Showing papers in "Journal of Mathematical Biology in 2011"

Predator–prey system with strong Allee effect in prey

Abstract: Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

A reaction-diffusion malaria model with incubation period in the vector population.

Abstract: Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction-diffusion model. We then define the basic reproduction ratio R₀ and show that R₀ serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.

A note on a paper by Erik Volz: SIR dynamics in random networks

Abstract: Recent work by Volz (J Math Biol 56:293–310, 2008) has shown how to calculate the growth and eventual decay of an SIR epidemic on a static random network, assuming infection and recovery each happen at constant rates. This calculation allows us to account for effects due to heterogeneity and finiteness of degree that are neglected in the standard mass-action SIR equations. In this note we offer an alternate derivation which arrives at a simpler—though equivalent—system of governing equations to that of Volz. This new derivation is more closely connected to the underlying physical processes, and the resulting equations are of comparable complexity to the mass-action SIR equations. We further show that earlier derivations of the final size of epidemics on networks can be reproduced using the same approach, thereby providing a common framework for calculating both the dynamics and the final size of an epidemic spreading on a random network. Under appropriate assumptions these equations reduce to the standard SIR equations, and we are able to estimate the magnitude of the error introduced by assuming the SIR equations.

Optimal control of epidemics with limited resources.

Abstract: We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.

Effective degree network disease models

Abstract: An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.

Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent

Abstract: Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals—each with many genes—splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are four species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendant branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.

Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation

Abstract: The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.

Exact epidemic models on graphs using graph-automorphism driven lumping

Abstract: The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.

Persistence in fluctuating environments

Abstract: Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions. One day is fine, the next is black.—The Clash

The surface finite element method for pattern formation on evolving biological surfaces.

Abstract: In this article we propose models and a numerical method for pattern formation on evolving curved surfaces. We formulate reaction-diffusion equations on evolving surfaces using the material transport formula, surface gradients and diffusive conservation laws. The evolution of the surface is defined by a material surface velocity. The numerical method is based on the evolving surface finite element method. The key idea is based on the approximation of Γ by a triangulated surface Γh consisting of a union of triangles with vertices on Γ. A finite element space of functions is then defined by taking the continuous functions on Γh which are linear affine on each simplex of the polygonal surface. To demonstrate the capability, flexibility, versatility and generality of our methodology we present results for uniform isotropic growth as well as anisotropic growth of the evolution surfaces and growth coupled to the solution of the reaction-diffusion system. The surface finite element method provides a robust numerical method for solving partial differential systems on continuously evolving domains and surfaces with numerous applications in developmental biology, tumour growth and cell movement and deformation.

A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Abstract: Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide (NO) have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which may reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways, (ii) the concentrations in the tracheo-bronchial lining fluid, (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, the model illuminates the discrepancies between observed and theoretically predicted blood-breath ratios of acetone during resting conditions, i.e., in steady state. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases and thus is expected to have general relevance for a wider range of blood-borne inert gases. The chief intention of the present modeling study is to provide mechanistic relationships for further investigating the exhalation kinetics of acetone and other water-soluble species. This quantitative approach is a first step towards new guidelines for breath gas analyses of volatile organic compounds, similar to those for nitric oxide.

An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution.

Abstract: In this paper, we present a mathematical description for excitable biological membranes, in particular neuronal membranes. We aim to model the (spatio-) temporal dynamics, e.g., the travelling of an action potential along the axon, subject to noise, such as ion channel noise. Using the framework of Piecewise Deterministic Processes (PDPs) we provide an exact mathematical description—in contrast to pseudo-exact algorithms considered in the literature—of the stochastic process one obtains coupling a continuous time Markov chain model with a deterministic dynamic model of a macroscopic variable, that is coupling Markovian channel dynamics to the time-evolution of the transmembrane potential. We extend the existing framework of PDPs in finite dimensional state space to include infinite-dimensional evolution equations and thus obtain a stochastic hybrid model suitable for modelling spatio-temporal dynamics. We derive analytic results for the infinite-dimensional process, such as existence, the strong Markov property and its extended generator. Further, we exemplify modelling of spatially extended excitable membranes with PDPs by a stochastic hybrid version of the Hodgkin–Huxley model of the squid giant axon. Finally, we discuss the advantages of the PDP formulation in view of analytical and numerical investigations as well as the application of PDPs to structurally more complex models of excitable membranes.

On selection dynamics for competitive interactions.

Abstract: In this paper, we are interested in an integro-differential model that describe the evolution of a population structured with respect to a continuous trait. Under some assumption, we are able to find an entropy for the system, and show that some steady solutions are globally stable. The stability conditions we find are coherent with those of Adaptive Dynamics.

Modeling the adaptive immune response in HBV infection

Abstract: The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R 0 < 1, (2) and the existence of four endemic steady states when R 0 > 1. The local stability of these steady states depends on functions of R 0. Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.

Global analysis on delay epidemiological dynamic models with nonlinear incidence

Abstract: In this paper, we derive and study the classical SIR, SIS, SEIR and SEI models of epidemiological dynamics with time delays and a general incidence rate. By constructing Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is shown. This analysis extends and develops further our previous results and can be applied to the other biological dynamics, including such as single species population delay models and chemostat models with delay response.

Differential geometry based solvation model II: Lagrangian formulation

Abstract: Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent–solute interaction potential. The nonpolar solvation model is completed with a Poisson–Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent–solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace–Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent–solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein–protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein–protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

Abstract: In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M c . However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above M c always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M c , which is forbidden in the parabolic–elliptic case.

Genealogy with seasonality, the basic reproduction number, and the influenza pandemic.

Abstract: La reproductivite nette R0 est utilisee en biologie des populations et notamment en epidemiologie depuis plusieurs decennies. Mais on n'a propose une definition convenant au cas des modeles avec coefficients periodiques qu'il y a quelques annees. La definition fait intervenir le rayon spectral d'un operateur integral. Comme dans l'etude des modeles epidemiques structures dans un environnement constant, il est bon d'expliquer la signification biologique de ce rayon spectral. On montre dans cet article que R0 pour les modeles periodiques est encore un taux asymptotique de croissance par generation. On insiste aussi sur la difference entre ce R0 theorique pour les modeles periodiques et la ≪ reproductivite nette ≫ obtenue en ajustant une exponentielle au debut d'une courbe epidemique. Les etudes recentes sur la pandemie de grippe H1N1 n'ont pas pris en compte cette difference.

The minimal model of the hypothalamic–pituitary–adrenal axis

Abstract: This paper concerns ODE modeling of the hypothalamic-pituitary- adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the 'worst case parameters' is performed.

A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics

Abstract: The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space–time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non–linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a ‘moment closure’; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood superposition approximation, or ‘power-3’ closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.

Sustained oscillations for density dependent Markov processes

Abstract: Simulations of models of epidemics, biochemical systems, and other bio-systems show that when deterministic models yield damped oscillations, stochastic counterparts show sustained oscillations at an amplitude well above the expected noise level A characterization of damped oscillations in terms of the local linear structure of the associated dynamics is well known, but in general there remains the problem of identifying the stochastic process which is observed in stochastic simulations Here we show that in a general limiting sense the stochastic path describes a circular motion modulated by a slowly varying Ornstein–Uhlenbeck process Numerical examples are shown for the Volterra predator–prey model, Sel’kov’s model for glycolysis, and a damped linear oscillator

Oligomorphic dynamics for analyzing the quantitative genetics of adaptive speciation

Abstract: Ecological interaction, including competition for resources, often causes frequency-dependent disruptive selection, which, when accompanied by reproductive isolation, may act as driving forces of adaptive speciation. While adaptive dynamics models have added new perspectives to our understanding of the ecological dimensions of speciation processes, it remains an open question how best to incorporate and analyze genetic detail in such models. Conventional approaches, based on quantitative genetics theory, typically assume a unimodal character distribution and examine how its moments change over time. Such approximations inevitably fail when a character distribution becomes multimodal. Here, we propose a new approximation, oligomorphic dynamics, to the quantitative genetics of populations that include several morphs and that therefore exhibit multiple peaks in their character distribution. To this end, we first decompose the character distribution into a sum of unimodal distributions corresponding to individual morphs. Characterizing these morphs by their frequency (fraction of individuals belonging to each morph), position (mean character of each morph), and width (standard deviation of each morph), we then derive the coupled eco-evolutionary dynamics of morphs through a double Taylor expansion. We show that the demographic, convergence, and evolutionary stability of a population’s character distribution correspond, respectively, to the asymptotic stability of frequencies, positions, and widths under the oligomorphic dynamics introduced here. As first applications of oligomorphic dynamics theory, we analytically derive the effects (a) of the strength of disruptive selection on waiting times until speciation, (b) of mutation on conditions for speciation, and (c) of the fourth moments of competition kernels on patterns of speciation.

A discrete time neural network model with spiking neurons: II: Dynamics with noise

Abstract: We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky Integrate-and-Fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics.

Effects of anisotropic interactions on the structure of animal groups

Abstract: This paper proposes an agent-based model which reproduces different structures of animal groups. The shape and structure of the group is the effect of simple interaction rules among individuals: each animal deploys itself depending on the position of a limited number of close group mates. The proposed model is shown to produce clustered formations, as well as lines and V-like formations. The key factors which trigger the onset of different patterns are argued to be the relative strength of attraction and repulsion forces and, most important, the anisotropy in their application.

A stability analysis of the power-law steady state of marine size spectra.

Abstract: This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects.

Abstract: We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.

Necessary and sufficient conditions for the existence of an optimisation principle in evolution.

Abstract: This paper presents a necessary condition for the existence of a numerical quantity optimised by evolution by natural selection, which also turns out to be a sufficient condition under rather general conditions. As a corollary, a related criterion with a particularly intuitive graphical interpretation in terms of pairwise invadability plots is obtained.

Species abundance distributions in neutral models with immigration or mutation and general lifetimes

Abstract: We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate λ. Thus, the population size is a homogeneous, binary Crump-Mode-Jagers process (which is not necessarily a Markov process). We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate μ (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at constant rate θ. We are interested in the species abundance distribution, i.e., in the numbers, denoted I(n)(k) in the immigration model and A(n)(k) in the mutation model, of species represented by k individuals, k = 1, 2, . . . , n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I(t)(k); k ≥ 1) of species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher's log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by Ewens' sampling formula. In particular, I(n)(k) converges as n → ∞ to a Poisson r.v. with mean γ/k, where γ : = μ/λ. In the mutation model, as n → ∞, we obtain the almost sure convergence of n (-1) A(n)(k) to a nonrandom explicit constant. In the case of a critical, linear birth-death process, this constant is given by Fisher's log-series, namely n(-1) A(n)(k) converges to α(k)/k, where α : = λ/(λ + θ). In both models, the abundances of the most abundant species are briefly discussed.

Stability of differential susceptibility and infectivity epidemic models

Abstract: We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio [Formula: see text] and the existence and uniqueness of an endemic equilibrium when [Formula: see text] . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when [Formula: see text] . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626-644, 2005; Math Biosci Eng 3:89-100, 2006).

Epidemic growth rate and household reproduction number in communities of households, schools and workplaces

Abstract: In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.