Modelling influenza strain competition dynamics and transmission fitness

Tese de mestrado, Bioestatística, Universidade de Lisboa, Faculdade de Ciências, 2019A estimação de diferenças de aptidão (ou fitness) entre indivíduos, particularmente de espécies microbiais patogénicas, é uma área muito ativa de investigação. Nesta área, dado o aumento constante da evolução de resistência a drogas, mecanismos de evasão de vacinas e emergência viral, tem sido crucial entender as diferenças de aptidão viral. Recentemente, têm sido propostas várias abordagens experimentais e de modelação com o objetivo de quantificar a diversidade viral. Isto é particularmente importante quando se pretende comparar patogénios resistentes ou sensíveis a tratamentos, e quando se procura planear medidas de contenção e prevenção. Esta tese é motivada por uma série de experiências de transmissão de duas estirpes de influenza, o vírus responsável pela gripe. A gripe é uma doença infeciosa que afeta populações animais e humanas, e é uma causa significativa de morbilidade e mortalidade no mundo. Estima-se que ocorram até 650 000 mortes durante eventos epidémicos anuais. O vírus é transmitido através de aerossóis espalhados por espirros ou tosse. É comum a infeção estar associada a complicações causadas por outros agentes, como a coinfecção com a bactéria que causa a pneumonia. A prevenção de epidemias gripais graves é feita recorrendo à ação de antivirais, no entanto, a resistência aos mesmos está a aumentar. Através de alterações genómicas, novas estirpes do vírus podem emergir, havendo o risco de algumas serem resistentes ao reconhecimento do sistema imunitário ou dos antivirais. No caso de a resistência não acarretar custos de fitness de transmissão, existe o potencial de causar um evento pandémico. A estimação de aptidão viral é usualmente baseada em métodos estatísticos que comparam o fitness replicativo relativo de duas estirpes, em culturas de células, tecidos ou hospedeiros individuais. De modo a associar a aptidão num hospedeiro com a aptidão de transmissão entre hospedeiros, uma abordagem experimental, conhecida como misturas competitivas, foi proposta por Hurt et al. (2010). Estas experiências envolveram a infeção de furões com uma mistura de uma estirpe de gripe suscetível a um antiviral comum e uma estirpe resistente ao mesmo, e a subsequente medição diária das proporções relativas destas estirpes de modo a investigar se um vírus se está a replicar mais rapidamente que o outro. Os dados obtidos neste estudo servem como ponto de partida para este trabalho. Para quantificar as diferenças entre estirpes suscetíveis e estirpes resistentes, foi proposto mais tarde por McCaw et al (2011) um modelo matemático que traduz essas diferenças em termos de um coeficiente baseado nas suas taxas de crescimento. Esse modelo é simplista e prevê apenas cenários em que uma estirpe leva a outra estirpe à extinção. Estes autores estimaram uma ligeira vantagem da estirpe resistente, isto é, tem uma taxa de crescimento maior que a estirpe suscetível. No entanto, os dados têm uma grande variabilidade, indicando que provavelmente a estirpe resistente não conduz sempre a estirpe suscetível à extinção. Além disso, este modelo não é capaz de explicar a coexistência de ambas as estirpes. Nem todos os vírus são capazes de se transmitir de um hospedeiro para outro, um conceito denominado de bottleneck de transmissão, refletido pelo parâmetro N, o número total de vírus que se transmitem, independentemente da estirpe. Esse modelo prevê um bottleneck estreito, isto é, poucas partículas virais no total são transmitidas, N ≈ 4. Isto permite explicar a grande variabilidade observada nas proporções relativas das duas estirpes, no entanto é inconsistente com estimativas atuais para o número de vírus transmitidos entre hospedeiros. A heterogeneidade num hospedeiro, isto é, a co-circulação de diferentes estirpes de gripe, é comum, e resulta em competição pelos recursos e espaço do hospedeiro. Dadas estas preocupações, é de uma grande relevância modelar e ganhar conhecimento mais aprofundado das dinâmicas de competição entre estirpes de gripe e como isso afeta as suas capacidades de transmissão entre hospedeiros. O principal objetivo desta tese é aplicar um modelo alternativo a estes dados de transmissão de misturas de estirpes que permita explicar os dados recorrendo às dinâmicas de crescimento e de competição, e com mais flexibilidade para estimar o número de vírus transmitidos. Nesta tese, apresentamos um modelo matemático baseado nas dinâmicas de competição entre estirpes de gripes suscetíveis e resistentes a antivirais. O nosso modelo, baseado nas equações de competição de Lotka-Volterra, é aplicado aos dados experimentais de misturas de estirpes gripais, com o objetivo de compreender como os mecanismos de competição intra- e interestirpe afetam a aptidão relativa de transmissão entre hospedeiros. No Capítulo 2 introduzimos o modelo e ilustramos as suas previsões num hospedeiro. Aí mostramos como este modelo, ao contrário de abordagens clássicas baseadas apenas num coeficiente de aptidão, não está limitado a cenários de exclusão competitiva, isto é, a estirpe resistente leva a suscetível à extinção ou vice-versa. Dois novos cenários ecológicos emergem: coexistência estável de ambas as estirpes, e um cenário de bi-estabilidade, no qual, dependendo das condições iniciais, o sistema colapsa para um dos cenários de exclusão competitiva. No Capítulo 3 fazemos a ponte entre as dinâmicas num hospedeiro e as subsequentes dinâmicas de transmissão entre hospedeiros. Também validamos as previsões do modelo com recurso a simulações. No Capítulo 4 o modelo é ajustado aos dados de transmissão de misturas de estirpes de influenza (McCaw et al (2011)) através de optimização em R. A incerteza à volta das estimativas do modelo é quantificada recorrendo a simulações. Estas simulações baseiam-se em gerar dados artificiais equivalentes aos dados originais, em que simulam a variabilidade observada causada por erro experimental (gerando dados de forma uniforme em redor das observações) ou pelo efeito de bottleneck (gerando dados através de reamostragem das condições iniciais com um modelo Binomial). Este último método tem a vantagem de não só permitir quantificar a incerteza dos parâmetros do modelo, como também obter uma estimativa do número total de vírus transmitidos entre hospedeiros, N. A nossa proposta é capaz de inferir com precisão parâmetros de crescimento e de competição, e prevê neste contexto específico um cenário de coexistência das duas estirpes virais, isto é, que a estirpe suscetível e a resistente são transmitidas em conjunto, com base nestes dados de experiências de misturas competitivas. O nosso estudo tem implicações para a epidemiologia e modelação matemática, e aprofunda o conhecimento dos resultados experimentais de misturas competitivas no geral. Ao permitir a possibilidade de competição dependente da frequência e hierarquias entre estirpes, este modelo expande o alcance de cenários ecológicos que podem ser capturados, incluindo coexistência e bi-estabilidade, antes da ativação do sistema imune. A coexistência mútua resultante das dinâmicas de competição pode ajudar a explicar a heterogeneidade viral observada ao nível populacional. Adicionalmente, o modelo prevê um número de vírus transmitidos, N ≈ 230, compatível com a literatura recente de influenza em humanos. Quanto mais flexível um modelo é para capturar não-linearidade nos dados, menos hipóteses existem para atribuir flutuações observadas nos dados a pura variabilidade causada por um bottleneck estreito. Esta tese deve servir como proof-of-concept, sendo, no entanto, a abordagem geral o suficiente para ser aplicada a cenários ecológicos entre outras estirpes ou outras espécies que compitam entre si.There have been proposed in the recent years many experimental and modelling approaches to quantify viral diversity. This is particularly important when comparing drug-resistant with drug-sensitive pathogens and when designing control measures. We present a general mathematical and statistical framework that focuses on the competition dynamics between two strains of influenza. Managing influenza outbreaks has been done extensively over the years using antivirals, however resistance is on the rise. Via mutational changes, new strains of virus can emerge that are resistant to these antivirals. It is important to quantify fitness differences of such mutants with wild-type strains, to predict epidemiological outcomes and design control measures. The resistance to antivirals can sometimes carry no cost of transmission fitness, having the potential to cause a pandemic event. Given these concerns, it is of major relevance to model and gain an understanding of the dynamics of competing influenza strains within host, and how this affects their relative transmissibility between hosts. Our model is based on the Lotka-Volterra equations and is applied to data from competitive mixture experiments, with the aim is to comprehend how the intra- and inter-strain competition affects the relative strain transmission between hosts. The model is validated through a simulation approach and the parameter uncertainty is quantified using observations from the simulation procedure. Our framework can accurately infer parameter values and, for this data, predicts a scenario of coexistence of the antiviral susceptible and antiviral resistant strains. Additionally, we predict, compared with previous estimates, a relatively higher transmission bottleneck size, i.e. a total number of virions transmitted between hosts of approximately 230. This thesis serves as a proof-of-principle, with the model being general enough to be applied to a variety of ecological interactions that involve competition and allow for results beyond competitive exclusion

Diversity, variability and persistence elements for a non-equilibrium theory of eco-evolutionary dynamics

Natural ecosystems persist in variable environments by virtue of a suite of traits that span from the individual to the community, and from the ecological to the evolutionary scenarios. How these internal characteristics operate to allow living beings to cope with the uncertainty present in their environments is the subject matter of quantitative theoretical ecology. Under the framework of structural realism, the present dissertation project has advocated for the strategy of mathematical modeling as a strategy of abstraction. The goal is to explore if a range of natural ecosystems display the features of complex systems, and evaluate whether these features provide insights into how they persist in their current environments, and how might they cope with changing environments in the future. A suite of inverse, linear and non-linear dynamical mathematical models, including non-equilibrium catastrophe models, and structured demographic approaches is applied to five case studies of natural systems fluctuating in the long-term in diverse scenarios: phytoplankton in the global ocean, a mixotrophic plankton food web in a marine coastal environment, a wintering waterfowl community in a major Mediterranean biodiversity hot-spot, a breeding colony of a keystone avian scavenger in a mountainous environment and the shorebird community inhabiting the coast of UK. In all case studies, there is strong evidence that ecosystems are able to closely track their common environment through several strategies. For example, in global phytoplankton communities, a latitudinal gradient in the positive impact of functional diversity on community stability counteracts the increasing environmental variability with latitude. Mixotrophy, by linking several feeding strategies in a food web, internally drives community dynamics to the edge of instability while maximizing network complexity. In contrast, an externally generated major perturbation, operating through planetary climatic disruptions, induce an abrupt regime shift between alternative stable states in the wintering waterfowl community. Overall, the natural systems studied are shown to posses features of complex systems: connectivity, autonomy, emergence, non-equilibrium, non-linearity, self-organization and coevolution. In rapidly changing environments, these features are hypothesized to allow natural system to robustly respond to stress and disturbances to a large extent. At the same time, future scenarios will be probably characterized by conditions never experienced before by the studied systems. How will they respond to them, is an open question. Based on the results of this dissertation, future research directions in theoretical quantitative ecology will likely benefit from non-autonomous dynamical system approaches, where model parameters are a function of time, and from the deeper exploration of global attractors and the non-equilibriumness of dynamical systems

Food Webs, Models and Species Extinctions in a Stochastic Environment

In light of the current global mass extinction of species, ecologists are facing great challenges. In order to reverse the path towards additional extinctions early warning systems to guide management actions need to be developed. However, considering the countless species to monitor and the complexity of interactions affecting species abundances in ecological communities, this is not an easy task. Before this goal can be reached our understanding of how community structure and species interactions interact and affect the risk of extinction of single species needs to be increased. Thus the primary aim of the present thesis is to study this interaction and contribute to a theoretical basis for the identification of extinction prone species. In paper II it is concluded that spectral analysis of population time series may function as a tool to predict extinctions at an early stage. More specifically, I show that extinction risk of producer species in food webs under influence of uncorrelated environmental stochasticity increases with intensified red-shift of population time series. However, this relationship is strictly context-dependent, which means that a producer with red dynamics might survive in one type of food web, but the same producer species with a similar magnitude of spectral redness can go extinct in another food web where the interactions with other species are arranged in a different manner. Then I turn to look at which species might be more prone to become endangered or to go extinct in food webs experiencing various types of uncorrelated environmental stochasticity. In paper I I show that producer species are more likely to reach endangered population levels (according to The World Conservation Union, IUCN, criterion), whereas paper III demonstrates that consumer species more frequently go extinct. This seemingly contradiction may be explained by characteristics inherent to many producer species (e.g. high growth rate, short generation time) that enable them to recover from low population levels and thus escape extinction. Furthermore, in both the second and the third paper I show that the structure of food webs as well as the presence, position and direction of a strong interaction between two species in a food web play significant roles in the likelihood of a species reaching endangered population levels or going extinct. In paper IV I show that small and condensed food webs are likely to express fundamentally different dynamics compared to large and well-resolved versions of the same natural food webs. Starting from a well-resolved version of a real food web, local dynamics of the ecological system change in a non-linear manner, during gradual lumping of the functionally most similar species into aggregated species (or trophospecies), Here it is also suggested that functional redundancy exists in natural food webs. This may imply support for the ?insurance hypothesis? since sequential extinction of one of the species in the functionally most similar pair of species initially did not generate any significant changes in local dynamics of the system. To sum up, in this thesis I present a prototype of a predictive tool to discover species at risk of going extinct. I also present directions to which type of species to look for and what type of structures and interactions to pay attention to when searching for presumptive victims of extinction in ecological systems. However, the features of the ecological models I have used for my research are in many cases incomplete. For example, my food webs contain relatively few species without competitive interactions subjected to only uncorrelated environmental variability. Further research will have to test the generality of the results and the robustness of the conclusions drawn from them

Continuous coexistence or discrete species? A new review of an old question

Question: Is the coexistence of a continuum of species or ecological types possible in real-world communities? Or should one expect distinctly different species? Mathematical methods: We study whether the coexistence of species in a continuum of ecological types is (a) dynamically stable (against changes in population densities) and (b) structurally robust (against changes in population dynamics). Since most of the reviewed investigations are based on Lotka-Volterra models, we carefully explain which of the presented conclusions are model-independent. mathematical conclusions: Seemingly plausible models with dynamically stable continuous- coexistence solutions do exist. However, these models either depend on biologically unrealistic mathematical assumptions (e.g. non-differentiable ingredient functions) or are structurally unstable (i.e. destroyable by arbitrarily small modifications to those ingredient functions). The dynamical stability of a continuous-coexistence solution, if it exists, requires positive definiteness of the model's competition kernel. Biological conclusions: While the classical expectation of fixed limits to similarity is mathematically naive, the fundamental discreteness of species is a natural consequence of the basic structure of ecological interactio

Ecological Invasion, Roughened Fronts, and a Competitor's Extreme Advance: Integrating Stochastic Spatial-Growth Models

Both community ecology and conservation biology seek further understanding of factors governing the advance of an invasive species. We model biological invasion as an individual-based, stochastic process on a two-dimensional landscape. An ecologically superior invader and a resident species compete for space preemptively. Our general model includes the basic contact process and a variant of the Eden model as special cases. We employ the concept of a "roughened" front to quantify effects of discreteness and stochasticity on invasion; we emphasize the probability distribution of the front-runner's relative position. That is, we analyze the location of the most advanced invader as the extreme deviation about the front's mean position. We find that a class of models with different assumptions about neighborhood interactions exhibit universal characteristics. That is, key features of the invasion dynamics span a class of models, independently of locally detailed demographic rules. Our results integrate theories of invasive spatial growth and generate novel hypotheses linking habitat or landscape size (length of the invading front) to invasion velocity, and to the relative position of the most advanced invader.Comment: The original publication is available at www.springerlink.com/content/8528v8563r7u2742

Species assembly in model ecosystems, I: Analysis of the population model and the invasion dynamics

Recently we have introduced a simplified model of ecosystem assembly (Capitan et al., 2009) for which we are able to map out all assembly pathways generated by external invasions in an exact manner. In this paper we provide a deeper analysis of the model, obtaining analytical results and introducing some approximations which allow us to reconstruct the results of our previous work. In particular, we show that the population dynamics equations of a very general class of trophic-level structured food-web have an unique interior equilibrium point which is globally stable. We show analytically that communities found as end states of the assembly process are pyramidal and we find that the equilibrium abundance of any species at any trophic level is approximately inversely proportional to the number of species in that level. We also find that the per capita growth rate of a top predator invading a resident community is key to understand the appearance of complex end states reported in our previous work. The sign of these rates allows us to separate regions in the space of parameters where the end state is either a single community or a complex set containing more than one community. We have also built up analytical approximations to the time evolution of species abundances that allow us to determine, with high accuracy, the sequence of extinctions that an invasion may cause. Finally we apply this analysis to obtain the communities in the end states. To test the accuracy of the transition probability matrix generated by this analytical procedure for the end states, we have compared averages over those sets with those obtained from the graph derived by numerical integration of the Lotka-Volterra equations. The agreement is excellent.Comment: 16 pages, 8 figures. Revised versio

Differential Equations arising from Organising Principles in Biology

This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations

Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities

Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics