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In this paper we examine the mechanisms by which a given viral disease spreads successfully in a geographical region. We propose that the key factors that cause pandemics are of social nature, whose influence can be modeled by mobility parameters that take into account the different ways in which the disease can be transmitted. We use a new SEIR(S) model that is able to reproduce the temporal behavior of a given disease within a local homogeneous community. The geographical spread of the disease is studied by defining a two dimensional grid of cells, weighted by the population density, and a SEIR(S) model in each cell. The cells are coupled with their nearest neighbors, simulating terrestrial communications, and with long range links representing airline routes. We propose that the mechanisms of transmission depend on the length scale. We simulate other transportation events as thermal noise, which turns out to be a key factor. Numerical simulations are presented to illustrate the results of the model. Actual data on the population, communication networks and incidence of influenza pandemic in Mexico in 2009 are used to validate the model. The agreement with actual data is satisfactory. Furthermore, it accounts for the effect of social distancing measures implemented in Mexico during the 2144 Rafael A. Barrio et al. early stages of the pandemic. Since the model can predict the development of viral epidemics with few data from the early stages, it can be used to predict the effect of actions to be taken to reduce and mitigate the risks of pandemics.

ResumenEn la actividad eléctrica cardiaca, distintos tipos de onda viajan a través del corazón. Presentamos un modelo de la actividad eléctrica del corazón, proponemos que los frentes de onda homogéneos que se propagan en el corazón son, de hecho, ondas solitarias, o “solitones”. Usamos un conjunto general de ecuaciones de reacción-difusión conocido como el modelo de Barrio-Varea-Aragón-Maini (BVAM)[1], el cual presenta una abundancia de bifurcaciones no lineales, siendo capaces de encontrar la ruta al caos usando un mapeo de las ecuaciones de amplitud a la dinámica de la ecuación compleja de Ginzburg-Landau. Estudiamos numéricamente la dinámica de los frentes de onda en el modelo BVAM para describir los mecanismos que conducen a la fibrilación en el corazón y comparamos los resultados con datos experimentales.AbstractIn cardiac electrical activity, different types of waves meander through the heart. We present a model of the electrical activity of the heart that proposes that the homogeneous wave fronts propagating through the heart are in fact solitons. We use a general set of reaction-diffusion equations known as the Barrio-Varea-Aragón-Maini (BVAM) model[1] that presents a wealth of non-linear bifurcations, and we are able to follow the route to chaos, using a mapping of the amplitude equations to the dynamics of the complex Ginzburg-Landau equation. We study the dynamics of wave fronts numerically in the BVAM model to describe the mechanisms leading to heart fibrillation and compare the findings with experimental data.

For many years Turing systems have been proposed to account for spatial and spatiotemporal pattern formation in chemistry and biology. We extend the study of Turing systems to investigate the rô1e of boundary conditions, domain shape, non-linearities, and coupling of such systems. We show that such modifications lead to a wide variety of patterns that bear a striking resemblance to pigmentation patterns in fish, particularly those involving stripes, spots and transitions between them. Using the Turing system as a metaphor for activator-inhibitor models we conclude that such a mechanism, with the aforementioned modifications, may play a rô1e in fish patterning.

We review the general properties of non-linear systems and show the basic techniques, used universally, to study the symmetry breaking and bifurcation properties. We exemplify these characteristics by using a Turing system that is general enough as to present many of the universal features of non-linear systems. We then show some interesting applications to various problems that we have treated in the past.

We study the stability of planar, cylindrical and spherical interfaces with respect to shape and width fluctuations for a model amphiphile solution described by a free energy density functional with square-gradient and square-Laplacian terms. That is, we determine the stability matrix when the stationary state consists of an interface with given geometry that separates two immiscible solvent phases. From the spectrum and the related eigenfunctions of this matrix we establish where lamellar and micellar domain-structured phases occur, and contrast our results with those for a simple square-gradient fluid model for which these phases are always unstable. We also characterize some instability properties such as the buckling of lamella, the undulation of cylindrical structures and the nucleation of micelles.

The density fluctuations about the equilibrium structure of fluids confined by parallel planar walls are analyzed for the cases of identical and symmetrically opposed fields at the walls. We determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) for conditions both above and below the wetting transition temperature Tw of the semi-infinite system and corroborate in all cases that the equilibrium configurations are stable. We identify the fluctuations close to the walls and in the middle of the slab and discuss their effect when the wall separation L diverges. For competing walls above Tw the localized fluctuation with lowest eigenvalue describes the displacements of the incipient wetting films that become unimpeded interfacial translations for L→∞. Below Tw the fluctuations with lowest eigenvalue correspond to stiffer deformations extended across the slab. For identical walls above Tw coexisting states display incipient prewetting films and the lowest eigenvalue describes the nature of their growth as L increases. We also calculate the pair correlation function for the inhomogeneous states and, for symmetrically opposed walls, we obtain standard Ornstein–Zernike (OZ) behavior at the walls, but find significant deviations from this law at the interface-like region in the middle of the slab. To model fluids with short-ranged forces we use a ferromagnetic Ising-type Hamiltonian in mean-field approximation.

We study the stability of cylindrical and spherical interfaces with respect to density fluctuations within the square-gradient approximation. That is, we determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) when the stationary state is a cylindrical or spherical droplet of a stable phase embedded in the metastable phase. For these geometries the stationary states are unstable, some of the eigenvalues are negative and their eigenfunctions represent those fluctuations that are amplified in a process where the equilibrium state is reached. At early times, in a simple model A kinetics, the eigenfunctions represent the fluctuations that grow or decay with a simple exponential law and with a characteristic time that is proportional to the inverse of the eigenvalues. Our results agree with the stability criteria obtained from the Laplace equation, that is, the nucleation of critical droplets and in the case of cylinders also the Rayleigh instability. In the limit of infinite radius we recover the known results for the planar interface between two stable phases. The modes with lowest energy correspond to the customary capillary waves, while other modes with higher energy associated to changes in the interfacial width are shown to be related to a novel interfacial coefficient.

Symmetric patterns were recently obtained in the famous Faraday experiment by using a special liquid. A model is developed to explain the robustness of the observed patterns. A dissipation mechanism is introduced phenomenologically and shown to be responsible for the stabilization of patterns with several symmetries. The resulting system of nonlinear equations resembles the well-known Turing equations used to study pattern formation in biological systems.

The spinodal decomposition of a simple fluid or binary alloy is studied within the Cahn model of conserved order parameter for a slab geometry generated by competing walls. We restrict our calculations to short-range surface forces and consider parameter values that correspond to both nonwet and wet equilibrium states. We find that, after a rapid development of thin films at the walls, phase patterns form and grow into lamellar-like domains parallel to the wall surfaces. The stabilization of this geometry allows for the study of the later stages of phase separation through the use of the associated dynamical-system phase portraits of the nonlinear problem characteristic of descriptions of inhomogeneities in one spatial direction. We observe that coarsening in the late stage slows down markedly and comes to a standstill because the driving force for the kinetic equation vanishes almost everywhere for the layered structure. The infrequent coarsening events at long times can be monitored via the accompanying large and clearly defined disjoining pressure peaks.

In this paper we address the problem of pattern formation in confined Turing systems in two dimensions, when one assumes the enhancement of the concentration of one of the chemicals at some of the confining surfaces. This model is suitable to study biological systems, such as the skin patterns shown by some marine fish. We also study numerically the dynamical growth of the system by changing the size of the confined region while dynamical diffusion and reaction phenomena take place. This idea is tested in two different models. This allows one to estimate the robustness of stripe formation.

A prominent feature of the phase behavior of aqueous solutions of nonionic amphiphiles is a phase separation into dilute and concentrated micellar liquids above a lower critical solution temperature T c . Experiment has revealed a strong asymmetry of the coexistence curve, a high sensitivity of the (effective) critical indexes on deviations of the critical composition, and an unusual excess adsorption at the liquid-liquid interface as indicated by ellipsometric measurements. Here, we review and extend the study of a micellar solution model designed to describe amphiphiles in polar solvents and, as we illustrate, it reproduces the properties mentioned above. The model allows for elimination of the degrees of freedom within the aggregates, and this leads to an equivalence with the ordinary Ising model but with temperature and concentration-dependent field and coupling. The latter is identified with the effective intermicellar interaction.

We derive explicit expressions for the stress tensor for general inhomogeneities in a one-component simple fluid in terms of density gradients and moments of the direct correlation function. The expressions follow from the change in grand potential ΔΩV that takes place in a selected portion of fluid of volume V as an arbitrary strain is applied to it. We employ the free energy density functional approach to determine ΔΩV, and consider two model mean-field density functionals, one spatially non-local and the other a local Landau form that contains a squared-gradient and a squared-laplacian terms. In the first case the expression for the stress tensor is spatially non-local, and in the latter case the tensor can be written down in the form of a volume σ and a surface contribution τ. These two terms can be modified by the transformation of volume into surface terms, or vice versa, of the fixed change ΔΩV, and in this way symmetric forms for both σ and τ can be obtained. We observe that the surface term τ and, more generally, the non-locality of the stress tensor are features that arise as the finite range of molecular interactions is restored in a local description by incorporation of higher order terms in the derivatives of the density. Our results corroborate expressions for the elastic bending free energy terms of curved interfaces derived previously.

We report on a study of the free energy of a spherical interface described by a van der Waals density functional with a squared-Laplacian term. We examine the bulk, the surface tension and the bending rigidity terms, and find the position for the dividing surface that satisfies the Laplace equation generalized to nonvanishing bending energy. In doing this we have made explicit the connection between two previously derived but dissimilar sets of expressions for the interfacial coefficients that stem from the same free energy model (one by Romero-Rochin et al. (Phys. Rev. A 44 (1991) 8417; Phys. Rev. E 46 (1993) 1600) and the other by Gompper and Zschocke (Europhys. Lett. 18 (1991) 731) and by Blokhuis and Bedeaux (Mol. Phys. 80 (1993) 705).

We propose a model to describe the physical mechanisms by which chemical substances separate in the vicinity of a membrane. We assume that the adsorption of the different components of a complex liquid on a membrane is governed by interactions that couple them to the spontaneous curvature of the membrane. This problem is relevant to many fields in science, as cell constriction and division, micelles with cosurfactants, holometamorphosis, and morphogenesis in general.

We use density functional theory to analyse the structure of the stress tensor σ for fluid interfaces of arbitrary shape, and determine the spatially non-local components of σ that correspond to a description with a semi-orthogonal set of coordinates (ˆn, t 1, t 2). We then study a local, van der Waals-type, density functional with squared-Laplacian term and arrive at closed and general expressions for the surface tension γ, the bending constants κ and κ and the spontaneous curvature c 0. These expressions have the same general form for all interfacial shapes, but their precise values are curvature-dependent. In the case of γ this dependence is that already known and measured by the Tolman length δ. We discuss and compare our results with those of others and resolve existing discrepancies.

In this paper we propose a model to describe the mechanisms by which undifferentiated cells attain gene configurations underlying cell fate determination during morphogenesis. Despite the complicated mechanisms that surely intervene in this process, it is clear that the fundamental fact is that cells obtain spatial and temporal information that bias their destiny. Our main hypothesis assumes that there is at least one macroscopic field that breaks the symmetry of space at a given time. This field provides the information required for the process of cell differentiation to occur by being dynamically coupled to a signal transduction mechanism that, in turn, acts directly upon the gene regulatory network (GRN) underlying cell-fate decisions within cells. We illustrate and test our proposal with a GRN model grounded on experimental data for cell fate specification during organ formation in early Arabidopsis thaliana flower development. We show that our model is able to recover the multigene configurations characteristic of sepal, petal, stamen and carpel primordial cells arranged in concentric rings, in a similar pattern to that observed during actual floral organ determination. Such pattern is robust to alterations of the model parameters and simulated failures predict altered spatio-temporal patterns that mimic those described for several mutants. Furthermore, simulated alterations in the physical fields predict a pattern equivalent to that found in Lacandonia schismatica, the only flowering species with central stamens surrounded by carpels.

An extended capillary-wave model is derived from density functional theory and the predictions of the usual capillary-wave model are reexamined. The model is expressed in terms of the direct correlation function C(r, r') of the equilibrium fluid. A small wavevector expansion of C(r, r') reduces the extended model to the usual one but it is shown that higher-order terms also play a role in determining interfacial properties in the limit of vanishing gravity. The meaning of the root-mean-square fluctuation W of the distortions of the Gibbs dividing surface and its relation to the actual width Ω of the density profile are specified. For spatial dimensions d < 3 the predictions of the extended and the usual models agree. Explicit calculations for d = 3 show W ∼ In (Lc/a) with Lc the classical capillary length and a a length proportional to the fourth transverse moment of C(r, r'). Adoption of a scaling hypothesis near g → 0, similar to that proposed by Weeks, shows that a ∼ Lc and W does not diverge as g → 0. It is argued that this does not necessarily imply that the width Ω is also finite, although this remains a possibility. Extensions to Week's hypothesis are also examined.

We investigate the properties of the line, say the x-axis, at which two or more interfaces meet, and derive an exact expression for its excess free energy in terms of the gradients of the densities ϱa = ϱa(y,z) and the direct correlation function C(2)ab(|x−x'|; y, y'; z, z'). We evaluate the line tension τ in mean field and for various surface enhancements g at vanishing substrate field h1, and include the interesting case when the line inhomogeneity persists beyond bulk criticality. The critical index for τ is determined for different surface criticalities.

The standard wetting transition consists of the transformation of a microscopically thin two-dimensional interface into a macroscopically thick structure composed of two interfaces separated by a bulk phase. We consider the one-dimensional analog of this phenomenon, when a contact line among three or more phases decomposes into two contact lines separated by an interface. We uncover a wetting transition for the contact line, which occurs at surface two-phase coexistence, as a function of a line or edge field. This is exemplified by means of a lattice mean-field calculation for an Ising model bounded by two surfaces that meet in an edge.

We derive the interfacial-curvature free energy for a simple fluid from density-functional theory, and find a form matching that for a two-dimensional shearless elastic media. The fourth moment of the direct correlation function and the density gradient determine the bending moduli \ensuremath{\kappa} and \ensuremath{\kappa}\ifmmode\bar\else\textasciimacron\fi{}, while the spontaneous curvature ${\mathit{c}}_{0}$ is given by asymmetry in the density. We obtain the critical indices of these quantities and corrections to the Laplace equation for curved interfaces.

The long wavelength behavior of the Ornstein-Zernike direct correlation function for nonuniform fluids c(r,r′) has provided important results that comprise our current understanding of fluid interfaces. As we know, the surface tension is obtained from the second moment of c(r,r′), and long-ranged correlations parallel to the surface stand among the main predictions. Also, conventional capillary wave theory leads to its notorious divergent results for the interfacial width when the surface tension cost of thermally excited fluctuations is considered. Here we discuss the consequences of the higher moments of c(r,r′) in the density functional theory of fluid interfaces. We find that in an appropriately extended capillary wave model the interfacial width is finite in the absence of external fields. We derive the expression for the elastic curvature energy and find that the bending moduli are given by the fourth moment of c(r,r′). We obtain too in the same fashion the line tension that originates when interfaces meet.

We describe two types of curvature-related interfacial transitions occuring in soluble amphiphile monolayers under compression. The first type corresponds to the buckling of areas with sizes up to the monolayer's de Genres--Taupin persistence length squared, and starts to occur when the interfacial tension is about one-half its bare value. The second type corresponds to the transformation of the simply connected monolayer into either a dispersion of droplets or an interconnected bicontinuous structure. The resulting type of volume-spanning state is determined by the sign of the saddle-splay bending constant

In this paper, we examine the dynamics of reaction-diffusion systems with fractional time derivatives. It is shown that in these conditions diffusion is anomalous, in the sense that the mean-square displacement $⟨{r}^{2}⟩\ensuremath{\sim}{t}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}l1$, a situation known as subdiffusion. We study the conditions for the appearance of a diffusion-driven instability and show that the restrictive conditions for a Turing instability are relaxed. This implies that systems whose kinetics are not of the activator-inhibitor kind can have a Turing instability and a modulated final state. We demonstrate our results with numerical calculations in two dimensions using a generic Turing model.

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.

In this paper we study the solutions of a generalized reaction-diffusion system with a bistable reaction term, and considering directional anomalous diffusion. We use the well-known properties of fractional derivatives to model asymmetric anomalous diffusion, and obtain traveling wave solutions that propagate in a direction that depends on the metastability of the front, the fractional exponent and the asymmetry of the diffusion.

We study numerically the dynamics, in two dimensions, of phase separation in ternary mixtures with dipolar interactions which lead to the formation of modulated phases. We distinguish three different modulated phases: a hexagonal phase of droplets, a lamellar phase, and a hexagonal phase of bubbles. Inside the crystal structures an additional phase separation occurs ``coloring'' the texture. The dynamics in the droplet phase mixes the two kinds of droplets of different composition. The lamellar phase does not evolve toward parallel lamellae, and the phase separation inside the channels proceeds until they reach a grain boundary. The hexagonal bubble phase is never formed due to the phase separation that forms an interface of bubbles which blocks the contact between the two phases. In its place we find an unsuspected lamellar phase.

We study numerically the dynamics of phase separation in ternary mixtures at a triple point. For the full range of compositions and for different interaction parameters, the long time growth is in accord with a universal law. The early time behavior is governed by the structure of the spinodal region, including the possibility of a two step separation and decomposition originating at a surface and propagating into the bulk. The appearance of the domains is governed by the wetting properties of the mixture and the growth of a wetting layer follows again the universal law; a result that we can interpret with a simple phenomenological model.

We describe two types of curvature-related interfacial transitions occurring in soluble amphiphile monolayers. The first type corresponds to the buckling of areas with sizes up to the monolayer’s persistence length squared, and starts to occur when the interfacial tension is about one-half its bare value. A relationship is hinted at between the properties of this transition and the stability of foam films. The second, topologically-driven, transition separates a simply-connected surface state and a multiply-disconnected (or multiply connected) volume-filling state. Its order parameter is given by the surface’s genus (or number of micelles). The three-dimensional gain in configurational entropy of surface fragmentation masks the singularity and provides the known continuous properties of amphiphile solutions at the critical micellar concentration. The resulting structure, droplet dispersion or interconnected bicontinuous, is determined by the sign of the monolayer’s saddle-splay bending constant.

The surface energy σ of an ordering AB alloy in a bcc lattice cleaved along a (001) plane has been investigated. Segregation to the surface, say of species A,, below the critical temperature T c leads to two kinds of surface behavior according to whether the lattice ends on an A- or a B-rich sublattice layer. In the former case the composition profiles for alternating layers are smooth at all T ≤ T c , whereas in the second case the low T profiles show an antiphase boundary (APB) parallel to the surface which enriches the surface of A atoms. At a higher temperature T u ≤ T c an unpinning transition occurs and the APB travels to the bulk. We point out the relevance of this phenomenon for domain coarsening induced by surface treatment of materials.

We have calculated the pressure coefficients of the main gaps in Si, using the Empirical Pseudopotential Method (EPM). We find a trend toward metallization at high hydrostatic pressures. The deformation potential at the top of the valence band for uniaxial stress along the (0, 0, 1) direction is also obtained. All of the calculated pressure coefficients are in good agreement with experiment.

A statistical mechanical treatment for the van der Waals fluid mixture is developed, based on Widom’s potential‐distribution theory for nonuniform fluids. For the specific choice of Kac pair interactions, in the van der Waals limit, a set of coupled second‐order differential equations determine the density profiles. The solution of these equations and the determination of the interfacial tension are discussed in terms of a mechanical analogy.

Orthobaric densities of a square well fluid are obtained by means of a molecular dynamics simulation. Hard wall boundary conditions in one direction and periodic boundary conditions in the other two, added to a mean density close to the critical have two effects: a fast spinodal decomposition that condenses a slab of liquid at the center of the box plus a better sampling of the whole orthobaric curve. Results for the liquid densities compare well with previously obtained data while vapor densities are more sensitive to capillary effects. These finite size effects are analyzed within a lattice gas model, especially those produced by changes in box size and mean density. The method is well suited to predict orthobaric and miscibility curves for binary mixtures once the orthobaric curves of each molecular species have been properly adjusted to experiment.

Transition de rugosite de l'interphase dans un melange modele constitue de copolymeres bisequences amphiphiles de deux solvants non miscibles et d'un cosurfactant monomerique

We analyze the critical behavior of the magnetization at antiphase boundaries (APB s) of magnetic alloys. For an Ising-like model alloy in the mean-field approximation the symmetry-breaking APB s induce, within a certain range of bulk compositions, loca1 magnetic order above the Curie temperature. This purely interfacial critical behavior explains the self-reversal thermoremanent magnetism of the ilmenite (FeTi02) haematite (Fe203) solid-solution series (Ilm„Hemi ).