pathogen single-host model of pathogen interactions that simultaneously incorporates these mechanisms. We are then able to mechanistically explore how immunoecological processes mediate interactions between diseases for a pool of susceptible individuals. We show that the precise nature of the interaction can induce either competitive or cooperative associations between pathogens. Understanding the dynamic implications of multi-pathogen associations has potentially important public health consequences. Such a framework may be especially helpful in disentangling the effects of partially cross-immunizing infections that affect populations with a pre-disposition towards immunosuppression such as children and the elderly.

Shaping the phylogenetic tree of influenza by cross-immunity

Cross-immunity among related strains can account for the selection producing the slender phylogenetic tree of influenza A and B in humans. Using a model of seasonal influenza epidemics with drift (Andreasen, 2003. Dynamics of annual influenza A epidemics with immuno-selection. J. Math. Biol. 46, 504-536), and assuming that two mutants arrive in the host population sequentially, we determine the threshold condition for the establishment of the second mutant in the presence of partial cross-protection caused by the first mutant and their common ancestors. For fixed levels of cross-protection, the chance that the second mutant establishes increases with rho the basic reproduction ratio and some temporary immunity may be necessary to explain the slenderness of flu's phylogenetic tree. In the presence of moderate levels of temporary immunity, an asymmetric situation can arise in the season after the two mutants were introduced and established: if the offspring of the new mutant arrives before the offspring of the resident type, then the mutant-line may produce a massive epidemic suppressing the original lineage. However, if the original lineage arrives first then both strains may establish and the phylogenetic tree may bifurcate.

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.