Cholera dynamics with Bacteriophage infection: A mathematical study

Mathematical modeling of waterborne diseases, such as cholera, including a biological control using Bacteriophage viruses in the aquatic reservoirs is of great relevance in epidemiology. In this paper, our aim is twofold: at first, to understand the cholera dynamics in the region around a water body; secondly, to understand how the spread of Bacteriophage infection in the cholera bacterium V. cholerae controls the disease in the human population. For this purpose, we modify the model proposed by Codeço, for the spread of cholera infection in human population and the one proposed by Beretta and Kuang, for the spread of Bacteriophage infection in the bacteria population [1, 2]. We first discuss the feasibility and local asymptotic stability of all the possible equilibria of the proposed model. Further, in the numerical investigation, we have found that the parameter ϕ, called the phage adsorption rate, plays an important role. There is a critical value, ϕ_c, at which the model possess Hopf-bifurcation. For lower values than ϕ_c, the equilibrium E^* is unstable and periodic solutions are observed, while above ϕ_c, the equilibrium E^* is locally asymptotically stable, and further shown to be also globally asymptotically stable. We investigate the effect of the various parameters on the dynamics of the infected humans by means of numerical simulations.     

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability p _c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p _c ~ π ²/(18?log?n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is ?(log?n)^?3/2+o(1), and moreover determining it up to a poly(log?log?n)-factor. The exponent ?3/2 corrects numerical predictions from the physics literature.     

Bifurcations in a predator–prey model with general logistic growth and exponential fading memory

A multiparameter predator–prey system generalizing the model introduced in [6] is considered. The system studied in this paper corresponds to the type of models with exponential fading memory where the logistic per capita rate growth of the prey is given by an arbitrary function of class C^k, k ≥ 3. We prove that the model has a Hopf bifurcation and that there exist open sets in the parameter space such that the system exhibits singular attractors and asymptotically stable limit cycles. A numerical simulation is conducted in order to show the existence of critical attractor elements.As pointed out by Ayala et al. in [14], the Lotka–Volterra model of interspecific competition, which is based on the logistic theory of population growth and assumes that the intra and interspecific competitive interactions between species are linear, does not explain satisfactorily the population dynamics of some species. This is due to fact that the model does not take into account some important features of the population, which affect its dynamics. The model introduced in this paper provides independent conditions of these facts, for the existence of a Hopf bifurcation and the asymptotically stable limit cycles.     

A temperature-calibrated continuum model for vibrational analysis of the fullerene family using molecular dynamics simulations

In the present study, a model was proposed to determine the elastic properties of the family of fullerenes at different temperatures (between 300 and 2000 K) using a combination of molecular dynamics simulation and continuum shell theory. The fullerenes molecules examined here are eight spherical fullerenes, including C₆₀, C₈₀, C₁₈₀, C₂₄₀, C₂₆₀, C₃₂₀, C₅₀₀, and C₇₂₀. First, the breathing mode frequency and the radius of gyration of the molecules were obtained at different temperatures by molecular dynamics simulations using AIREBO potential. Then, these data were used in a continuum model to obtain the elastic coefficients of these closed clusters of carbon in terms of temperature changes. As another result of this paper is finding a nearly linear relationship between the changes in radius and breathing mode frequency of molecules versus temperature variations. Validation of the results was accomplished by comparing the results with the available laboratory as well as quantum mechanics results.     

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

陶瓷/金属复合靶板受可变形弹体撞击的理论分析模型

针对弹体撞击陶瓷/金属复合靶板的问题,将弹体的变形、陶瓷面板的碎裂和金属背板的变形结合起来,建立了新的可变形弹体垂直撞击陶瓷/金属靶板的理论分析模型．模型中计入了弹体刚性区长度和运动速度、塑性变形区长度、横截面积和运动速度的变化以及弹体对靶板的侵入速度和深度;对陶瓷面板考虑了陶瓷锥体积和抗压强度的变化;对金属背板的变形,根据其塑性变形功、外力功及其动能守恒原理,得到金属背板的运动方程．最后对具体算例进行了分析,得到了各物理量随时间的变化,给出了一些有价值的规律．结果表明,模型能较好地描述撞击过程中的有关规律;与实验结果和数值模拟结果进行对比,吻合较好,说明了模型的有效性．     

Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates

In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number R₀. It is shown that, if R₀?1, then the disease free equilibrium is globally asymptotically stable and the disease dies out; if R₀>1, there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results.     

Detection of alkoxy-complexes using visual indicators in non-aqueous solvents

The titrations of metal chlorides and alkali metal alkoxides have been carried out in anhydrous non-aqueous solvent using visual indicators. Basic indicators such as thymolphthalein, thymol blue,m-cresol purple and phenolphthalein have been found to be suitable in detection of double alkoxides. By contrast, the acidic indicators,e.g., bromo-cresol purple, bromo-cresol green, bromo-phenol blue and bromo-thymol blue were able to detect simple alkoxides only. The formation of a number of alkoxy complexes of the type M [M′ (OR)₄], M [M″₂ (OR) ⁱ ] and M′ (OR)₃ (where M=Li, Na or K; M′=Al, Ga or Fe; M″=Sn; R=Me, Et, or Pr ⁱ ) has been detected.     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Consensus formation of two-level opinion dynamics

Opinion dynamics have received significant attention in recent years. This paper proposes a bounded confidence opinion model for a group of agents with two different confidence levels. Each agent in the population is endowed with a confidence interval around her opinion with radius αd or (1-α)d, where α ∈ (0,1/2] represents the differentiation of confidence levels. We analytically derived the critical confidence bound d_c = 1/(4α) for the two-level opinion dynamics on ?. A single opinion cluster is formed with probability 1 above this critical value regardless of the ratio p of agents with high/low confidence. Extensive numerical simulations are performed to illustrate our theoretical results. Noticed is a clear impact of p on the collective behavior: more agents with high confidence lead to harder agreement. It is also experimentally revealed that the sharpness of the threshold d_c increases with α but does not depend on p.     

Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient

A diffusive epidemic model for H1N1 influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. Mortality rate for infective individuals in the population is assumed to be greater than natural mortality rate. Latent, infectious and immune periods are assumed to be constants throughout this study. The numerical solutions of this model are carried out under three different initial populations distribution. In order to investigate the effect of the disease transmission coefficient on the spread of disease, β is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x  . The threshold quantity (_R0)$({\mathfrak{R}}_0)$ that governs the disease dynamics is derived. Numerical simulation shows that the system supports the existence of sustained and damped oscillations depending on initial populations distribution, the disease transmission rate and diffusion.     

Complex dynamics in a prey predator system with multiple delays

The complex dynamics is explored in a prey predator system with multiple delays. Holling type-II functional response is assumed for prey dynamics. The predator dynamics is governed by modified Leslie-Gower scheme. The existence of periodic solutions via Hopf-bifurcation with respect to both delays are established. An algorithm is developed for drawing two-parametric bifurcation diagram with respect to two delays. The domain of stability with respect to τ₁ and τ₂ is thus obtained. The complex dynamical behavior of the system outside the domain of stability is evident from the exhaustive numerical simulation. Direction and stability of periodic solutions are also determined using normal form theory and center manifold argument.     

A combined molecular dynamics-phase-field modelling approach to fracture

In order to better understand and ease the determination of material and model parameters required for the macroscopic modelling of brittle fracture, a bottom-up comparative study between molecular dynamics (MD) simulations and the continuum phase-field modelling (PFM) is carried out. In particular, based on the MD simulations of fracture of a highly brittle material, a number of PFM parameters such as the width of the transition zone between the damaged and the undamaged material, the crack resistance and the elasticity modulus are estimated. This study opens the door for an efficient way for multi-scale modelling of fracture. To illustrate this approach, a comparative two-dimensional numerical initial-boundary-value problem (IBVP) for the highly brittle aragonite (CaCO₃) is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

X-ray spectroscopic study of chemical bonding in MnSe2 and CoSe2

X-ray absorption study of two intermetallic compounds MnSe₂ and CoSe₂ has been carried out using a Cauchois type bent crystal spectrograph. The metal K absorption edges in both the compounds are found to shift toward the high energy side with respect to the discontinuities in the pure metals. On the other hand, the Se K absorption edge in both these compounds is found to shift toward the lower energies. Emission study of the compound MnSe₂ has shown that the Mn Kβ ₅ band in this compound is shifted toward the high energy side with respect to that in the pure metal. From the magnetic data and the results obtained in this work it is possible to obtain the chemical bonding pictures in these compounds. For MnSe₂ the bondings sp ³ d ² for the metal atom andsp ³ for the metalloid atom have been suggested. For CoSe₂ the bondings appear to bed ² sp ³ for the metal atom andsp ³ for the metalloid atom. These bondings are compatible with the pyrite type structure of these compounds. It is possible to explain the electrical behaviour of the compounds on the basis of these bonding pictures.     

Mathematical analysis of a virus dynamics model with general incidence rate and cure rate

The rate of infection in many virus dynamics models is assumed to be bilinear in the virus and uninfected target cells. In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and cure rate is studied. Global dynamics of the model is established. We prove that the virus is cleared and the disease dies out if the basic reproduction number R₀≤1 while the virus persists in the host and the infection becomes endemic if R₀>1.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.