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Documents  Pardoux, Etienne | enregistrements trouvés : 24

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Research talks;Mathematics in Science and Technology;Probability and Statistics

A new type of a simple iterated game with natural biological motivation is introduced. Two individuals are chosen at random from a population. They must survive a certain number of steps. They start together, but if one of them dies the other one tries to survive on its own. The only payoff is to survive the game. We only allow two strategies: cooperators help the other individual, while defectors do not. There is no strategic complexity. There are no conditional strategies. Depending on the number of steps we recover various forms of stringent and relaxed cooperative dilemmas. We derive conditions for the evolution of cooperation.
Specifically, we describe an iterated game between two players, in which the payoff is to survive a number of steps. Expected payoffs are probabilities of survival. A key feature of the game is that individuals have to survive on their own if their partner dies. We consider individuals with simple, unconditional strategies. When both players are present, each step is a symmetric two-player game. As the number of iterations tends to infinity, all probabilities of survival decrease to zero. We obtain general, analytical results for n-step payoffs and use these to describe how the game changes as n increases. In order to predict changes in the frequency of a cooperative strategy over time, we embed the survival game in three different models of a large, well-mixed population. Two of these models are deterministic and one is stochastic. Offspring receive their parent’s type without modification and fitnesses are determined by the game. Increasing the number of iterations changes the prospects for cooperation. All models become neutral in the limit $(n \rightarrow \infty)$. Further, if pairs of cooperative individuals survive together with high probability, specifically higher than for any other pair and for either type when it is alone, then cooperation becomes favored if the number of iterations is large enough. This holds regardless of the structure of pairwise interactions in a single step. Even if the single-step interaction is a Prisoner’s Dilemma, the cooperative type becomes favored. Enhanced survival is crucial in these iterated evolutionary games: if players in pairs start the game with a fitness deficit relative to lone individuals, the prospects for cooperation can become even worse than in the case of a single-step game.
A new type of a simple iterated game with natural biological motivation is introduced. Two individuals are chosen at random from a population. They must survive a certain number of steps. They start together, but if one of them dies the other one tries to survive on its own. The only payoff is to survive the game. We only allow two strategies: cooperators help the other individual, while defectors do not. There is no strategic complexity. There ...

91A80 ; 91A40 ; 91A22 ; 91A12 ; 91A20 ; 92D15

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Research talks;Partial Differential Equations;Mathematics in Science and Technology

In this talk, I will focus on a Fokker-Planck equation modeling interacting neurons in a network where each neuron is governed by an Integrate and Fire dynamic type. When the network is excitatory, neurons that discharge, instantaneously increased the membrane potential of the neurons of the network with a speed which is proportional to the amplitude of the global activity of the network. The self-excitable nature of these neurons in the case of excitatory networks leads to phenomena of blow-up, once the proportion of neurons that are close to their action potential is too high. In this talk, we are interested in understanding the regimes where solutions globally exist. By new methods of entropy and upper-solution, we give criteria where the phenomena of blow-up can not appear and specify, in some cases, the asymptotic behavior of the solution.

integrate-and-fire - neural networks - Fokker-Planck equation - blow-up
In this talk, I will focus on a Fokker-Planck equation modeling interacting neurons in a network where each neuron is governed by an Integrate and Fire dynamic type. When the network is excitatory, neurons that discharge, instantaneously increased the membrane potential of the neurons of the network with a speed which is proportional to the amplitude of the global activity of the network. The self-excitable nature of these neurons in the case of ...

92B20 ; 82C32 ; 35Q84

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Research talks;Mathematics in Science and Technology;Probability and Statistics;Topology

A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric spaces and give some applications of coalescent point processes to the phylogeny of bird species.

However, these models of diversification assume that species are exchangeable particles, and this always leads to the same (Yule) tree shape in distribution. Here, we propose a non-exchangeable, individual-based, point mutation model of diversification, where interspecific pairwise competition is only felt from the part of individuals belonging to younger species. As the initial (meta)population size grows to infinity, the properly rescaled dynamics of species lineages converge to a one-parameter family of coalescent trees interpolating between the caterpillar tree and the Kingman coalescent.

Keywords: ultrametric tree, inference, phylogenetic tree, phylogeny, birth-death process, population dynamics, evolution
A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric ...

60J80 ; 60J85 ; 92D15 ; 92D25 ; 54E45 ; 54E70

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Research talks;Mathematics in Science and Technology;Probability and Statistics

In recent years, new pandemic threats have become more and more frequent (SARS, bird flu, swine flu, Ebola, MERS, nCoV...) and analyses of data from the early spread more and more common and rapid. Particular interest is usually focused on the estimation of $ R_{0}$ and various methods, essentially based estimates of exponential growth rate and generation time distribution, have been proposed. Other parameters, such as fatality rate, are also of interest. In this talk, various sources of bias arising because observations are made in the early phase of spread will be discussed and also possible remedies proposed. In recent years, new pandemic threats have become more and more frequent (SARS, bird flu, swine flu, Ebola, MERS, nCoV...) and analyses of data from the early spread more and more common and rapid. Particular interest is usually focused on the estimation of $ R_{0}$ and various methods, essentially based estimates of exponential growth rate and generation time distribution, have been proposed. Other parameters, such as fatality rate, are also of ...

92B05 ; 92B15 ; 62P10

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Research talks;Mathematics in Science and Technology;Probability and Statistics

In this presentation, we shall discuss the reconstruction of demographic parameters based on the genetic variability observed within a sample of individual DNA. In the family of models that we consider, the statistics describing this genetic diversity (number of mutations, distribution of the mutations amongst individuals in the sample) depend on a more or less coarse ‘resolution of (i.e., level of information on) the hidden genealogical tree that relates the sampled individuals. Considering the optimal resolution thus allows to greatly improve the exploration of the space of possible genealogies when computing the likelihood of demographic parameters, compared to classical methods based on full labelled trees such as Kingmans coalescent. We shall focus on two examples, based on works with Raazesh Sainudiin (Uppsala Univ.) and with Julia Palacios (Stanford Univ.), Sohini Ramachandran (Brown Univ.) and John Wakeley (Harvard Univ.). In this presentation, we shall discuss the reconstruction of demographic parameters based on the genetic variability observed within a sample of individual DNA. In the family of models that we consider, the statistics describing this genetic diversity (number of mutations, distribution of the mutations amongst individuals in the sample) depend on a more or less coarse ‘resolution of (i.e., level of information on) the hidden genealogical tree ...

92D15 ; 92D20 ; 60J10 ; 60J27

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Research talks;Combinatorics;Mathematics in Science and Technology;Probability and Statistics

A phylogenetic tree that has been reconstructed from a given gene can describe a different evolutionary history from its underlying species tree. The reasons for this include: error in inferring the gene tree, incomplete lineage sorting, lateral gene transfer, and the absence of the gene in certain species. In this talk, I discuss probabilistic models and mathematical results that help address basic questions concerning the consistency and efficiency of different methods for inferring a species phylogeny from gene trees. A phylogenetic tree that has been reconstructed from a given gene can describe a different evolutionary history from its underlying species tree. The reasons for this include: error in inferring the gene tree, incomplete lineage sorting, lateral gene transfer, and the absence of the gene in certain species. In this talk, I discuss probabilistic models and mathematical results that help address basic questions concerning the consistency and ...

92D15 ; 92C37 ; 92C80 ; 05C05

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Research talks;Mathematics in Science and Technology;Probability and Statistics

Maladapted individuals can only colonise a new habitat if they can evolve a positive growth rate fast enough to avoid extinction - evolutionary rescue. We use the infinitesimal model to follow the evolution of the growth rate, and find that the probability that a single migrant can establish depends on just two parameters: the mean and genetic variance of fitness. With continued migration, establishment is inevitable. However, above a threshold migration rate, the population may be trapped in a sink state, in which adaptation is held back by gene flow. By assuming a constant genetic variance, we develop a diffusion approximation for the joint distribution of population size and trait mean. Maladapted individuals can only colonise a new habitat if they can evolve a positive growth rate fast enough to avoid extinction - evolutionary rescue. We use the infinitesimal model to follow the evolution of the growth rate, and find that the probability that a single migrant can establish depends on just two parameters: the mean and genetic variance of fitness. With continued migration, establishment is inevitable. However, above a threshold ...

92D15 ; 92D10 ; 92D25

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Outreach;Mathematics Education and Popularization of Mathematics;Mathematics in Science and Technology

Il y a cent ans, Sir Ronald Ross tentait de convaincre ses collègues médecins que l'épidémiologie doit être étudiée avec l'aide des mathématiques. Le but de cet exposé est d'expliquer pourquoi les mathématiques sont essentielles pour combattre les épidémies, et de donner quelques indications sur les avancées récentes de la modélisation mathématique en épidémiologie.

00A06 ; 00A08 ; 92C60 ; 92D30

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Research talks;Partial Differential Equations

The determination of the shape of an obstacle from its effects on known acoustic waves is an important problem in many technologies such as sonar, geophysical exploration and medical imaging. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because of its ill-posed and nonlinear nature. Its investigation requires the understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods. The main goal of this work is the development of an efficient procedure for retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics of the surface of the obstacle. We propose a solution methodology based on a regularized Newton-type method. The solution of the considered IOP by the proposed iterative method incurs, at each iteration, the solution of a linear system whose entries are the Fréchet derivatives of the elasto-acoustic field with respect to the shape parameters. We prove that these derivatives are solutions of the same direct elasto-acoustic scattering problem that differs only in the transmission conditions on the surface of the scatterer. Furthermore, the computational efficiency of the IOP solver depends mainly on the computational efficiency of the solution of the forward problems that arise at each Newton iteration. We propose to solve the direct scattering-type problems using a finite-element method based on discontinuous Galerkin approximations equipped with curved element boundaries. Numerical results will be presented to illustrate the salient features of this computational methodology and highlight its performance characteristics.

acoustics - shape derivative - inverse obstacle problem - Fréchet derivatives - inverse elasto-acoustic scattering problems
The determination of the shape of an obstacle from its effects on known acoustic waves is an important problem in many technologies such as sonar, geophysical exploration and medical imaging. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because of its ill-posed and nonlinear nature. Its investigation requires the understanding of the theory for the associated direct scattering problem, and the ...

65N21 ; 76Q05 ; 35R30

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Research talks;Partial Differential Equations;Mathematical Physics

We consider the nonlinear Schrödinger equation in the partially periodic setting $\mathbb{R}^d\times \mathbb{T}$. We present some recent results obtained in collaboration with N. Tzvetkov concerning the Cauchy theory and the long-time behavior of the solutions.

nonlinear Schrödinger equation - Cauchy theory - scattering

35Q55 ; 35B40 ; 35P25

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Research talks;Partial Differential Equations

This talk is devoted to the study of the following inverse boundary value problem: given a Riemannian manifold with boundary determine the magnetic potential in a dynamical Schrödinger equation in a magnetic field from the observations made at the boundary.

inverse problem - Schrödinger equation - magnetic field

35R30 ; 35Q55 ; 35R35 ; 35Q60

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Research talks;Probability and Statistics

In this talk I will present a stochastic model for the excitability of a neuron in a network. The neuron described by an Hodgkin-Huxley type model receives from the network a random input which is a perturbation of a periodic deterministic signal. For such a model we study ergodicity properties. Then, we prove limit theorems in order to be able to estimate characteristics of the sequence of spiking times. This talk is based on a joint work with R. Hoepfner (Univ. Mainz) and E. Loecherbach (Univ. Cergy-Pontoise).

Hodgkin-Huxley model - ergodicity - limit theorems - estimation
In this talk I will present a stochastic model for the excitability of a neuron in a network. The neuron described by an Hodgkin-Huxley type model receives from the network a random input which is a perturbation of a periodic deterministic signal. For such a model we study ergodicity properties. Then, we prove limit theorems in order to be able to estimate characteristics of the sequence of spiking times. This talk is based on a joint work with ...

60J60 ; 60J25 ; 60H07

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Research talks;Mathematics in Science and Technology;Probability and Statistics

We analyse patterns of genetic variability of populations in the presence of a large seed bank with the help of a new coalescent structure called seed bank coalescent. This ancestral process appears naturally as scaling limit of the genealogy of large populations that sustain seed banks, if the seed bank size and individual dormancy times are of the same order as the active population. Mutations appear as Poisson process on the active lineages, and potentially at reduced rate also on the dormant lineages. The presence of ‘dormant’ lineages leads to qualitatively altered times to the most recent common ancestor and non-classical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with seed bank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seed bank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Commonly employed distance statistics, in the presence and absence of a seed bank, are compared. The effect of a seed bank on the expected site-frequency spectrum is also investigated. Our results indicate that the presence of a large seed bank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect the presence of a large seed bank in genetic data. Joint work with Bjarki Eldon, Adrián González Casanova, Noemi Kurt, Maite Wilke-Berenguer We analyse patterns of genetic variability of populations in the presence of a large seed bank with the help of a new coalescent structure called seed bank coalescent. This ancestral process appears naturally as scaling limit of the genealogy of large populations that sustain seed banks, if the seed bank size and individual dormancy times are of the same order as the active population. Mutations appear as Poisson process on the active lineages, ...

92D10 ; 60K35 ; 62P10

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Research talks;Mathematics in Science and Technology;Probability and Statistics

Recent population genomics studies focus prevalently on the aspects of demography and adaptation, whereas age structure (for example, in plants via the maintenance of seed banks) has attracted less attention. Germ banking, that is, seed or egg dormancy, is a prevalent and important life-history trait in plants and invertebrates, which buffers against environmental variability and modulates species extinction in fragmented habitats. I will here summarize our recent findings investigating the intertwined effect of germ banking, time-varying population size and selection on genetic polymorphism in the wild tomato species. First, we examine the effect of seed banking on within species variability and local adaptation in the wild tomato Solanum chilense. Population genetic analyses and statistical inference of past demography was conducted on pooled-sequencing from 30 genes from an exhaustive sampling of 23 populations over Chile and Peru. We reveal a north-south colonization associated with relaxed purifying selection in the south as shown by a decrease of genetic variation and an increasing proportion of nonsynonymous polymorphism from north to south and population substructure with at least four genetic groups. We also uncover 1) a decreasing proportion of adaptive amino acid substitutions from north to south suggesting that adaptation is favoured in large populations, while 2) signatures of local adaptation predominantly occur in the smaller populations from the marginal ranges in the south. These results combined with additional germination data suggest that colonization of new habitats was accompanied by local adaptation for shorter seed banks in the marginal populations, shaping in return the available nucleotide diversity and effectiveness of purifying and positive selection. Second, we use ABC and polymorphism data to estimate population divergence times between two wild tomato species in presence of seed banks. We show that unknown seed banking also impedes our knowledge of the speciation process. Joint work with Katharina B. Böndel, Wolfgang Stephan. Recent population genomics studies focus prevalently on the aspects of demography and adaptation, whereas age structure (for example, in plants via the maintenance of seed banks) has attracted less attention. Germ banking, that is, seed or egg dormancy, is a prevalent and important life-history trait in plants and invertebrates, which buffers against environmental variability and modulates species extinction in fragmented habitats. I will here ...

92D40 ; 92C80 ; 92D15

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Research talks;Mathematics in Science and Technology;Probability and Statistics

Pervasive natural selection can strongly influence observed patterns of genetic variation, but these effects remain poorly understood when multiple selected variants segregate in nearby regions of the genome. Classical population genetics fails to account for interference between linked mutations, which grows increasingly severe as the density of selected polymorphisms increases. I will describe a simple limit that emerges when interference is common, in which the fitness effects of individual mutations play a relatively minor role. Instead, similar to models of quantitative genetics, molecular evolution is determined by the variance in fitness within the population, defined over an effectively asexual segment of the genome (a "linkage block"). I will describe how we can exploit this insensitivity in a new "coarse-grained" coalescent framework, which approximates the effects of many weakly selected mutations with a smaller number of strongly selected mutations that create the same variance in fitness. This approximation generates accurate and efficient predictions for silent site variability when interference is common. However, these results suggest that there is reduced power to resolve individual selection pressures when interference is sufficiently widespread, since a broad range of parameters possess nearly identical patterns of silent site variability. Pervasive natural selection can strongly influence observed patterns of genetic variation, but these effects remain poorly understood when multiple selected variants segregate in nearby regions of the genome. Classical population genetics fails to account for interference between linked mutations, which grows increasingly severe as the density of selected polymorphisms increases. I will describe a simple limit that emerges when interference is ...

92D10 ; 92D15

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Research talks;Mathematics in Science and Technology;Probability and Statistics

Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the transfer of plasmids in bacteria. The transfer rates are either density-dependent (DD) or frequency-dependent (FD) or of Michaelis-Menten form (MM). Our model allows eco-evolutionary feedbacks. In the first part we present a two-traits (alleles or kinds of plasmids, etc.) model with horizontal transfer without mutation and study a large population limit. It’s a ODEs system. We show that the phase diagrams are different in the (DD), (FD) and (MM) cases. We interpret the results for the impact of horizontal transfer on the maintenance of polymorphism and the invasion or elimination of pathogens strains. We also propose a diffusive approximation of adaptation with transfer. In a second part, we study the impact of the horizontal transfer on the evolution. We explain why it can drastically affect the evolutionary outcomes. Joint work with S. Billiard,P. Collet, R. Ferrière, C.V. Tran. Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the ...

60J75 ; 60J80 ; 92D25 ; 92D15

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Research talks;Combinatorics;Dynamical Systems and Ordinary Differential Equations;Mathematics Education and Popularization of Mathematics

Mathematical models of infectious disease transmission are increasingly used to guide public health and policy decisions. Hence, it is important that every effort is made to ensure that models are ‘correct’, made difficult by the frequent need to simulate a model numerically. The best we can do in most cases is to be able to replicate a model i.e. generate the same results from the same inputs (model plus parameters), or failing that, reproduce results that are similar. This can be achieved by sharing the computer code, and/or providing a sufficiently detailed description of the model. I will illustrate that it is often difficult to replicate or reproduce results of modeling publications, using case studies that highlight some of the many causes of this failure. I will argue that the FAIR principles proposed for data - that they should be Findable, Accessible, Interoperable and Reusable - are equally valid for modeling studies, and go a long way towards ensuring reproducibility. I will present Epirecipes (http://epirecip.es) a FAIR platform that both allows models to be replicated exactly, while fostering the idea that a wide variety of approaches are needed to ensure the robustness of model results. The added value from this platform includes resources for teaching, acting as a ‘Rosetta Stone’ - allowing models from one computer language to be ported to another, and as a repository of best practices, potential pitfalls, and technical tricks that are all too often tucked away in papers or textbooks. As quoted from ‘The Turing Way’ (https://the-turing-way.netlify.com), a handbook for reproducible science, reproducing models of infectious disease should be ‘too easy not to do’. Mathematical models of infectious disease transmission are increasingly used to guide public health and policy decisions. Hence, it is important that every effort is made to ensure that models are ‘correct’, made difficult by the frequent need to simulate a model numerically. The best we can do in most cases is to be able to replicate a model i.e. generate the same results from the same inputs (model plus parameters), or failing that, reproduce ...

97B10 ; 97D40 ; 97M60

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Research talks;Probability and Statistics

Low-dimensional compartment models for biological systems can be fitted to time series data using Monte Carlo particle filter methods. As dimension increases, for example when analyzing a collection of spatially coupled populations, particle filter methods rapidly degenerate. We show that many independent Monte Carlo calculations, each of which does not attempt to solve the filtering problem, can be combined to give a global filtering solution with favorable theoretical scaling properties under a weak coupling condition. The independent Monte Carlo calculations are called islands, and the operation carried out on each island is called adapted simulation, so the complete algorithm is called an adapted simulation island filter. We demonstrate this methodology and some related algorithms on a model for measles transmission within and between cities. Low-dimensional compartment models for biological systems can be fitted to time series data using Monte Carlo particle filter methods. As dimension increases, for example when analyzing a collection of spatially coupled populations, particle filter methods rapidly degenerate. We show that many independent Monte Carlo calculations, each of which does not attempt to solve the filtering problem, can be combined to give a global filtering solution ...

60G35 ; 60J20 ; 62M02 ; 62M05 ; 62M20 ; 62P10 ; 65C35

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Research talks;Dynamical Systems and Ordinary Differential Equations

Antibiotic resistance is a serious public health concern. Responding to this problem effectively requires characterising the factors (i.e. evolutionary and ecological processes) that determine resistance frequencies. At present, we do not have ecologically plausible models of resistance that are able to replicate observed trends - we are therefore unable to make credible predictions about resistance dynamics. In this talk, I will present work motivated by three tends observed in Streptococcus pneumoniae resistance data: the stable coexistence of antibiotic sensitivity and resistance, variation between resistance frequencies between pneumococcal lineages and correlation in resistance to different antibiotics. I will propose that variation in the fitness benefit gained from resistance arising from variation in the duration of carriage of pneumococcal lineages is a parsimonious explanation for all three trends. This eco-evolutionary framework could allow more accurate prediction of future resistance levels and play a role in informing strategies to prevent the spread of resistance. Antibiotic resistance is a serious public health concern. Responding to this problem effectively requires characterising the factors (i.e. evolutionary and ecological processes) that determine resistance frequencies. At present, we do not have ecologically plausible models of resistance that are able to replicate observed trends - we are therefore unable to make credible predictions about resistance dynamics. In this talk, I will present work ...

92A15 ; 92D30 ; 92D40

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Research talks;Probability and Statistics

In an epidemic model, the basic reproduction number $ R_{0}$ is a function of the parameters (such as infection rate) measuring disease infectivity. In a large population, if $ R_{0}> 1$, then the disease can spread and infect much of the population (supercritical epidemic); if $ R_{0}< 1$, then the disease will die out quickly (subcritical epidemic), with only few individuals infected.
For many epidemics, the dynamics are such that $ R_{0}$ can cross the threshold from supercritical to subcritical (for instance, due to control measures such as vaccination) or from subcritical to supercritical (for instance, due to a virus mutation making it easier for it to infect hosts). Therefore, near-criticality can be thought of as a paradigm for disease emergence and eradication, and understanding near-critical phenomena is a key epidemiological challenge.
In this talk, we explore near-criticality in the context of some simple models of SIS (susceptible-infective-susceptible) epidemics in large homogeneous populations.
In an epidemic model, the basic reproduction number $ R_{0}$ is a function of the parameters (such as infection rate) measuring disease infectivity. In a large population, if $ R_{0}> 1$, then the disease can spread and infect much of the population (supercritical epidemic); if $ R_{0}< 1$, then the disease will die out quickly (subcritical epidemic), with only few individuals infected.
For many epidemics, the dynamics are such that $ R_{0}$ can ...

92D30 ; 05C80 ; 92D25 ; 60J28

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