Mathematical Modelling of Malaria with Treatment

This paper proposes a Susceptible-Infective-Susceptible (SIS) model to study the malaria transmission with treatment by considering logistic growth of mosquito population. In this work, it is assumed that the treatment rate is proportional to the number of infectives below the capacity and is constant when the number of infectives is greater than the capacity. We find that the system exhibits backward bifurcation if the capacity is small and it gives bi-stable equilibria which makes the system more sensitive to the initial conditions. The existence and stability of the equilibria of the model are discussed in detail and numerical simulations are presented to illustrate the numerical results.     

In search for the optimal strategy in population dynamics

A unification of recently proposed models describing population dynamics is presented. We study the effect of different factors, like environmental conditions, concentration of individuals in a given area and migration strategies, on population dynamics. Moreover, we show that a population occupying a smaller area is more susceptible to extinction, which is a well known biological fact. We solve the model using Monte Carlo simulations and the mean-field approach. Constructing flow diagrams we find the optimal strategy in population dynamics. Received 10 June 2001 and Received in final form 22 November 2001     

Modelling collective opinion formation by means of active Brownian particles

The concept of active Brownian particles is used to model a collective opinion formation process. It is assumed that individuals in community create a two-component communication field that influences the change of opinions of other persons and/or can induce their migration. The communication field is described by a reaction-diffusion equation, the opinion change of the individuals is given by a master equation, while the migration is described by a set of Langevin equations, coupled by the communication field. In the mean-field limit holding for fast communication we derive a critical population size, above which the community separates into a majority and a minority with opposite opinions. The existence of external support (e.g. from mass media) changes the ratio between minority and majority, until above a critical external support the supported subpopulation exists always as a majority. Spatial effects lead to two critical “social” temperatures, between which the community exists in a metastable state, thus fluctuations below a certain critical wave number may result in a spatial opinion separation. The range of metastability is particularly determined by a parameter characterizing the individual response to the communication field. In our discussion, we draw analogies to phase transitions in physical systems. Received 26 November 1999     

Low temperature thermodynamics of inverse square spin models in one dimension

We present a field-theoretic renormalization group calculation in two loop order for classical O(N)-models with an inverse square interaction in the vicinity of their lower critical dimensionality one. The magnetic susceptibility at low temperatures is shown to diverge like with a=(N-2)/(N-1) and . From a comparison with the exactly solvable Haldane-Shastry model we find that the same temperature dependence applies also to ferromagnetic quantum spin chains. Received: 20 February 1998 / Revised: 27 April 1998 / Accepted: 30 April 1998     

Anomalous temperature dependence of resistivity in quasi-one-dimensional conductors in a strong magnetic field

We present a heuristic, semiphenomenological model of the anomalous temperature (T) dependence of resistivity recently observed experimentally in the quasi-one-dimensional (Q1D) organic conductors of the family in moderately strong magnetic fields. We suggest that a Q1D conductor behaves like an insulator (), when its effective dimensionality is one, and like a metal (), when its effective dimensionality is greater than one. Applying a magnetic field reduces the effective dimensionality of the system and switches the temperature dependence of resistivity between the insulating and metallic laws depending on the magnitude and orientation of the magnetic field. We critically analyze whether various microscopic models suggested in literature can produce such a behavior and find that none of the models is fully satisfactory. In particular, we perform detailed analytical and numerical calculations within the scenario of magnetic-field-induced spin-density-wave precursor effect suggested by Gor'kov and find that the theoretical results do not agree with the experimental observations. Received 20 October 1998     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

Molecular-dynamics simulation of a glassy polymer melt: Incoherent scattering function

We present simulation results for a model polymer melt, consisting of short, nonentangled chains, in the supercooled state. The analysis focuses on the monomer dynamics, which is monitored by the incoherent intermediate scattering function. The scattering function is recorded over six decades in time and for many different wave-vectors which range from the size of a chain to about three times the maximum position of the static structure factor. The lowest temperatures studied are slightly above , the critical temperature of mode-coupling theory (MCT), where was determined from a quantitative analysis of the - and -relaxations. We find evidence for the space-time factorization theorem in the -relaxation regime, and for the time-temperature superposition principle in the -regime, if the temperature is not too close to . The wave-vector (q-) dependence of the nonergodicity parameter, of the critical amplitude, and the -relaxation time are in qualitative agreement with calculations for hard spheres. For q larger than the maximum of the structure factor the -relaxation time already agrees fairly well with the asymptotic MCT-prediction . The behavior of the relaxation time at small q can be rationalized by the validity of the Gaussian approximation and the value of the Kohlrausch stretching exponent, as suggested in neutron-scattering experiments. Received 30 October 1998     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

Stretching DNA: Role of electrostatic interactions

The effect of electrostatic interactions on the stretching of DNA is investigated using a simple worm like chain model. In the limit of small force there are large conformational fluctuations which are treated using a self-consistent variational approach. For small values of the external force f, we find the extension scales as where is the Debye screening length. In the limit of large force the electrostatic effects can be accounted for within the semiflexible chain model of DNA by assuming that only small excursions from rod-like conformations are possible. In this regime the extension approaches the contour length as where f is the magnitude of the external force. The theory is used to analyze experiments that have measured the extension of double-stranded DNA subject to tension at various salt concentrations. The theory reproduces nearly quantitatively the elastic response of DNA at small and large values of f and for all concentration of the monovalent counterions. The limitations of the theory are also pointed out. Received 13 October 1998 and Received in final form 9 June 1999     

Critical behavior of a three-state Potts model on a Voronoi lattice

We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8 000 sites. We consider the effect of an exponential decay of the interactions with the distance, , with a>0, and observe that this system seems to have critical exponents and which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio remains essentially the same. We find numerical evidences (although not conclusive, due to the small range of system size) that the specific heat on this random system behaves as a power-law for a=0 and as a logarithmic divergence for a=0.5 and a=1.0 Received 5 April 2000     

Lehmann rotation of cholesteric droplets subjected to a temperature gradient: Role of the concentration of chiral molecules

We present a systematic study of the Lehmann rotation of cholesteric droplets subjected to a temperature gradient when the concentration of chiral molecules is changed. The liquid crystal chosen is an eutectic mixture of 8CB and 8OCB doped with a small amount of the chiral molecule R811. The angular velocity of the droplets strongly depend on their size and on the concentration of chiral molecules. The Lehmann coefficient is estimated by using three different methods. Our results are consistent with a Lehmann coefficient proportional to the concentration of chiral molecules. We additionally show the existence of a critical size of the droplets below which they change texture and stop rotating.     

Evidence for N1/3 dependence of the sticking cross-section of atoms on small and medium-size van der Waals clusters

This paper introduces a simple model describing the cluster growth in supersonic expansions. The predicted terminal mean cluster size is compared to the available data in the case of argon. The agreement between the model and the experimental results requires that the cross-section describing the sticking of an atom on a cluster of size N scales like with in the range 0.34-0.44, well below the predicted by the simplest geometrical scaling argument. We explain this unexpected result in two steps. First, using Monte Carlo simulations, we check that the potential between an atom and a cluster is accurately represented by the Gspann and Vollmar potential, even at finite temperature. Then, using Langevin's approximation, we show that the sticking cross-section scales like N ^1/3 for small to moderate N values and switches to the geometric scaling N ^2/3 for very large N values. The crossover between these two scalings occurs when for argon, but the mean exponent over the size range 1-10⁴ is 0.46. This N scaling of the sticking cross-section should play an important role whenever condensation is important as it modifies the kinetics of the early stages. Received 20 May 1999 and Revised in final form 22 July 1999     

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL ^d lattice scales like . I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.     

Griffiths phase manifestation in disordered dielectrics

We predict the existence of a Griffiths phase in dielectrics with a concentrational crossover between dipole glass (electric analog of spin glass) and ferroelectricity. Particular representatives of the above substances are KTaO₃:Li, Nb, Na, or relaxor ferroelectrics like Pb_1–xLa_xZr_0.65Ti_0.35O₃. Since this phase exists above the ferroelectric phase-transition temperature (but below that temperature for ordered substances), we call it a “para-glass phase”. We assert that the difference between paraelectric and para-glass phases in the above substances is the existence of clusters (inherent to the “ordinary” Griffiths phase of Ising magnets) of correlated dipoles. We show that randomness plays a decisive role in the Griffiths (para-glass) phase formation: this phase does not exist in a mean field approximation. To investigate the Griffiths phase properties, we calculate the density of Yang-Lee (YL) zeros in the partition function and find that it has “tails” inherent to the Griffiths phase in the above temperature interval. We perform calculations on the basis of our self-consistent equation for the long-range order parameter in an external electric field. This equation has been derived in the framework of the random field theory. The latter automatically incorporates both short-range (due to indirect interaction via transverse optical phonons of the host lattice) and long-range (ordinary dipole-dipole) interactions between impurity dipoles, so that the problem of long-range interaction considerations does not appear in it. Received 17 May 2000     

Fluctuations, response and aging dynamics in a simple glass-forming liquid out of equilibrium

By means of molecular dynamics computer simulations we investigate the out of equilibrium relaxation dynamics of a simple glass former, a binary Lennard-Jones system, after a quench to low temperatures. We find that one-time quantities, such as the energy or the structure factor, show only a weak time dependence. By comparing the out of equilibrium structure factor with equilibrium data we find evidence that during the aging process the system remains in that part of phase space that mode-coupling theory classifies as liquid like. Two-times correlation functions show a strong time and waiting time dependence. For large and times corresponding to the early -relaxation regime the correlators approach the Edwards-Anderson value by means of a power-law in time. For large but fixed values of the relaxation dynamics in the -relaxation regime seems to be independent of the observable and temperature. The -relaxation shows a power-law dependence on time with an exponent which is independent of but depends on the observable. We find that at long times the correlation functions can be expressed as and compute the function h(t). This function is found to show a t-dependence which is a bit stronger than a logarithm and to depend on the observable considered. If the system is quenched to very low temperatures the relaxation dynamics at long times shows fast drops as a function of time. We relate these drops to relatively local rearrangements in which part of the sample relaxes its stress by a collective motion of 50-100 particles. Finally we discuss our measurements of the time dependent response function. We find that at long times the correlation functions and the response are not related by the usual fluctuation dissipation theorem but that this relation is similar to the one found for spin glasses with one step replica symmetry breaking. Received 17 May 1999     

Local impurity phase pinning and pinning force in charge density waves

Starting from the static Fukuyama-Lee-Rice equation for a three-dimensional incommensurate charge density wave (CDW) in quasi one-dimensional conductors a solvable model for local phase pinning by impurities is defined and studied. We find that average CDW energy and average pinning force show critical behaviour with respect to the pinning parameter h. Specifically the pinning force exhibits a threshold at h=1 with exponent . Our model exemplifies a general concept of local impurity pinning in which the force exerted by the impurity on the periodic CDW structure becomes multivalued and metastable states appear beyond a threshold. It is found that local impurity pinning becomes less effective at low temperatures and may eventually cease completely. These results are independent of spatial dimensionality as expected for local impurity pinning. Comparison with Larkin's model is also made. Received 8 July 1998     

Model of evolution with sexual and non-sexual reproduction

Using a previously introduced model (Refs. [9, 10]) of biological evolution, we study the role of the reproduction pattern on the fate of an evolving population. Each individual is under the selectional pressure from the environment and random harmful mutations. The habitat (“climate") is changing periodically. Evolution of populations following three reproduction patterns are compared - an asexual one (without recombination) and two with recombination - asexual (meiotic parthenogenesis) and sexual. We show, via Monte-Carlo simulations, that sexual reproduction leads to a better adaptation to the environment, slightly better survival rates for the individuals and higher probability that the population will not become extinct in difficult external conditions. The benefits of sexual reproduction are enhanced by higher birth rates and lower mutation rates. In the case of low birth rates and high mutation rates there is a small preference for the meiotic parthenogenesis. Received 9 August 1999