A two-tiered model for simulating the ecological and evolutionary dynamics of rapidly evolving viruses, with an application to influenza

Predicting elimination of evolving virus variants

As the SARS-CoV-2 virus spreads around the world new variants are appearing regularly. Although some countries have achieved very swift and successful vaccination campaigns, on a global scale the vast majority of the population is unvaccinated and new variants are proving more resistant to the current set of vaccines. We present a simple model of disease spread, which includes the evolution of new variants of a novel virus and varying vaccine effectiveness to these new strains. We show that rapid vaccine updates to target new strains are more effective than slow updates and containing spread through non-pharmaceutical interventions is vital while these vaccines are delivered. Finally, when measuring the key model inputs, e.g. the rate at which new mutations and variants of concern emerge, is difficult we show how an observable model output, the number of new variants that have been seen, is strongly correlated with the probability the virus is eliminated.

Ecological and evolutionary dynamics of multi-strain RNA viruses

Potential interactions among co-circulating viral strains in host populations are often overlooked in the study of virus transmission. However, these interactions probably shape transmission dynamics by influencing host immune responses or altering the relative fitness among co-circulating strains. In this Review, we describe multi-strain dynamics from ecological and evolutionary perspectives, outline scales in which multi-strain dynamics occur and summarize important immunological, phylogenetic and mathematical modelling approaches used to quantify interactions among strains. We also discuss how host-pathogen interactions influence the co-circulation of pathogens. Finally, we highlight outstanding questions and knowledge gaps in the current theory and study of ecological and evolutionary dynamics of multi-strain viruses.

A Review on Equine Influenza from a Human Influenza Perspective

Influenza A viruses (IAVs) have a main natural reservoir in wild birds. IAVs are highly contagious, continually evolve, and have a wide host range that includes various mammalian species including horses, pigs, and humans. Furthering our understanding of host-pathogen interactions and cross-species transmissions is therefore essential. This review focuses on what is known regarding equine influenza virus (EIV) virology, pathogenesis, immune responses, clinical aspects, epidemiology (including factors contributing to local, national, and international transmission), surveillance, and preventive measures such as vaccines. We compare EIV and human influenza viruses and discuss parallels that can be drawn between them. We highlight differences in evolutionary rates between EIV and human IAVs, their impact on antigenic drift, and vaccine strain updates. We also describe the approaches used for the control of equine influenza (EI), which originated from those used in the human field, including surveillance networks and virological analysis methods. Finally, as vaccination in both species remains the cornerstone of disease mitigation, vaccine technologies and vaccination strategies against influenza in horses and humans are compared and discussed.

Mutation induced infection waves in diseases like COVID-19

After more than 6 million deaths worldwide, the ongoing vaccination to conquer the COVID-19 disease is now competing with the emergence of increasingly contagious mutations, repeatedly supplanting earlier strains. Following the near-absence of historical examples of the long-time evolution of infectious diseases under similar circumstances, models are crucial to exemplify possible scenarios. Accordingly, in the present work we systematically generalize the popular susceptible-infected-recovered model to account for mutations leading to repeatedly occurring new strains, which we coarse grain based on tools from statistical mechanics to derive a model predicting the most likely outcomes. The model predicts that mutations can induce a super-exponential growth of infection numbers at early times, which self-amplify to giant infection waves which are caused by a positive feedback loop between infection numbers and mutations and lead to a simultaneous infection of the majority of the population. At later stages-if vaccination progresses too slowly-mutations can interrupt an ongoing decrease of infection numbers and can cause infection revivals which occur as single waves or even as whole wave trains featuring alternative periods of decreasing and increasing infection numbers. This panorama of possible mutation-induced scenarios should be tested in more detailed models to explore their concrete significance for specific infectious diseases. Further, our results might be useful for discussions regarding the importance of a release of vaccine-patents to reduce the risk of mutation-induced infection revivals but also to coordinate the release of measures following a downwards trend of infection numbers.

Equine Influenza

Horses are the third major mammalian species, along with humans and swine, long known to be subject to acute upper respiratory disease from influenza A virus infection. The viruses responsible are subtype H7N7, which is believed extinct, and H3N8, which circulates worldwide. The equine influenza lineages are clearly divergent from avian influenza lineages of the same subtypes. Their genetic evolution and potential for interspecies transmission, as well as clinical features and epidemiology, are discussed. Equine influenza is spread internationally and vaccination is central to control efforts. The current mechanism of international surveillance and virus strain recommendations for vaccines is described.

Dynamic interactions of influenza viruses in Hong Kong during 1998-2018

Influenza epidemics cause substantial morbidity and mortality every year worldwide. Currently, two influenza A subtypes, A(H1N1) and A(H3N2), and type B viruses co-circulate in humans and infection with one type/subtype could provide cross-protection against the others. However, it remains unclear how such ecologic competition via cross-immunity and antigenic mutations that allow immune escape impact influenza epidemic dynamics at the population level. Here we develop a comprehensive model-inference system and apply it to study the evolutionary and epidemiological dynamics of the three influenza types/subtypes in Hong Kong, a city of global public health significance for influenza epidemic and pandemic control. Utilizing long-term influenza surveillance data since 1998, we are able to estimate the strength of cross-immunity between each virus-pairs, the timing and frequency of punctuated changes in population immunity in response to antigenic mutations in influenza viruses, and key epidemiological parameters over the last 20 years including the 2009 pandemic. We find evidence of cross-immunity in all types/subtypes, with strongest cross-immunity from A(H1N1) against A(H3N2). Our results also suggest that A(H3N2) may undergo antigenic mutations in both summers and winters and thus monitoring the virus in both seasons may be important for vaccine development. Overall, our study reveals intricate epidemiological interactions and underscores the importance of simultaneous monitoring of population immunity, incidence rates, and viral genetic and antigenic changes.

Why Are CD8 T Cell Epitopes of Human Influenza A Virus Conserved?

Universal influenza vaccines against the conserved epitopes of influenza A virus have been proposed to minimize the burden of seasonal outbreaks and prepare for the pandemics. However, it is not clear how rapidly T cell-inducing vaccines will select for viruses that escape these T cell responses. Our mathematical models explore the factors that contribute to the conservation of CD8 T cell epitopes and how rapidly the virus will evolve in response to T cell-inducing vaccines. We identify the key biological parameters to be measured and questions that need to be addressed in future studies.

The high degree of conservation of CD8 T cell epitopes of influenza A virus (IAV) may allow for the development of T cell-inducing vaccines that provide protection across different strains and subtypes. This conservation is not fully explained by functional constraint, since an additional mutation(s) can compensate for the replicative fitness loss of IAV escape variants. Here, we propose three additional mechanisms that contribute to the conservation of CD8 T cell epitopes of IAV. First, influenza-specific CD8 T cells may protect predominantly against severe pathology rather than infection and may have only a modest effect on transmission. Second, polymorphism of the human major histocompatibility complex class I (MHC-I) gene restricts the advantage of an escape variant to only a small fraction of the human population who carry the relevant MHC-I alleles. Finally, infection with CD8 T cell escape variants may result in a compensatory increase in the responses to other epitopes of IAV. We use a combination of population genetics and epidemiological models to examine how the interplay between these mechanisms affects the rate of invasion of IAV escape variants. We conclude that for a wide range of biologically reasonable parameters, the invasion of an escape variant virus will be slow, with a timescale of a decade or more. The results suggest T cell-inducing vaccines do not engender the rapid evolution of IAV. Finally, we identify key parameters whose measurement will allow for more accurate quantification of the long-term effectiveness and impact of universal T cell-inducing influenza vaccines.

IMPORTANCE Universal influenza vaccines against the conserved epitopes of influenza A virus have been proposed to minimize the burden of seasonal outbreaks and prepare for the pandemics. However, it is not clear how rapidly T cell-inducing vaccines will select for viruses that escape these T cell responses. Our mathematical models explore the factors that contribute to the conservation of CD8 T cell epitopes and how rapidly the virus will evolve in response to T cell-inducing vaccines. We identify the key biological parameters to be measured and questions that need to be addressed in future studies.

Effect of time varying transmission rates on the coupled dynamics of epidemic and awareness over a multiplex network

A non-linear stochastic model is presented to study the effect of time variation of transmission rates on the co-evolution of epidemics and its corresponding awareness over a two layered multiplex network. In the model, the infection transmission rate of a given node in the epidemic layer depends upon its awareness probability in the awareness layer. Similarly, the infection information transmission rate of a node in the awareness layer depends upon its infection probability in the epidemic layer. The spread of disease resulting from physical contacts is described in terms of a Susceptible Infected Susceptible process over the epidemic layer and the spread of information about the disease outbreak is described in terms of an Unaware Aware Unaware process over the virtual interaction mediated awareness layer. The time variation of the transmission rates and the resulting co-evolution of these mutually competing processes are studied in terms of a network topology dependent parameter ( α ). Using a second order linear theory, it is shown that in the continuous time limit, the co-evolution of these processes can be described in terms of damped and driven harmonic oscillator equations. From the results of a Monte-Carlo simulation, it is shown that for a suitable choice of the parameter ( α ) , the two processes can either exhibit sustained oscillatory or damped dynamics. The damped dynamics corresponds to the endemic state. Furthermore, for the case of an endemic state, it is shown that the inclusion of the awareness layer significantly lowers the disease transmission rate and reduces the size of the epidemic. The infection probability of the nodes in the endemic state is found to have a dependence on both the transmission rates and on their absolute degrees in each of the network layers and on the relative differences between their degrees in the respective layers.

Intermediate levels of vaccination coverage may minimize seasonal influenza outbreaks

For most pathogens, vaccination reduces the spread of the infection and total number of cases; thus, public policy usually advocates maximizing vaccination coverage. We use simple mathematical models to explore how this may be different for pathogens, such as influenza, which exhibit strain variation. Our models predict that the total number of seasonal influenza infections is minimized at an intermediate (rather than maximal) level of vaccination, and, somewhat counter-intuitively, further increasing the level of the vaccination coverage may lead to higher number of influenza infections and be detrimental to the public interest. This arises due to the combined effects of: competition between multiple co-circulating strains; limited breadth of protection afforded by the vaccine; and short-term strain-transcending immunity following natural infection. The study highlights the need for better quantification of the components of vaccine efficacy and longevity of strain-transcending cross-immunity in order to generate nuanced recommendations for influenza vaccine coverage levels.

Evolution-informed forecasting of seasonal influenza A (H3N2)

Interpandemic or seasonal influenza A, currently subtypes H3N2 and H1N1, exacts an enormous annual burden both in terms of human health and economic impact. Incidence prediction ahead of season remains a challenge largely because of the virus' antigenic evolution. We propose a forecasting approach that incorporates evolutionary change into a mechanistic epidemiological model. The proposed models are simple enough that their parameters can be estimated from retrospective surveillance data. These models link amino acid sequences of hemagglutinin epitopes with a transmission model for seasonal H3N2 influenza, also informed by H1N1 levels. With a monthly time series of H3N2 incidence in the United States for more than 10 years, we demonstrate the feasibility of skillful prediction for total cases ahead of season, with a tendency to underpredict monthly peak epidemic size, and an accurate real-time forecast for the 2016/2017 influenza season.

A Metric on Phylogenetic Tree Shapes

The shapes of evolutionary trees are influenced by the nature of the evolutionary process but comparisons of trees from different processes are hindered by the challenge of completely describing tree shape. We present a full characterization of the shapes of rooted branching trees in a form that lends itself to natural tree comparisons. We use this characterization to define a metric, in the sense of a true distance function, on tree shapes. The metric distinguishes trees from random models known to produce different tree shapes. It separates trees derived from tropical versus USA influenza A sequences, which reflect the differing epidemiology of tropical and seasonal flu. We describe several metrics based on the same core characterization, and illustrate how to extend the metric to incorporate trees’ branch lengths or other features such as overall imbalance. Our approach allows us to construct addition and multiplication on trees, and to create a convex metric on tree shapes which formally allows computation of average tree shapes.

Prediction of influenza B vaccine effectiveness from sequence data

Influenza is a contagious respiratory illness that causes significant human morbidity and mortality, affecting 5-15% of the population in a typical epidemic season. Human influenza epidemics are caused by types A and B, with roughly 25% of human cases due to influenza B. Influenza B is a single-stranded RNA virus with a high mutation rate, and both prior immune history and vaccination put significant pressure on the virus to evolve. Due to the high rate of viral evolution, the influenza B vaccine component of the annual influenza vaccine is updated, roughly every other year in recent years. To predict when an update to the vaccine is needed, an estimate of expected vaccine effectiveness against a range of viral strains is required. We here introduce a method to measure antigenic distance between the influenza B vaccine and circulating viral strains. The measure correlates well with effectiveness of the influenza B component of the annual vaccine in humans between 1979 and 2014. We discuss how this measure of antigenic distance may be used in the context of annual influenza vaccine design and prediction of vaccine effectiveness.

Quantifying the risk of pandemic influenza virus evolution by mutation and re-assortment

Large outbreaks of zoonotic influenza A virus (IAV) infections may presage an influenza pandemic. However, the likelihood that an airborne-transmissible variant evolves upon zoonotic infection or co-infection with zoonotic and seasonal IAVs remains poorly understood, as does the relative importance of accumulating mutations versus re-assortment in this process. Using discrete-time probabilistic models, we determined quantitative probability ranges that transmissible variants with 1-5 mutations and transmissible re-assortants evolve after a given number of zoonotic IAV infections. The systematic exploration of a large population of model parameter values was designed to account for uncertainty and variability in influenza virus infection, epidemiological and evolutionary processes. The models suggested that immunocompromised individuals are at high risk of generating IAV variants with pandemic potential by accumulation of mutations. Yet, both immunocompetent and immunocompromised individuals could generate high viral loads of single and double mutants, which may facilitate their onward transmission and the subsequent accumulation of additional 1-2 mutations in newly-infected individuals. This may result in the evolution of a full transmissible genotype along short chains of contact transmission. Although co-infection with zoonotic and seasonal IAVs was shown to be a rare event, it consistently resulted in high viral loads of re-assortants, which may facilitate their onward transmission among humans. The prevention or limitation of zoonotic IAV infection in immunocompromised and contact individuals, including health care workers, as well as vaccination against seasonal IAVs-limiting the risk of co-infection-should be considered fundamental tools to thwart the evolution of a novel pandemic IAV by accumulation of mutations and re-assortment.

The effects of a deleterious mutation load on patterns of influenza A/H3N2's antigenic evolution in humans

Recent phylogenetic analyses indicate that RNA virus populations carry a significant deleterious mutation load. This mutation load has the potential to shape patterns of adaptive evolution via genetic linkage to beneficial mutations. Here, we examine the effect of deleterious mutations on patterns of influenza A subtype H3N2's antigenic evolution in humans. By first analyzing simple models of influenza that incorporate a mutation load, we show that deleterious mutations, as expected, act to slow the virus's rate of antigenic evolution, while making it more punctuated in nature. These models further predict three distinct molecular pathways by which antigenic cluster transitions occur, and we find phylogenetic patterns consistent with each of these pathways in influenza virus sequences. Simulations of a more complex phylodynamic model further indicate that antigenic mutations act in concert with deleterious mutations to reproduce influenza's spindly hemagglutinin phylogeny, co-circulation of antigenic variants, and high annual attack rates.

DOI:http://dx.doi.org/10.7554/eLife.07361.001

Each year, up to 15% of the world's population experience symptoms of an influenza infection, also commonly known as flu. The most common culprit is a strain of the virus called influenza type A subtype H3N2. One reason that so many people become infected each year is that this virus evolves rapidly. Within a few years, proteins on the surface of the virus known as antigens become less recognizable to the immune system of a person who has been previously infected. This means that the person can become ill with the virus again because their immune system cannot mount an effective response to the evolved virus strain.

Influenza virus strains evolve rapidly because their genetic material accumulates mutations quickly. Although some of these mutations are beneficial to the virus, other mutations are harmful and reduce the ability of the virus to spread. Sometimes beneficial mutations may occur alongside harmful ones, but it is not known how the harmful mutations affect the evolution of the virus.

Here, Koelle and Rasmussen used computer models of H3N2 influenza to examine the effect of harmful mutations on the evolution of this virus population. The models show that harmful mutations limit how quickly the antigens can evolve. Also, the presence of these harmful mutations effectively acts as a sieve: they allow only large changes in the antigens to establish in the virus population.

The models suggest that there are three routes by which large changes in the antigens on H3N2 viruses may occur. The first is by a single mutation that has a big effect on the antigens in viruses that only carry a few harmful mutations, but these large mutations would not happen very often. Another route may be through more common mutations that have only a small or moderate benefit, which would allow the virus to become more common in the population before it acquires a beneficial mutation with a much greater effect. The third possibility is that a large beneficial mutation may arise in viruses that have many harmful mutations. These harmful mutations may initially limit the ability of the virus to spread, but over time, some of these harmful mutations may then be lost.

Koelle and Rasmussen found that the computer models could recreate the patterns of virus evolution that have been observed in real strains of H3N2. Researchers use predictions of influenza evolution to help them decide which virus strains should be included in flu vaccines each year. Koelle and Rasmussen findings indicate that harmful mutations should be considered when making these predictions.

DOI:http://dx.doi.org/10.7554/eLife.07361.002

The Role of Human Transportation Networks in Mediating the Genetic Structure of Seasonal Influenza in the United States

Recent studies have demonstrated the importance of accounting for human mobility networks when modeling epidemics in order to accurately predict spatial dynamics. However, little is known about the impact these movement networks have on the genetic structure of pathogen populations and whether these effects are scale-dependent. We investigated how human movement along the aviation and commuter networks contributed to intra-seasonal genetic structure of influenza A epidemics in the continental United States using spatially-referenced hemagglutinin nucleotide sequences collected from 2003–2013 for both the H3N2 and H1N1 subtypes. Comparative analysis of these transportation networks revealed that the commuter network is highly spatially-organized and more heavily traveled than the aviation network, which instead is characterized by high connectivity between all state pairs. We found that genetic distance between sequences often correlated with distance based on interstate commuter network connectivity for the H1N1 subtype, and that this correlation was not as prevalent when geographic distance or aviation network connectivity distance was assessed against genetic distance. However, these patterns were not as apparent for the H3N2 subtype at the scale of the continental United States. Finally, although sequences were spatially referenced at the level of the US state of collection, a community analysis based on county to county commuter connections revealed that commuting communities did not consistently align with state geographic boundaries, emphasizing the need for the greater availability of more specific sequence location data. Our results highlight the importance of utilizing host movement data in characterizing the underlying genetic structure of pathogen populations and demonstrate a need for a greater understanding of the differential effects of host movement networks on pathogen transmission at various spatial scales.

The rapid, long-distance spread of human pathogens such as seasonal influenza A across modern transportation networks presents a tremendous challenge for public health. Previous work based on influenza-like illness reports has demonstrated that commuters play an important role in the transmission of influenza across the United States. However, genetic structuring of influenza populations within a single season has not previously been detected. Here, we use sequence data collected over multiple seasons to investigate how human movement along the aviation and commuter networks in the United States contributes to influenza transmission at the regional scale. We confirm that commuters can play an integral role in interstate influenza transmission, but found that this pattern was specific to the influenza A subtype under investigation. We additionally show that strong county-to-county commuter flows do not necessarily fall within state boundaries, emphasizing the need for more precise spatial data to be associated with publically available sequences. Our results demonstrate that genetic structure does exist for influenza populations during the course of a single season at the regional scale and highlight the need to incorporate host movement patterns when studying spatial population structure.

'SEEDY' (Simulation of Evolutionary and Epidemiological Dynamics): An R Package to Follow Accumulation of Within-Host Mutation in Pathogens

Genome sequencing is an increasingly common component of infectious disease outbreak investigations. However, the relationship between pathogen transmission and observed genetic data is complex, and dependent on several uncertain factors. As such, simulation of pathogen dynamics is an important tool for interpreting observed genomic data in an infectious disease outbreak setting, in order to test hypotheses and to explore the range of outcomes consistent with a given set of parameters. We introduce ‘seedy’, an R package for the simulation of evolutionary and epidemiological dynamics (http://cran.r-project.org/web/packages/seedy/). Our software implements stochastic models for the accumulation of mutations within hosts, as well as individual-level disease transmission. By allowing variables such as the transmission bottleneck size, within-host effective population size and population mixing rates to be specified by the user, our package offers a flexible framework to investigate evolutionary dynamics during disease outbreaks. Furthermore, our software provides theoretical pairwise genetic distance distributions to provide a likelihood of person-to-person transmission based on genomic observations, and using this framework, implements transmission route assessment for genomic data collected during an outbreak. Our open source software provides an accessible platform for users to explore pathogen evolution and outbreak dynamics via simulation, and offers tools to assess observed genomic data in this context.

Prior population immunity reduces the expected impact of CTL-inducing vaccines for pandemic influenza control

Vaccines that trigger an influenza-specific cytotoxic T cell (CTL) response may aid pandemic control by limiting the transmission of novel influenza A viruses (IAV). We consider interventions with hypothetical CTL-inducing vaccines in a range of epidemiologically plausible pandemic scenarios. We estimate the achievable reduction in the attack rate, and, by adopting a model linking epidemic progression to the emergence of IAV variants, the opportunity for antigenic drift. We demonstrate that CTL-inducing vaccines have limited utility for modifying population-level outcomes if influenza-specific T cells found widely in adults already suppress transmission and prove difficult to enhance. Administration of CTL-inducing vaccines that are efficacious in "influenza-experienced" and "influenza-naive" hosts can likely slow transmission sufficiently to mitigate a moderate IAV pandemic. However if neutralising cross-reactive antibody to an emerging IAV are common in influenza-experienced hosts, as for the swine-variant H3N2v, boosting CTL immunity may be ineffective at reducing population spread, indicating that CTL-inducing vaccines are best used against novel subtypes such as H7N9. Unless vaccines cannot readily suppress transmission from infected hosts with naive T cell pools, targeting influenza-naive hosts is preferable. Such strategies are of enhanced benefit if naive hosts are typically intensively mixing children and when a subset of experienced hosts have pre-existing neutralising cross-reactive antibody. We show that CTL-inducing vaccination campaigns may have greater power to suppress antigenic drift than previously suggested, and targeting adults may be the optimal strategy to achieve this when the vaccination campaign does not have the power to curtail the attack rate. Our results highlight the need to design interventions based on pre-existing cellular immunity and knowledge of the host determinants of vaccine efficacy, and provide a framework for assessing the performance requirements of high-impact CTL-inducing vaccines.

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.

Pathogen evolution and the immunological niche

Host immunity is a major driver of pathogen evolution and thus a major determinant of pathogen diversity. Explanations for pathogen diversity traditionally assume simple interactions between pathogens and the immune system, a view encapsulated by the susceptible-infected-recovered (SIR) model. However, there is growing evidence that the complexity of many host-pathogen interactions is dynamically important. This revised perspective requires broadening the definition of a pathogen's immunological phenotype, or what can be thought of as its immunological niche. After reviewing evidence that interactions between pathogens and host immunity drive much of pathogen evolution, I introduce the concept of a pathogen's immunological phenotype. Models that depart from the SIR paradigm demonstrate the utility of this perspective and show that it is particularly useful in understanding vaccine-induced evolution. This paper highlights questions in immunology, evolution, and ecology that must be answered to advance theories of pathogen diversity.

A predictive fitness model for influenza

The seasonal human influenza A/H3N2 virus undergoes rapid evolution, which produces significant year-to-year sequence turnover in the population of circulating strains. Adaptive mutations respond to human immune challenge and occur primarily in antigenic epitopes, the antibody-binding domains of the viral surface protein haemagglutinin. Here we develop a fitness model for haemagglutinin that predicts the evolution of the viral population from one year to the next. Two factors are shown to determine the fitness of a strain: adaptive epitope changes and deleterious mutations outside the epitopes. We infer both fitness components for the strains circulating in a given year, using population-genetic data of all previous strains. From fitness and frequency of each strain, we predict the frequency of its descendent strains in the following year. This fitness model maps the adaptive history of influenza A and suggests a principled method for vaccine selection. Our results call for a more comprehensive epidemiology of influenza and other fast-evolving pathogens that integrates antigenic phenotypes with other viral functions coupled by genetic linkage.

What can mathematical models bring to the control of equine influenza?

Mathematical modelling of infectious disease is increasingly regarded as an important tool in the development of disease prevention and control measures. This article brings together key findings from various modelling studies conducted over the past 10 years that are of relevance to those on the front line of the battle against equine influenza.

The Summary is available in Chinese – see Supporting information.

How the dynamics and structure of sexual contact networks shape pathogen phylogenies

The characteristics of the host contact network over which a pathogen is transmitted affect both epidemic spread and the projected effectiveness of control strategies. Given the importance of understanding these contact networks, it is unfortunate that they are very difficult to measure directly. This challenge has led to an interest in methods to infer information about host contact networks from pathogen phylogenies, because in shaping a pathogen's opportunities for reproduction, contact networks also shape pathogen evolution. Host networks influence pathogen phylogenies both directly, through governing opportunities for evolution, and indirectly by changing the prevalence and incidence. Here, we aim to separate these two effects by comparing pathogen evolution on different host networks that share similar epidemic trajectories. This approach allows use to examine the direct effects of network structure on pathogen phylogenies, largely controlling for confounding differences arising from population dynamics. We find that networks with more heterogeneous degree distributions yield pathogen phylogenies with more variable cluster numbers, smaller mean cluster sizes, shorter mean branch lengths, and somewhat higher tree imbalance than networks with relatively homogeneous degree distributions. However, in particular for dynamic networks, we find that these direct effects are relatively modest. These findings suggest that the role of the epidemic trajectory, the dynamics of the network and the inherent variability of metrics such as cluster size must each be taken into account when trying to use pathogen phylogenies to understand characteristics about the underlying host contact network.

Towards multiscale modeling of influenza infection

Aided by recent advances in computational power, algorithms, and higher fidelity data, increasingly detailed theoretical models of infection with influenza A virus are being developed. We review single scale models as they describe influenza infection from intracellular to global scales, and, in particular, we consider those models that capture details specific to influenza and can be used to link different scales. We discuss the few multiscale models of influenza infection that have been developed in this emerging field. In addition to discussing modeling approaches, we also survey biological data on influenza infection and transmission that is relevant for constructing influenza infection models. We envision that, in the future, multiscale models that capitalize on technical advances in experimental biology and high performance computing could be used to describe the large spatial scale epidemiology of influenza infection, evolution of the virus, and transmission between hosts more accurately.

Viral phylodynamics

Viral phylodynamics is defined as the study of how epidemiological, immunological, and evolutionary processes act and potentially interact to shape viralphylogenies. Since the coining of the term in 2004, research on viral phylodynamics has focused on transmission dynamics in an effort to shed light on how these dynamics impact viral genetic variation. Transmission dynamics can be considered at the level of cells within an infected host, individual hosts within a population, or entire populations of hosts. Many viruses, especially RNA viruses, rapidly accumulate genetic variation because of short generation times and high mutation rates. Patterns of viral genetic variation are therefore heavily influenced by how quickly transmission occurs and by which entities transmit to one another. Patterns of viral genetic variation will also be affected by selection acting on viral phenotypes. Although viruses can differ with respect to many phenotypes, phylodynamic studies have to date tended to focus on a limited number of viral phenotypes. These include virulence phenotypes, phenotypes associated with viral transmissibility, cell or tissue tropism phenotypes, and antigenic phenotypes that can facilitate escape from host immunity. Due to the impact that transmission dynamics and selection can have on viral genetic variation, viral phylogenies can therefore be used to investigate important epidemiological, immunological, and evolutionary processes, such as epidemic spread[2], spatio-temporal dynamics including metapopulation dynamics[3], zoonotic transmission, tissue tropism[4], and antigenic drift[5]. The quantitative investigation of these processes through the consideration of viral phylogenies is the central aim of viral phylodynamics.

Navigating the devious course of evolution: the importance of mechanistic models for identifying eco-evolutionary dynamics in nature

In proposing his genetic feedback mechanism, David Pimentel was one of the first biologists to argue that the reciprocal interplay of ecological and evolutionary dynamics is an important process regulating population dynamics and ultimately affecting community composition. Although the past decade has seen an increase in research activity on these so-called eco-evolutionary dynamics, there remains a conspicuous lack of compelling natural examples of such feedback. Here we argue that this lack may be due to an inherent difficulty in detecting eco-evolutionary dynamics in nature. By examining models of virulence evolution, host resistance evolution, and antigenic evolution, we show that the influence of evolution on ecological dynamics can often be obscured by other ecological processes that yield similar dynamics. We then show, however, that mechanistic models can be used to navigate this, in Pimentel's words, "devious" course of evolution when effectively combined with empirical data. We argue that these models, improving upon Pimentel's original mathematical models, will therefore play an increasingly important role in identifying more subtle, but possibly ubiquitous, eco-evolutionary dynamics in nature. To highlight the importance of identifying these potentially subtle dynamics in nature, we end by considering our ability to anticipate the effect of population control strategies in the presence of these eco-evolutionary feedbacks.

The evolutionary dynamics of receptor binding avidity in influenza A: a mathematical model for a new antigenic drift hypothesis

The most salient feature of influenza evolution in humans is its antigenic drift. This process is characterized by structural changes in the virus's B-cell epitopes and ultimately results in the ability of the virus to evade immune recognition and thereby reinfect previously infected hosts. Until recently, amino acid substitutions in epitope regions of the viral haemagglutinin were thought to be positively selected for their ability to reduce antibody binding and therefore were thought to be responsible for driving antigenic drift. However, a recent hypothesis put forward by Hensley and co-workers posits that cellular receptor binding avidity is the dominant phenotype under selection, with antigenic drift being a side effect of these binding avidity changes. Here, we present a mathematical formulation of this new antigenic drift model and use it to show how rates of antigenic drift depend on epidemiological parameters. We further use the model to evaluate how two different vaccination strategies can impact antigenic drift rates and ultimately disease incidence levels. Finally, we discuss the assumptions present in the model formulation, predictions of the model, and future work that needs to be done to determine the consistency of this hypothesis with known patterns of influenza's genetic and antigenic evolution.

Phylodynamic inference and model assessment with approximate bayesian computation: influenza as a case study

A key priority in infectious disease research is to understand the ecological and evolutionary drivers of viral diseases from data on disease incidence as well as viral genetic and antigenic variation. We propose using a simulation-based, Bayesian method known as Approximate Bayesian Computation (ABC) to fit and assess phylodynamic models that simulate pathogen evolution and ecology against summaries of these data. We illustrate the versatility of the method by analyzing two spatial models describing the phylodynamics of interpandemic human influenza virus subtype A(H3N2). The first model captures antigenic drift phenomenologically with continuously waning immunity, and the second epochal evolution model describes the replacement of major, relatively long-lived antigenic clusters. Combining features of long-term surveillance data from the Netherlands with features of influenza A (H3N2) hemagglutinin gene sequences sampled in northern Europe, key phylodynamic parameters can be estimated with ABC. Goodness-of-fit analyses reveal that the irregularity in interannual incidence and H3N2's ladder-like hemagglutinin phylogeny are quantitatively only reproduced under the epochal evolution model within a spatial context. However, the concomitant incidence dynamics result in a very large reproductive number and are not consistent with empirical estimates of H3N2's population level attack rate. These results demonstrate that the interactions between the evolutionary and ecological processes impose multiple quantitative constraints on the phylodynamic trajectories of influenza A(H3N2), so that sequence and surveillance data can be used synergistically. ABC, one of several data synthesis approaches, can easily interface a broad class of phylodynamic models with various types of data but requires careful calibration of the summaries and tolerance parameters.

The infectious disease dynamics of many viral pathogens like influenza, norovirus and coronavirus are inextricably tied to their evolution. This interaction between evolutionary and ecological processes complicates our ability to understand the infectious disease behavior of rapidly evolving pathogens. Most statistical methods for the analysis of these “phylodynamics” require that the likelihood of the data can be explicitly calculated. Currently, this is not possible for many phylodynamic models, so that questions on the interaction between viral variants cannot be well-addressed within this framework. Simulation-based statistical methods circumvent likelihood calculations. Considering interpandemic human influenza A virus subtype H3N2, we here illustrate the effectiveness of these methods to fit and assess complex phylodynamic models against both sequence and surveillance data. We find that combining molecular genetic and epidemiological data is key to estimate phylodynamic parameters reliably. Moreover, the information in the available data taken together is enough to expose quantitative model inconsistencies. Methods such as ABC which can combine sequence and surveillance data appear to be well-suited to fit and assess mechanistic hypotheses on the phylodynamics of RNA viruses.

Escherichia coli adapts to tetracycline resistance plasmid (pBR322) by mutating endogenous potassium transport: in silico hypothesis testing

Antibiotic resistance exerts a metabolic cost on bacteria and presumably a fitness disadvantage in the absence of antibiotics. However, several studies have shown that bacteria can evolve to eliminate this cost. Escherichia coli can adapt to the plasmid pBR322 carrying the tetA tetracycline-resistance gene (codes for the TetA efflux pump) by a chromosome mutation, which requires an intact tetA gene on the plasmid. The TetA pump can mediate potassium uptake. Here, the hypothesis that TetA replaces the endogenous K(+) uptake system Trk is explored using a multi-level modeling approach that explicitly resolves relevant intracellular processes (e.g., metabolism and K(+) uptake) and simulates individual bacteria in competition. The general behavior of the model is consistent with observations from the literature (e.g., growth rate and K(+) limitation). In competition experiments without tetracycline, the model correctly predicts the fitness advantage of naive susceptible over naive resistant, evolved resistant over naive resistant and evolved resistant over evolved susceptible strains. Trk takes up about 10 times the K(+) required, which costs energy. TetA takes up less K(+) , which is more efficient and leads to the evolution of the Trk mutant. The evolved Trk mutant relies on TetA to take up K(+) , and thus, carrying the plasmid is advantageous even in the absence of the antibiotic.

Clonal interference in the evolution of influenza

The seasonal influenza A virus undergoes rapid evolution to escape human immune response. Adaptive changes occur primarily in antigenic epitopes, the antibody-binding domains of the viral hemagglutinin. This process involves recurrent selective sweeps, in which clusters of simultaneous nucleotide fixations in the hemagglutinin coding sequence are observed about every 4 years. Here, we show that influenza A (H3N2) evolves by strong clonal interference. This mode of evolution is a red queen race between viral strains with different beneficial mutations. Clonal interference explains and quantifies the observed sweep pattern: we find an average of at least one strongly beneficial amino acid substitution per year, and a given selective sweep has three to four driving mutations on average. The inference of selection and clonal interference is based on frequency time series of single-nucleotide polymorphisms, which are obtained from a sample of influenza genome sequences over 39 years. Our results imply that mode and speed of influenza evolution are governed not only by positive selection within, but also by background selection outside antigenic epitopes: immune adaptation and conservation of other viral functions interfere with each other. Hence, adapting viral proteins are predicted to be particularly brittle. We conclude that a quantitative understanding of influenza's evolutionary and epidemiological dynamics must be based on all genomic domains and functions coupled by clonal interference.

The impact of host immune status on the within-host and population dynamics of antigenic immune escape

Antigenically evolving pathogens such as influenza viruses are difficult to control owing to their ability to evade host immunity by producing immune escape variants. Experimental studies have repeatedly demonstrated that viral immune escape variants emerge more often from immunized hosts than from naive hosts. This empirical relationship between host immune status and within-host immune escape is not fully understood theoretically, nor has its impact on antigenic evolution at the population level been evaluated. Here, we show that this relationship can be understood as a trade-off between the probability that a new antigenic variant is produced and the level of viraemia it reaches within a host. Scaling up this intra-host level trade-off to a simple population level model, we obtain a distribution for variant persistence times that is consistent with influenza A/H3N2 antigenic variant data. At the within-host level, our results show that target cell limitation, or a functional equivalent, provides a parsimonious explanation for how host immune status drives the generation of immune escape mutants. At the population level, our analysis also offers an alternative explanation for the observed tempo of antigenic evolution, namely that the production rate of immune escape variants is driven by the accumulation of herd immunity. Overall, our results suggest that disease control strategies should be further assessed by considering the impact that increased immunity--through vaccination--has on the production of new antigenic variants.

Canalization of the evolutionary trajectory of the human influenza virus

BACKGROUND

Since its emergence in 1968, influenza A (H3N2) has evolved extensively in genotype and antigenic phenotype. However, despite strong pressure to evolve away from human immunity and to diversify in antigenic phenotype, H3N2 influenza shows paradoxically limited genetic and antigenic diversity present at any one time. Here, we propose a simple model of antigenic evolution in the influenza virus that accounts for this apparent discrepancy.

RESULTS

In this model, antigenic phenotype is represented by a N-dimensional vector, and virus mutations perturb phenotype within this continuous Euclidean space. We implement this model in a large-scale individual-based simulation, and in doing so, we find a remarkable correspondence between model behavior and observed influenza dynamics. This model displays rapid evolution but low standing diversity and simultaneously accounts for the epidemiological, genetic, antigenic, and geographical patterns displayed by the virus. We find that evolution away from existing human immunity results in rapid population turnover in the influenza virus and that this population turnover occurs primarily along a single antigenic axis.

CONCLUSIONS

Selective dynamics induce a canalized evolutionary trajectory, in which the evolutionary fate of the influenza population is surprisingly repeatable. In the model, the influenza population shows a 1- to 2-year timescale of repeatability, suggesting a window in which evolutionary dynamics could be, in theory, predictable.

A simple stochastic model with environmental transmission explains multi-year periodicity in outbreaks of avian flu

Avian influenza virus reveals persistent and recurrent outbreaks in North American wild waterfowl, and exhibits major outbreaks at 2–8 years intervals in duck populations. The standard susceptible-infected- recovered (SIR) framework, which includes seasonal migration and reproduction, but lacks environmental transmission, is unable to reproduce the multi-periodic patterns of avian influenza epidemics. In this paper, we argue that a fully stochastic theory based on environmental transmission provides a simple, plausible explanation for the phenomenon of multi-year periodic outbreaks of avian flu. Our theory predicts complex fluctuations with a dominant period of 2 to 8 years which essentially depends on the intensity of environmental transmission. A wavelet analysis of the observed data supports this prediction. Furthermore, using master equations and van Kampen system-size expansion techniques, we provide an analytical expression for the spectrum of stochastic fluctuations, revealing how the outbreak period varies with the environmental transmission.

Impact of cross-protective vaccines on epidemiological and evolutionary dynamics of influenza

Large-scale immunization has profoundly impacted control of many infectious diseases such as measles and smallpox because of the ability of vaccination campaigns to maintain long-term herd immunity and, hence, indirect protection of the unvaccinated. In the case of human influenza, such potential benefits of mass vaccination have so far proved elusive. The central difficulty is a considerable viral capacity for immune escape; new pandemic variants, as well as viral escape mutants in seasonal influenza, compromise the buildup of herd immunity from natural infection or deployment of current vaccines. Consequently, most current influenza vaccination programs focus mainly on protection of specific risk groups, rather than mass prophylactic protection. Here, we use epidemiological models to show that emerging vaccine technologies, aimed at broad-spectrum protection, could qualitatively alter this picture. We demonstrate that sustained immunization with such vaccines could--through potentially lowering transmission rates and improving herd immunity--significantly moderate both influenza pandemic and seasonal epidemics. More subtly, phylodynamic models indicate that widespread cross-protective immunization could slow the antigenic evolution of seasonal influenza; these effects have profound implications for a transition to mass vaccination strategies against human influenza, and for the management of antigenically variable viruses in general.

Mathematical Modeling of Viral Zoonoses in Wildlife

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.

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.

A dimensionless number for understanding the evolutionary dynamics of antigenically variable RNA viruses

Antigenically variable RNA viruses are significant contributors to the burden of infectious disease worldwide. One reason for their ubiquity is their ability to escape herd immunity through rapid antigenic evolution and thereby to reinfect previously infected hosts. However, the ways in which these viruses evolve antigenically are highly diverse. Some have only limited diversity in the long-run, with every emergence of a new antigenic variant coupled with a replacement of the older variant. Other viruses rapidly accumulate antigenic diversity over time. Others still exhibit dynamics that can be considered evolutionary intermediates between these two extremes. Here, we present a theoretical framework that aims to understand these differences in evolutionary patterns by considering a virus's epidemiological dynamics in a given host population. Our framework, based on a dimensionless number, probabilistically anticipates patterns of viral antigenic diversification and thereby quantifies a virus's evolutionary potential. It is therefore similar in spirit to the basic reproduction number, the well-known dimensionless number which quantifies a pathogen's reproductive potential. We further outline how our theoretical framework can be applied to empirical viral systems, using influenza A/H3N2 as a case study. We end with predictions of our framework and work that remains to be done to further integrate viral evolutionary dynamics with disease ecology.

Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection

Background

Recent studies have suggested that vaccination with seasonal influenza vaccine resulted in an apparent higher risk of infection with pandemic influenza H1N1 2009. A simple mathematical model incorporating strain competition and a hypothesised temporary strain-transcending immunity is constructed to investigate this observation. The model assumes that seasonal vaccine has no effect on the risk of infection with pandemic influenza.

Results

Results of the model over a range of reproduction numbers and effective vaccination coverage confirm this apparent increased risk in the Northern, but not the Southern, hemisphere. This is due to unvaccinated individuals being more likely to be infected with seasonal influenza (if it is circulating) and developing hypothesised temporary immunity to the pandemic strain. Because vaccinated individuals are less likely to have been infected with seasonal influenza, they are less likely to have developed the hypothesised temporary immunity and are therefore more likely to be infected with pandemic influenza. If the reproduction number for pandemic influenza is increased, as it is for children, an increase in the apparent risk of seasonal vaccination is observed. The maximum apparent risk effect is found when seasonal vaccination coverage is in the range 20-40%.

Conclusions

Only when pandemic influenza is recently preceded by seasonal influenza circulation is there a modelled increased risk of pandemic influenza infection associated with prior receipt of seasonal vaccine.

The impact of seasonal and year-round transmission regimes on the evolution of influenza A virus

Punctuated antigenic change is believed to be a key element in the evolution of influenza A; clusters of antigenically similar strains predominate worldwide for several years until an antigenically distant mutant emerges and instigates a selective sweep. It is thought that a region of East-Southeast Asia with year-round transmission acts as a source of antigenic diversity for influenza A and seasonal epidemics in temperate regions make little contribution to antigenic evolution. We use a mathematical model to examine how different transmission regimes affect the evolutionary dynamics of influenza over the lifespan of an antigenic cluster. Our model indicates that, in non-seasonal regions, mutants that cause significant outbreaks appear before the peak of the wild-type epidemic. A relatively large proportion of these mutants spread globally. In seasonal regions, mutants that cause significant local outbreaks appear each year before the seasonal peak of the wild-type epidemic, but only a small proportion spread globally. The potential for global spread is strongly influenced by the intensity of non-seasonal circulation and coupling between non-seasonal and seasonal regions. Results are similar if mutations are neutral, or confer a weak to moderate antigenic advantage. However, there is a threshold antigenic advantage, depending on the non-seasonal transmission intensity, beyond which mutants can escape herd immunity in the non-seasonal region and there is a global explosion in diversity. We conclude that non-seasonal transmission regions are fundamental to the generation and maintenance of influenza diversity owing to their epidemiology. More extensive sampling of viral diversity in such regions could facilitate earlier identification of antigenically novel strains and extend the critical window for vaccine development.