Waning immunity can drive repeated waves of infections

In infectious disease models, it is known that mechanisms such as births, seasonality in transmission and pathogen evolution can generate oscillations in infection numbers. We show how waning immunity is also a mechanism that is sufficient on its own to enable sustained oscillations. When previously infected or vaccinated individuals lose full protective immunity, they become partially susceptible to reinfections. This partial immunity subsequently wanes over time, making individuals more susceptible to reinfections and potentially more infectious if infected. Losses of full and partial immunity lead to a surge in infections, which is the precursor of oscillations. We present a discrete-time Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) model that features the waning of fully immune individuals (as a distribution of time at which individuals lose fully immunity) and the gradual loss of partial immunity (as increases in susceptibility and potential infectiousness over time). A special case of SIRWY is the discrete-time SIRS model with geometric distributions for waning and recovery. Its continuous-time analogue is the classic SIRS with exponential distributions, which does not produce sustained oscillations for any choice of parameters. We show that the discrete-time version can produce sustained oscillations and that the oscillatory regime disappears as discrete-time tends to continuous-time. A different special case of SIRWY is one with fixed times for waning and recovery. We show that this simpler model can also produce sustained oscillations. In conclusion, under certain feature and parameter choices relating to how exactly immunity wanes, fluctuations in infection numbers can be sustained without the need for any additional mechanisms.

A novel approach to finding conserved features in low-variability gene alignments characterises RNA motifs in SARS-CoV and SARS-CoV-2

Collections of genetic sequences belonging to related organisms contain information on the evolutionary constraints to which the organisms have been subjected. Heavily constrained regions can be investigated to understand their roles in an organism's life cycle, and drugs can be sought to disrupt these roles. In organisms with low genetic diversity, such as newly-emerged pathogens, it is key to obtain this information early to develop new treatments. Here, we present methods that ensure we can leverage all the information available in a low-signal, low-noise set of sequences, to find contiguous regions of relatively conserved nucleic acid. We demonstrate the application of these methods by analysing over 5 million genome sequences of the recently-emerged RNA virus SARS-CoV-2 and correlating these results with an analysis of 119 genome sequences of SARS-CoV. We propose the precise location of a previously described packaging signal, and discuss explanations for other regions of high conservation.

The importance of vaccinated individuals to population-level evolution of pathogens

Virus evolution shapes the epidemiological patterns of infectious disease, particularly via evasion of population immunity. At the individual level, host immunity itself may drive viral evolution towards antigenic escape. Using compartmental SIR-style models with imperfect vaccination, we allow the probability of immune escape to differ in vaccinated and unvaccinated hosts. As the relative contribution to selection in these different hosts varies, the overall effect of vaccination on the antigenic escape pressure at the population level changes. We find that this relative contribution to escape is important for understanding the effects of vaccination on the escape pressure and we draw out some fairly general patterns. If vaccinated hosts do not contribute much more than unvaccinated hosts to the escape pressure, then increasing vaccination always reduces the overall escape pressure. In contrast, if vaccinated hosts contribute significantly more than unvaccinated hosts to the population level escape pressure, then the escape pressure is maximised for intermediate vaccination levels. Past studies find only that the escape pressure is maximal for intermediate levels with fixed extreme assumptions about this relative contribution. Here we show that this result does not hold across the range of plausible assumptions for the relative contribution to escape from vaccinated and unvaccinated hosts. We also find that these results depend on the vaccine efficacy against transmission, particularly through the partial protection against infection. This work highlights the potential value of understanding better how the contribution to antigenic escape pressure depends on individual host immunity.

Coughs, colds and "freshers' flu" survey in the University of Cambridge, 2007-2008

Universities provide many opportunities for the spread of infectious respiratory illnesses. Students are brought together into close proximity from all across the world and interact with one another in their accommodation, through lectures and small group teaching and in social settings. The COVID-19 global pandemic has highlighted the need for sufficient data to help determine which of these factors are important for infectious disease transmission in universities and hence control university morbidity as well as community spillover. We describe the data from a previously unpublished self-reported university survey of coughs, colds and influenza-like symptoms collected in Cambridge, UK, during winter 2007-2008. The online survey collected information on symptoms and socio-demographic, academic and lifestyle factors. There were 1076 responses, 97% from University of Cambridge students (5.7% of the total university student population), 3% from staff and <1% from other participants, reporting onset of symptoms between September 2007 and March 2008. Undergraduates are seen to report symptoms earlier in the term than postgraduates; differences in reported date of symptoms are also seen between subjects and accommodation types, although these descriptive results could be confounded by survey biases. Despite the historical and exploratory nature of the study, this is one of few recent detailed datasets of influenza-like infection in a university context and is especially valuable to share now to improve understanding of potential transmission dynamics in universities during the current COVID-19 pandemic.

Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Commentary on the use of the reproduction number R during the COVID-19 pandemic

Since the beginning of the COVID-19 pandemic, the reproduction number [Formula: see text] has become a popular epidemiological metric used to communicate the state of the epidemic. At its most basic, [Formula: see text] is defined as the average number of secondary infections caused by one primary infected individual. [Formula: see text] seems convenient, because the epidemic is expanding if [Formula: see text] and contracting if [Formula: see text]. The magnitude of [Formula: see text] indicates by how much transmission needs to be reduced to control the epidemic. Using [Formula: see text] in a naïve way can cause new problems. The reasons for this are threefold: (1) There is not just one definition of [Formula: see text] but many, and the precise definition of [Formula: see text] affects both its estimated value and how it should be interpreted. (2) Even with a particular clearly defined [Formula: see text], there may be different statistical methods used to estimate its value, and the choice of method will affect the estimate. (3) The availability and type of data used to estimate [Formula: see text] vary, and it is not always clear what data should be included in the estimation. In this review, we discuss when [Formula: see text] is useful, when it may be of use but needs to be interpreted with care, and when it may be an inappropriate indicator of the progress of the epidemic. We also argue that careful definition of [Formula: see text], and the data and methods used to estimate it, can make [Formula: see text] a more useful metric for future management of the epidemic.

SARS-CoV-2 infection in UK university students: lessons from September-December 2020 and modelling insights for future student return

In this paper, we present work on SARS-CoV-2 transmission in UK higher education settings using multiple approaches to assess the extent of university outbreaks, how much those outbreaks may have led to spillover in the community, and the expected effects of control measures. Firstly, we found that the distribution of outbreaks in universities in late 2020 was consistent with the expected importation of infection from arriving students. Considering outbreaks at one university, larger halls of residence posed higher risks for transmission. The dynamics of transmission from university outbreaks to wider communities is complex, and while sometimes spillover does occur, occasionally even large outbreaks do not give any detectable signal of spillover to the local population. Secondly, we explored proposed control measures for reopening and keeping open universities. We found the proposal of staggering the return of students to university residence is of limited value in terms of reducing transmission. We show that student adherence to testing and self-isolation is likely to be much more important for reducing transmission during term time. Finally, we explored strategies for testing students in the context of a more transmissible variant and found that frequent testing would be necessary to prevent a major outbreak.

Vaccine escape in a heterogeneous population: insights for SARS-CoV-2 from a simple model

As a countermeasure to the SARS-CoV-2 pandemic, there has been swift development and clinical trial assessment of candidate vaccines, with subsequent deployment as part of mass vaccination campaigns. However, the SARS-CoV-2 virus has demonstrated the ability to mutate and develop variants, which can modify epidemiological properties and potentially also the effectiveness of vaccines. The widespread deployment of highly effective vaccines may rapidly exert selection pressure on the SARS-CoV-2 virus directed towards mutations that escape the vaccine-induced immune response. This is particularly concerning while infection is widespread. By developing and analysing a mathematical model of two population groupings with differing vulnerability and contact rates, we explore the impact of the deployment of vaccines among the population on the reproduction ratio, cases, disease abundance and vaccine escape pressure. The results from this model illustrate two insights: (i) vaccination aimed at reducing prevalence could be more effective at reducing disease than directly vaccinating the vulnerable; (ii) the highest risk for vaccine escape can occur at intermediate levels of vaccination. This work demonstrates a key principle: the careful targeting of vaccines towards particular population groups could reduce disease as much as possible while limiting the risk of vaccine escape.

Epidemic interventions: insights from classic results

Analytical expressions and approximations from simple models have performed a pivotal role in our understanding of infectious disease epidemiology. During the current COVID-19 pandemic, while there has been proliferation of increasingly complex models, still the most basic models have provided the core framework for our thinking and interpreting policy decisions. Here, classic results are presented that give insights into both the role of transmission-reducing interventions (such as social distancing) in controlling an emerging epidemic, and also what would happen if insufficient control is applied. Though these are simple results from the most basic of epidemic models, they give valuable benchmarks for comparison with the outputs of more complex modelling approaches. This article is part of the theme issue 'Modelling that shaped the early COVID-19 pandemic response in the UK'.

Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study

BACKGROUND

The isolation of symptomatic cases and tracing of contacts has been used as an early COVID-19 containment measure in many countries, with additional physical distancing measures also introduced as outbreaks have grown. To maintain control of infection while also reducing disruption to populations, there is a need to understand what combination of measures-including novel digital tracing approaches and less intensive physical distancing-might be required to reduce transmission. We aimed to estimate the reduction in transmission under different control measures across settings and how many contacts would be quarantined per day in different strategies for a given level of symptomatic case incidence.

METHODS

For this mathematical modelling study, we used a model of individual-level transmission stratified by setting (household, work, school, or other) based on BBC Pandemic data from 40 162 UK participants. We simulated the effect of a range of different testing, isolation, tracing, and physical distancing scenarios. Under optimistic but plausible assumptions, we estimated reduction in the effective reproduction number and the number of contacts that would be newly quarantined each day under different strategies.

RESULTS

We estimated that combined isolation and tracing strategies would reduce transmission more than mass testing or self-isolation alone: mean transmission reduction of 2% for mass random testing of 5% of the population each week, 29% for self-isolation alone of symptomatic cases within the household, 35% for self-isolation alone outside the household, 37% for self-isolation plus household quarantine, 64% for self-isolation and household quarantine with the addition of manual contact tracing of all contacts, 57% with the addition of manual tracing of acquaintances only, and 47% with the addition of app-based tracing only. If limits were placed on gatherings outside of home, school, or work, then manual contact tracing of acquaintances alone could have an effect on transmission reduction similar to that of detailed contact tracing. In a scenario where 1000 new symptomatic cases that met the definition to trigger contact tracing occurred per day, we estimated that, in most contact tracing strategies, 15 000-41 000 contacts would be newly quarantined each day.

INTERPRETATION

Consistent with previous modelling studies and country-specific COVID-19 responses to date, our analysis estimated that a high proportion of cases would need to self-isolate and a high proportion of their contacts to be successfully traced to ensure an effective reproduction number lower than 1 in the absence of other measures. If combined with moderate physical distancing measures, self-isolation and contact tracing would be more likely to achieve control of severe acute respiratory syndrome coronavirus 2 transmission.

FUNDING

Wellcome Trust, UK Engineering and Physical Sciences Research Council, European Commission, Royal Society, Medical Research Council.

Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study

BACKGROUND

The isolation of symptomatic cases and tracing of contacts has been used as an early COVID-19 containment measure in many countries, with additional physical distancing measures also introduced as outbreaks have grown. To maintain control of infection while also reducing disruption to populations, there is a need to understand what combination of measures-including novel digital tracing approaches and less intensive physical distancing-might be required to reduce transmission. We aimed to estimate the reduction in transmission under different control measures across settings and how many contacts would be quarantined per day in different strategies for a given level of symptomatic case incidence.

METHODS

For this mathematical modelling study, we used a model of individual-level transmission stratified by setting (household, work, school, or other) based on BBC Pandemic data from 40 162 UK participants. We simulated the effect of a range of different testing, isolation, tracing, and physical distancing scenarios. Under optimistic but plausible assumptions, we estimated reduction in the effective reproduction number and the number of contacts that would be newly quarantined each day under different strategies.

RESULTS

We estimated that combined isolation and tracing strategies would reduce transmission more than mass testing or self-isolation alone: mean transmission reduction of 2% for mass random testing of 5% of the population each week, 29% for self-isolation alone of symptomatic cases within the household, 35% for self-isolation alone outside the household, 37% for self-isolation plus household quarantine, 64% for self-isolation and household quarantine with the addition of manual contact tracing of all contacts, 57% with the addition of manual tracing of acquaintances only, and 47% with the addition of app-based tracing only. If limits were placed on gatherings outside of home, school, or work, then manual contact tracing of acquaintances alone could have an effect on transmission reduction similar to that of detailed contact tracing. In a scenario where 1000 new symptomatic cases that met the definition to trigger contact tracing occurred per day, we estimated that, in most contact tracing strategies, 15 000-41 000 contacts would be newly quarantined each day.

INTERPRETATION

Consistent with previous modelling studies and country-specific COVID-19 responses to date, our analysis estimated that a high proportion of cases would need to self-isolate and a high proportion of their contacts to be successfully traced to ensure an effective reproduction number lower than 1 in the absence of other measures. If combined with moderate physical distancing measures, self-isolation and contact tracing would be more likely to achieve control of severe acute respiratory syndrome coronavirus 2 transmission.

FUNDING

Wellcome Trust, UK Engineering and Physical Sciences Research Council, European Commission, Royal Society, Medical Research Council.

Key questions for modelling COVID-19 exit strategies

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

Symbolic transfer entropy reveals the age structure of pandemic influenza transmission from high-volume influenza-like illness data

Existing methods to infer the relative roles of age groups in epidemic transmission can normally only accommodate a few age classes, and/or require data that are highly specific for the disease being studied. Here, symbolic transfer entropy (STE), a measure developed to identify asymmetric transfer of information between stochastic processes, is presented as a way to reveal asymmetric transmission patterns between age groups in an epidemic. STE provides a ranking of which age groups may dominate transmission, rather than a reconstruction of the explicit between-age-group transmission matrix. Using simulations, we establish that STE can identify which age groups dominate transmission even when there are differences in reporting rates between age groups and even if the data are noisy. Then, the pairwise STE is calculated between time series of influenza-like illness for 12 age groups in 884 US cities during the autumn of 2009. Elevated STE from 5 to 19 year-olds indicates that school-aged children were likely the most important transmitters of infection during the autumn wave of the 2009 pandemic in the USA. The results may be partially confounded by higher rates of physician-seeking behaviour in children compared to adults, but it is unlikely that differences in reporting rates can explain the observed differences in STE.

How you can help with COVID-19 modelling

Many physicists want to use their mathematical modelling skills to study the COVID-19 pandemic. Julia Gog, a mathematical epidemiologist, explains some ways to contribute.

A scale-free analysis of the HIV-1 genome demonstrates multiple conserved regions of structural and functional importance

HIV-1 replicates via a low-fidelity polymerase with a high mutation rate; strong conservation of individual nucleotides is highly indicative of the presence of critical structural or functional properties. Identifying such conservation can reveal novel insights into viral behaviour. We analysed 3651 publicly available sequences for the presence of nucleic acid conservation beyond that required by amino acid constraints, using a novel scale-free method that identifies regions of outlying score together with a codon scoring algorithm. Sequences with outlying score were further analysed using an algorithm for producing local RNA folds whilst accounting for alignment properties. 11 different conserved regions were identified, some corresponding to well-known cis-acting functions of the HIV-1 genome but also others whose conservation has not previously been noted. We identify rational causes for many of these, including cis functions, possible additional reading frame usage, a plausible mechanism by which the central polypurine tract primes second-strand DNA synthesis and a conformational stabilising function of a region at the 5′ end of env.

HIV-1 is a very rapidly mutating organism, however some parts of its genetic material change more than others. We looked for coding regions of HIV-1 that change relatively little, by turning the problem of finding such regions into a problem in signal processing, and solving this using a novel analytical approach that we recently described. We investigated why the regions we identified change less, including using the genetic code in the regions we found to prime an algorithm to predict their structures. In some cases there are already known functions for the features we found, in others they provide new insights into the properties of known regions, and in some cases we identify new regions that vary less for as yet unknown functional reasons.

Urbanization and humidity shape the intensity of influenza epidemics in U.S. cities

Influenza epidemics vary in intensity from year to year, driven by climatic conditions and by viral antigenic evolution. However, important spatial variation remains unexplained. Here we show predictable differences in influenza incidence among cities, driven by population size and structure. Weekly incidence data from 603 cities in the United States reveal that epidemics in smaller cities are focused on shorter periods of the influenza season, whereas in larger cities, incidence is more diffuse. Base transmission potential estimated from city-level incidence data is positively correlated with population size and with spatiotemporal organization in population density, indicating a milder response to climate forcing in metropolises. This suggests that urban centers incubate critical chains of transmission outside of peak climatic conditions, altering the spatiotemporal geometry of herd immunity.

Geographic transmission hubs of the 2009 influenza pandemic in the United States

A key issue in infectious disease epidemiology is to identify and predict geographic sites of epidemic establishment that contribute to onward spread, especially in the context of invasion waves of emerging pathogens. Conventional wisdom suggests that these sites are likely to be in densely-populated, well-connected areas. For pandemic influenza, however, epidemiological data have not been available at a fine enough geographic resolution to test this assumption. Here, we make use of fine-scale influenza-like illness incidence data derived from electronic medical claims records gathered from 834 3-digit ZIP (postal) codes across the US to identify the key geographic establishment sites, or "hubs", of the autumn wave of the 2009 A/H1N1pdm influenza pandemic in the United States. A mechanistic spatial transmission model is fit to epidemic onset times inferred from the data. Hubs are identified by tracing the most probable transmission routes back to a likely first establishment site. Four hubs are identified: two in the southeastern US, one in the central valley of California, and one in the midwestern US. According to the model, 75% of the 834 observed ZIP-level outbreaks in the US were seeded by these four hubs or their epidemiological descendants. Counter-intuitively, the pandemic hubs do not coincide with large and well-connected cities, indicating that factors beyond population density and travel volume are necessary to explain the establishment sites of the major autumn wave of the pandemic. Geographic regions are identified where infection can be statistically traced back to a hub, providing a testable prediction of the outbreak's phylogeography. Our method therefore provides an important way forward to reconcile spatial diffusion patterns inferred from epidemiological surveillance data and pathogen sequence data.

A new method for detecting signal regions in ordered sequences of real numbers, and application to viral genomic data

We present a fast, robust and parsimonious approach to detecting signals in an ordered sequence of numbers. Our motivation is in seeking a suitable method to take a sequence of scores corresponding to properties of positions in virus genomes, and find outlying regions of low scores. Suitable statistical methods without using complex models or making many assumptions are surprisingly lacking. We resolve this by developing a method that detects regions of low score within sequences of real numbers. The method makes no assumptions a priori about the length of such a region; it gives the explicit location of the region and scores it statistically. It does not use detailed mechanistic models so the method is fast and will be useful in a wide range of applications. We present our approach in detail, and test it on simulated sequences. We show that it is robust to a wide range of signal morphologies, and that it is able to capture multiple signals in the same sequence. Finally we apply it to viral genomic data to identify regions of evolutionary conservation within influenza and rotavirus.

Infectious Disease Surveillance in the Big Data Era: Towards Faster and Locally Relevant Systems

While big data have proven immensely useful in fields such as marketing and earth sciences, public health is still relying on more traditional surveillance systems and awaiting the fruits of a big data revolution. A new generation of big data surveillance systems is needed to achieve rapid, flexible, and local tracking of infectious diseases, especially for emerging pathogens. In this opinion piece, we reflect on the long and distinguished history of disease surveillance and discuss recent developments related to use of big data. We start with a brief review of traditional systems relying on clinical and laboratory reports. We then examine how large-volume medical claims data can, with great spatiotemporal resolution, help elucidate local disease patterns. Finally, we review efforts to develop surveillance systems based on digital and social data streams, including the recent rise and fall of Google Flu Trends. We conclude by advocating for increased use of hybrid systems combining information from traditional surveillance and big data sources, which seems the most promising option moving forward. Throughout the article, we use influenza as an exemplar of an emerging and reemerging infection which has traditionally been considered a model system for surveillance and modeling.

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.

Eight challenges in phylodynamic inference

The field of phylodynamics, which attempts to enhance our understanding of infectious disease dynamics using pathogen phylogenies, has made great strides in the past decade. Basic epidemiological and evolutionary models are now well characterized with inferential frameworks in place. However, significant challenges remain in extending phylodynamic inference to more complex systems. These challenges include accounting for evolutionary complexities such as changing mutation rates, selection, reassortment, and recombination, as well as epidemiological complexities such as stochastic population dynamics, host population structure, and different patterns at the within-host and between-host scales. An additional challenge exists in making efficient inferences from an ever increasing corpus of sequence data.

Seven challenges in modeling pathogen dynamics within-host and across scales

The population dynamics of infectious disease is a mature field in terms of theory and to some extent, application. However for microparasites, the theory and application of models of the dynamics within a single infected host is still an open field. Further, connecting across the scales--from cellular to host level, to population level--has potential to vastly improve our understanding of pathogen dynamics and evolution. Here, we highlight seven challenges in the following areas: transmission bottlenecks, heterogeneity within host, dynamic fitness landscapes within hosts, making use of next-generation sequencing data, capturing superinfection and when and how to model more than two scales.

Spatial Transmission of 2009 Pandemic Influenza in the US

The 2009 H1N1 influenza pandemic provides a unique opportunity for detailed examination of the spatial dynamics of an emerging pathogen. In the US, the pandemic was characterized by substantial geographical heterogeneity: the 2009 spring wave was limited mainly to northeastern cities while the larger fall wave affected the whole country. Here we use finely resolved spatial and temporal influenza disease data based on electronic medical claims to explore the spread of the fall pandemic wave across 271 US cities and associated suburban areas. We document a clear spatial pattern in the timing of onset of the fall wave, starting in southeastern cities and spreading outwards over a period of three months. We use mechanistic models to tease apart the external factors associated with the timing of the fall wave arrival: differential seeding events linked to demographic factors, school opening dates, absolute humidity, prior immunity from the spring wave, spatial diffusion, and their interactions. Although the onset of the fall wave was correlated with school openings as previously reported, models including spatial spread alone resulted in better fit. The best model had a combination of the two. Absolute humidity or prior exposure during the spring wave did not improve the fit and population size only played a weak role. In conclusion, the protracted spread of pandemic influenza in fall 2009 in the US was dominated by short-distance spatial spread partially catalysed by school openings rather than long-distance transmission events. This is in contrast to the rapid hierarchical transmission patterns previously described for seasonal influenza. The findings underline the critical role that school-age children play in facilitating the geographic spread of pandemic influenza and highlight the need for further information on the movement and mixing patterns of this age group.

The determinants of influenza spatial spread are not fully understood, in part due to the insufficient geographic resolution of incidence data. We address this using a fine-grained private sector electronic health database of insurance claims data from health encounters in the US during 2009. We used physician diagnoses codes to generate a dataset of the weekly number of office visits with diagnosed influenza-like illness for 271 US locations. Applying statistical and mathematical models to these disease data, we find that the main autumn wave of the 2009 pandemic in the US was remarkably spatially structured. Its onset in the South Eastern US precipitated a slow radial spread that took 3 months to diffuse across the country. These patterns were replicated by models that included short-distance spatial transmission between nearby locations and increased transmission rates when school was in session. Our results contrast with previous modelling studies that indicated that environmental factors, population sizes, and long-distance transmission events (air traffic) are major determinants in disease spread. We conclude that the 2009 pandemic autumn wave spread slowly because transmissibility of the influenza virus was relatively low and children (who travel long distance far less than adults) were the predominant sources of infection.

Identification of a novel splice variant form of the influenza A virus M2 ion channel with an antigenically distinct ectodomain

Segment 7 of influenza A virus produces up to four mRNAs. Unspliced transcripts encode M1, spliced mRNA2 encodes the M2 ion channel, while protein products from spliced mRNAs 3 and 4 have not previously been identified. The M2 protein plays important roles in virus entry and assembly, and is a target for antiviral drugs and vaccination. Surprisingly, M2 is not essential for virus replication in a laboratory setting, although its loss attenuates the virus. To better understand how IAV might replicate without M2, we studied the reversion mechanism of an M2-null virus. Serial passage of a virus lacking the mRNA2 splice donor site identified a single nucleotide pseudoreverting mutation, which restored growth in cell culture and virulence in mice by upregulating mRNA4 synthesis rather than by reinstating mRNA2 production. We show that mRNA4 encodes a novel M2-related protein (designated M42) with an antigenically distinct ectodomain that can functionally replace M2 despite showing clear differences in intracellular localisation, being largely retained in the Golgi compartment. We also show that the expression of two distinct ion channel proteins is not unique to laboratory-adapted viruses but, most notably, was also a feature of the 1983 North American outbreak of H5N2 highly pathogenic avian influenza virus. In identifying a 14th influenza A polypeptide, our data reinforce the unexpectedly high coding capacity of the viral genome and have implications for virus evolution, as well as for understanding the role of M2 in the virus life cycle.

Influenza A virus is a pathogen capable of infecting a wide range of avian and mammalian hosts, causing seasonal epidemics and pandemics in humans. In recent years, the unexpected coding capacity of the virus has begun to be unravelled, with the identification of three more protein products (PB1-F2, PB1-N40 and PA-X) on top of the 10 viral proteins originally identified 30 years ago. Here, we identify a 14th primary translation product, made from segment 7. Previously established protein products from segment 7 include the matrix (M1) and ion channel (M2) proteins. M2, made from a spliced transcript, has multiple roles in the virus lifecycle including in entry and budding. In a laboratory setting, it is possible to generate M2 deficient viruses, but these are highly attenuated. However, upon serial passage a virus lacking the M2 splice donor site quickly recovered wild type growth properties, without reverting the original mutation. Instead we found a compensatory single nucleotide mutation had upregulated another segment 7 mRNA. This mRNA encoded a novel M2-like protein with a variant extracellular domain, which we called M42. M42 compensated for loss of M2 in tissue culture cells and animals, although it displayed some differences in subcellular localisation. Our study therefore identifies a further novel influenza protein and gives insights into the evolution of the virus.

The role of social contacts and original antigenic sin in shaping the age pattern of immunity to seasonal influenza

Recent serological studies of seasonal influenza A in humans suggest a striking characteristic profile of immunity against age, which holds across different countries and against different subtypes of influenza. For both H1N1 and H3N2, the proportion of the population seropositive to recently circulated strains peaks in school-age children, reaches a minimum between ages 35–65, then rises again in the older ages. This pattern is little understood. Variable mixing between different age classes can have a profound effect on disease dynamics, and is hence the obvious candidate explanation for the profile, but using a mathematical model of multiple influenza strains, we see that age dependent transmission based on mixing data from social contact surveys cannot on its own explain the observed pattern. Instead, the number of seropositive individuals in a population may be a consequence of ‘original antigenic sin’; if the first infection of a lifetime dominates subsequent immune responses, we demonstrate that it is possible to reproduce the observed relationship between age and seroprevalence. We propose a candidate mechanism for this relationship, by which original antigenic sin, along with antigenic drift and vaccination, results in the age profile of immunity seen in empirical studies.

The way in which a population builds immunity to influenza affects outbreak size and the emergence of new strains. However, although age-specific immunity has been widely discussed for the 2009 influenza pandemic, the age profile of immunity to seasonal influenza remains little understood. In contrast to many infections, the proportion of people immune to recent strains peaks in school-age children then reaches a minimum between ages 35–65, before rising again in older age groups. Our results suggest that rather than variable mixing between different age groups being solely responsible, the pattern may be shaped by an effect known as ‘original antigenic sin’, by which the first infection of a lifetime dictates subsequent immune responses: instead of developing antibodies to every new virus that is encountered, the immune system may reuse the response to a similar virus it has already seen. The framework we describe, which extends theoretical models to allow for comparison with data, also opens the possibility of investigating the mechanisms behind patterns of immunity to other evolving pathogens.

Quantifying transmission of highly pathogenic and low pathogenicity H7N1 avian influenza in turkeys

Outbreaks of avian influenza in poultry can be devastating, yet many of the basic epidemiological parameters have not been accurately characterised. In 1999-2000 in Northern Italy, outbreaks of H7N1 low pathogenicity avian influenza virus (LPAI) were followed by the emergence of H7N1 highly pathogenic avian influenza virus (HPAI). This study investigates the transmission dynamics in turkeys of representative HPAI and LPAI H7N1 virus strains from this outbreak in an experimental setting, allowing direct comparison of the two strains. The fitted transmission rates for the two strains are similar: 2.04 (1.5-2.7) per day for HPAI, 2.01 (1.6-2.5) per day for LPAI. However, the mean infectious period is far shorter for HPAI (1.47 (1.3-1.7) days) than for LPAI (7.65 (7.0-8.3) days), due to the rapid death of infected turkeys. Hence the basic reproductive ratio, [Formula: see text] is significantly lower for HPAI (3.01 (2.2-4.0)) than for LPAI (15.3 (11.8-19.7)). The comparison of transmission rates and [Formula: see text] are critically important in relation to understanding how HPAI might emerge from LPAI. Two competing hypotheses for how transmission rates vary with population size are tested by fitting competing models to experiments with differing numbers of turkeys. A model with frequency-dependent transmission gives a significantly better fit to experimental data than density-dependent transmission. This has important implications for extrapolating experimental results from relatively small numbers of birds to the commercial poultry flock size, and for how control, including vaccination, might scale with flock size.

Dynamics of Salmonella infection of macrophages at the single cell level

Salmonella enterica causes a range of diseases. Salmonellae are intracellular parasites of macrophages, and the control of bacteria within these cells is critical to surviving an infection. The dynamics of the bacteria invading, surviving, proliferating in and killing macrophages are central to disease pathogenesis. Fundamentally important parameters, however, such as the cellular infection rate, have not previously been calculated. We used two independent approaches to calculate the macrophage infection rate: mathematical modelling of Salmonella infection experiments, and analysis of real-time video microscopy of infection events. Cells repeatedly encounter salmonellae, with the bacteria often remain associated with the macrophage for more than ten seconds. Once Salmonella encounters a macrophage, the probability of that bacterium infecting the cell is remarkably low: less than 5%. The macrophage population is heterogeneous in terms of its susceptibility to the first infection event. Once infected, a macrophage can undergo further infection events, but these reinfection events occur at a lower rate than that of the primary infection.

Transmission and dose-response experiments for social animals: a reappraisal of the colonization biology of Campylobacter jejuni in chickens

Dose-response experiments characterize the relationship between infectious agents and their hosts. These experiments are routinely used to estimate the minimum effective infectious dose for an infectious agent, which is most commonly characterized by the dose at which 50 per cent of challenged hosts become infected-the ID(50). In turn, the ID(50) is often used to compare between different agents and quantify the effect of treatment regimes. The statistical analysis of dose-response data typically makes the assumption that hosts within a given dose group are independent. For social animals, in particular avian species, hosts are routinely housed together in groups during experimental studies. For experiments with non-infectious agents, this poses no practical or theoretical problems. However, transmission of infectious agents between co-housed animals will modify the observed dose-response relationship with implications for the estimation of the ID(50) and the comparison between different agents and treatments. We derive a simple correction to the likelihood for standard dose-response models that allows us to estimate dose-response and transmission parameters simultaneously. We use this model to show that: transmission between co-housed animals reduces the apparent value of the ID(50) and increases the variability between replicates leading to a distinctive all-or-nothing response; in terms of the total number of animals used, individual housing is always the most efficient experimental design for ascertaining dose-response relationships; estimates of transmission from previously published experimental data for Campylobacter spp. in chickens suggest that considerable transmission occurred, greatly increasing the uncertainty in the estimates of dose-response parameters reported in the literature. Furthermore, we demonstrate that accounting for transmission in the analysis of dose-response data for Campylobacter spp. challenges our current understanding of the differing response of chickens with respect to host-age and in vivo passage of bacteria. Our findings suggest that the age-dependence of transmissibility between hosts-rather than their susceptibility to colonization-is the mechanism behind the 'lag-phase' reported in commercial flocks, which are typically found to be Campylobacter free for the first 14-21 days of life.

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.

Overlapping signals for translational regulation and packaging of influenza A virus segment 2

Influenza A virus segment 2 mRNA expresses three polypeptides: PB1, PB1-F2 and PB1-N40, from AUGs 1, 4 and 5 respectively. Two short open reading frames (sORFs) initiated by AUGs 2 and 3 are also present. To understand translational regulation in this system, we systematically mutated AUGs 1-4 and monitored polypeptide synthesis from plasmids and recombinant viruses. This identified sORF2 as a key regulatory element with opposing effects on PB1-F2 and PB1-N40 expression. We propose a model in which AUGs 1-4 are accessed by leaky ribosomal scanning, with sORF2 repressing synthesis of downstream PB1-F2. However, sORF2 also up-regulates PB1-N40 expression, most likely by a reinitiation mechanism that permits skipping of AUG4. Surprisingly, we also found that in contrast to plasmid-driven expression, viruses with improved AUG1 initiation contexts produced less PB1 in infected cells and replicated poorly, producing virions with elevated particle:PFU ratios. Analysis of the genome content of virus particles showed reduced packaging of the mutant segment 2 vRNAs. Overall, we conclude that segment 2 mRNA translation is regulated by a combination of leaky ribosomal scanning and reinitiation, and that the sequences surrounding the PB1 AUG codon are multifunctional, containing overlapping signals for translation initiation and for segment-specific packaging.

Measuring social networks in British primary schools through scientific engagement

Primary schools constitute a key risk group for the transmission of infectious diseases, concentrating great numbers of immunologically naive individuals at high densities. Despite this, very little is known about the social patterns of mixing within a school, which are likely to contribute to disease transmission. In this study, we present a novel approach where scientific engagement was used as a tool to access school populations and measure social networks between young (4-11 years) children. By embedding our research project within enrichment activities to older secondary school (13-15) children, we could exploit the existing links between schools to achieve a high response rate for our study population (around 90% in most schools). Social contacts of primary school children were measured through self-reporting based on a questionnaire design, and analysed using the techniques of social network analysis. We find evidence of marked social structure and gender assortativity within and between classrooms in the same school. These patterns have been previously reported in smaller studies, but to our knowledge no study has attempted to exhaustively sample entire school populations. Our innovative approach facilitates access to a vitally important (but difficult to sample) epidemiological sub-group. It provides a model whereby scientific communication can be used to enhance, rather than merely complement, the outcomes of research.

A Bayesian approach to analyse genetic variation within RNA viral populations

The development of modern and affordable sequencing technologies has allowed the study of viral populations to an unprecedented depth. This is of particular interest for the study of within-host RNA viral populations, where variation due to error-prone polymerases can lead to immune escape, antiviral resistance and adaptation to new host species. Methods to sequence RNA virus genomes include reverse transcription (RT) and polymerase chain reaction (PCR). RT-PCR is a molecular biology technique widely used to amplify DNA from an RNA template. The method itself relies on the in vitro synthesis of copy DNA from RNA followed by multiple cycles of DNA amplification. However, this method introduces artefactual errors that can act as confounding factors when the sequence data are analysed. Although there are a growing number of published studies exploring the intra- and inter-host evolutionary dynamics of RNA viruses, the complexity of the methods used to generate sequences makes it difficult to produce probabilistic statements about the likely sources of observed sequence variants. This complexity is further compounded as both the depth of sequencing and the length of the genome segment of interest increase. Here we develop a bayesian method to characterise and differentiate between likely structures for the background viral population. This approach can then be used to identify nucleotide sites that show evidence of change in the within-host viral population structure, either over time or relative to a reference sequence (e.g. an inoculum or another source of infection), or both, without having to build complex evolutionary models. Identification of these sites can help to inform the design of more focussed experiments using molecular biology tools, such as site-directed mutagenesis, to assess the function of specific amino acids. We illustrate the method by applying to datasets from experimental transmission of equine influenza, and a pre-clinical vaccine trial for HIV-1.

Genomic analysis of codon, sequence and structural conservation with selective biochemical-structure mapping reveals highly conserved and dynamic structures in rotavirus RNAs with potential cis-acting functions

Rotaviruses are a major cause of acute, often fatal, gastroenteritis in infants and young children world-wide. Virions contain an 11 segment double-stranded RNA genome. Little is known about the cis-acting sequences and structural elements of the viral RNAs. Using a database of 1621 full-length sequences of mammalian group A rotavirus RNA segments, we evaluated the codon, sequence and RNA structural conservation of the complete genome. Codon conservation regions were found in eight ORFs, suggesting the presence of functional RNA elements. Using ConStruct and RNAz programmes, we identified conserved secondary structures in the positive-sense RNAs including long-range interactions (LRIs) at the 5' and 3' terminal regions of all segments. In RNA9, two mutually exclusive structures were observed suggesting a switch mechanism between a conserved terminal LRI and an independent 3' stem-loop structure. In RNA6, a conserved stem-loop was found in a region previously reported to have translation enhancement activity. Biochemical structural analysis of RNA11 confirmed the presence of terminal LRIs and two internal helices with high codon and sequence conservation. These extensive in silico and in vitro analyses provide evidence of the conservation, complexity, multi-functionality and dynamics of rotavirus RNA structures which likely influence RNA replication, translation and genome packaging.

Genome packaging in influenza A virus

The negative-sense RNA genome of influenza A virus is composed of eight segments, which encode 12 proteins between them. At the final stage of viral assembly, these genomic virion (v)RNAs are incorporated into the virion as it buds from the apical plasma membrane of the cell. Genome segmentation confers evolutionary advantages on the virus, but also poses a problem during virion assembly as at least one copy of each of the eight segments is required to produce a fully infectious virus particle. Historically, arguments have been presented in favour of a specific packaging mechanism that ensures incorporation of a full genome complement, as well as for an alternative model in which segments are chosen at random but packaged in sufficient numbers to ensure that a reasonable proportion of virions are viable. The question has seen a resurgence of interest in recent years leading to a consensus that the vast majority of virions contain no more than eight segments and that a specific mechanism does indeed function to select one copy of each vRNA. This review summarizes work leading to this conclusion. In addition, we describe recent progress in identifying the specific packaging signals and discuss likely mechanisms by which these RNA elements might operate.

Campylobacter jejuni colonization and transmission in broiler chickens: a modelling perspective

Campylobacter jejuni is one of the most common causes of acute enteritis in the developed world. The consumption of contaminated poultry, where C. jejuni is believed to be a commensal organism, is a major risk factor. However, the dynamics of this colonization process in commercially reared chickens is still poorly understood. Quantification of these dynamics of infection at an individual level is vital to understand transmission within populations and formulate new control strategies. There are multiple potential routes of introduction of C. jejuni into a commercial flock. Introduction is followed by a rapid increase in environmental levels of C. jejuni and the level of colonization of individual broilers. Recent experimental and epidemiological evidence suggest that the celerity of this process could be masking a complex pattern of colonization and extinction of bacterial strains within individual hosts. Despite the rapidity of colonization, experimental transmission studies exhibit a highly variable and unexplained delay time in the initial stages of the process. We review past models of transmission of C. jejuni in broilers and consider simple modifications, motivated by the plausible biological mechanisms of clearance and latency, which could account for this delay. We show how simple mathematical models can be used to guide the focus of experimental studies by providing testable predictions based on our hypotheses. We conclude by suggesting that competition experiments could be used to further understand the dynamics and mechanisms underlying the colonization process. The population models for such competition processes have been extensively studied in other ecological and evolutionary contexts. However, C. jejuni can potentially adapt phenotypically through phase variation in gene expression, leading to unification of ecological and evolutionary time-scales. For a theoretician, the colonization dynamics of C. jejuni offer an experimental system to explore these 'phylodynamics', the synthesis of population dynamics and evolutionary biology.

Codon conservation in the influenza A virus genome defines RNA packaging signals

Genome segmentation facilitates reassortment and rapid evolution of influenza A virus. However, segmentation complicates particle assembly as virions must contain all eight vRNA species to be infectious. Specific packaging signals exist that extend into the coding regions of most if not all segments, but these RNA motifs are poorly defined. We measured codon variability in a large dataset of sequences to identify areas of low nucleotide sequence variation independent of amino acid conservation in each segment. Most clusters of codons showing very little synonymous variation were located at segment termini, consistent with previous experimental data mapping packaging signals. Certain internal regions of conservation, most notably in the PA gene, may however signify previously unidentified functions in the virus genome. To experimentally test the bioinformatics analysis, we introduced synonymous mutations into conserved codons within known packaging signals and measured incorporation of the mutant segment into virus particles. Surprisingly, in most cases, single nucleotide changes dramatically reduced segment packaging. Thus our analysis identifies cis-acting sequences in the influenza virus genome at the nucleotide level. Furthermore, we propose that strain-specific differences exist in certain packaging signals, most notably the haemagglutinin gene; this finding has major implications for the evolution of pandemic viruses.

Modelling dynamics of the type I interferon response to in vitro viral infection

Innate immunity is crucial in the early stages of resistance to novel viral infection. The family of cytokines known as the interferons (IFNs) forms an essential component of this system: they are responsible for signalling that an infection is underway and for promoting an antiviral response in susceptible cells. We construct a spatial stochastic model, parameterized by experimental data and informed by analytic approximation, to capture the dynamics of virus-IFN interaction during in vitro infection of Madin-Darby bovine kidney cell monolayers by Herpes simplex virus 1. The dose dependence of infection progression, subsequent monolayer destruction and IFN-beta production are investigated. Implications for in vivo infections, in particular the priming of susceptible cells by IFN-beta during infection, are considered.

Epidemic dynamics and antigenic evolution in a single season of influenza A

We use a mathematical model to study the evolution of influenza A during the epidemic dynamics of a single season. Classifying strains by their distance from the epidemic-originating strain, we show that neutral mutation yields a constant rate of antigenic evolution, even in the presence of epidemic dynamics. We introduce host immunity and viral immune escape to construct a non-neutral model. Our population dynamics can then be framed naturally in the context of population genetics, and we show that departure from neutrality is governed by the covariance between a strain's fitness and its distance from the original epidemic strain. We quantify the amount of antigenic evolution that takes place in excess of what is expected under neutrality and find that this excess amount is largest under strong host immunity and long epidemics.

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

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

Unifying the epidemiological and evolutionary dynamics of pathogens

A key priority for infectious disease research is to clarify how pathogen genetic variation, modulated by host immunity, transmission bottlenecks, and epidemic dynamics, determines the wide variety of pathogen phylogenies observed at scales that range from individual host to population. We call the melding of immunodynamics, epidemiology, and evolutionary biology required to achieve this synthesis pathogen "phylodynamics." We introduce a phylodynamic framework for the dissection of dynamic forces that determine the diversity of epidemiological and phylogenetic patterns observed in RNA viruses of vertebrates. A central pillar of this model is the Evolutionary Infectivity Profile, which captures the relationship between immune selection and pathogen transmission.

Population dynamics of rapid fixation in cytotoxic T lymphocyte escape mutants of influenza A

The dynamics of cellular immunity against pathogens, and its interaction with the human MHC system, is a key area for empirical research, both within individual hosts and in population genetic surveys. However, in contrast with humoral immunity, the dynamics of cellular immunity have not been modeled at the population level. Here, we address this lacuna with a model of recently observed dramatic invasions of cytotoxic T lymphocyte escape mutants in human influenza A. In particular, we offer an explanation for the rapid fixation of a HLA-B27 restricted cytotoxic T lymphocyte escape mutant on the nucleoprotein that emerged in the 1993-1994 season. We find that the dynamics within a single season of influenza do not provide a realistic description, but a model of the full annual dynamics can offer a possible explanation. Our model is deterministic for the winter epidemic, and stochastic for the summer period. An escape mutant that leads to a slightly longer infection in a small proportion of hosts has a substantial advantage through summer persistence. Furthermore, if a small number of founding cases are responsible for initiating each epidemic, then this effect of rapid mutant fixation is amplified.

Dynamics and selection of many-strain pathogens

Strain structure is of fundamental importance in the underlying dynamics of a number of pathogens. However, previous models have been too complex to accommodate many strains. This paper offers a solution to this problem, in the form of a simple model that is capable of capturing the dynamics of a large number of antigenic types that interact via host cross-immunity. We derive the structure of the model, which can manage the complexity of many strains by using a status-based formulation, assuming polarized immunity and cross-immunity act to reduced transmission probability. We then apply the model to address basic questions in strain dynamics, focusing particularly on the interpandemic dynamics of influenza. This model shows that strains have a tendency to "cluster." For a long infectious period, relative to host lifetime, clusters may coexist. By contrast, a short infectious period leads to a single dominant cluster at any given time. We show how the speed of cluster replacement depends on the specificity of cross-immunity and on the underlying pathogen mutation rate.

The onset of oscillatory dynamics in models of multiple disease strains

We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens-Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors.

A status-based approach to multiple strain dynamics

We present and investigate a new model for cross-immunity. Past models classify hosts according to their infection history. Here we represent hosts through their status: their current ability to respond to strains. This framework allows a different, a wider, and a more biologically interpretable range of forms of cross-immunity to be studied. Using this new form of cross-immunity we then consider a previously studied case of four strains, each of which confers partial immunity to two of the others. In this interesting special case, with applications to the genetic maintenance of strain diversity, we can make substantial analytical progress. We present methods for exploiting the symmetries of the system to show that only a particular invariant subspace need be considered for characterizing the dynamics of the whole system. A complete bifurcation structure is given for this subspace. In contrast to systems previously studied, this system does not exhibit sustained oscillations for any set of parameter values.