Narrow Results

Understanding the impact of human behavior on the spread of infectious diseases might be the key to developing better control strategies. Tuberculosis (TB) is an infectious disease caused by bacteria that mostly affects the lungs. TB remains a global health issue due to its high mortality. The paper proposes a spatiotemporal discrete tuberculosis model, based on the assumption that individuals can be classified as susceptible, exposed, infected, and recovered (SEIR). The objective of this work is to introduce a strategy of control that will reduce the number of exposed and infected individuals. Three controls are established to accomplish this. The first control is a public awareness campaign that will educate the public on the signs, symptoms, and treatments of tuberculosis, allowing them to seek treatment if they are at risk. The second control initiates chemoprophylaxis efforts for people who are latently infected, and the third control characterizes the treatment effort for people who are actively infected. We have shown the existence of optimal controls to give a characterization of controls in terms of states and adjoint functions by using Pontryagin’s maximum principle. Using numerical simulations, our results indicate that awareness campaigns should be combined with treatment and chemoprophylaxis techniques to reduce transmission. As a result, it demonstrates the efficacy of the suggested control strategies in reducing the impact of the disease.

We investigate a novel (d + 1)-dimensional discrete erosion model for d = 1, 2 and 3. The dynamics of the model is controlled by the physically motivated erosion mechanism. The coarse grained nature of this erosion process has been well compared with the Kardar–Parisi–Zhang (KPZ) equation. The kinetic roughening of the discrete model shows the same scaling behavior as that of the KPZ equation in the dimensions d = 1, 2. Moreover, in this present discrete model in (3 + 1)-dimension almost smooth interface has been obtained with vanishingly small roughness exponent, indicating the model belongs to the weak coupling regime of KPZ universality class.

In this present numerical work, we report a discrete erosion kind of model in (1 + 1)-dimension. Erosion and re-deposition phenomena with probabilities p and q(= 1 - p) are considered as two tunable parameters, which control the overall kinetic roughening behavior of the interface. Redeposition or diffusion dominated erosion like kinetic roughening model gives rise to nonuniversal growth exponent, which varies continuously with respect to erosion probability. However, universal character is restored for the roughness exponent with the value of 0.5 in (1 + 1)-dimension with respect to p. Due to nonuniversal nature of growth exponent, we observe a significant modification to the scaling behavior of surface width with respect to erosion probability. For low erosion probability (≲ 0.1) a power law like divergence has been observed of the correlation growth time. This can be argued as limiting behavior of a generalized functional behavior of crossover time with erosion probability.

A two-dimensional lattice-Boltzmann model with a hexagonal lattice is developed to simulate a boiling two-phase flow microscopically. Liquid-gas phase transition and bubble dynamics, including bubble formation, growth and deformation, are modeled by using an interparticle potential based on the van der Waals equation of state. Thermohydrodynamics is incorporated into the model by adding extra velocities to define temperature. The lattice-Boltzmann model is solved by a finite difference scheme so that numerical stability can be ensured at large discontinuity across the liquid-gas phase boundary and the narrow phase interface thickness can be attained. It is shown from numerical simulations that the model has the ability to reproduce phase transition, bubble dynamics and thermohydrodynamics while assuring numerical instability and narrow phase interface.

In this paper, we propose an exploited single-species discrete population model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or harvesting effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, pitchfork and tangent bifurcation, nonunique dynamics (meaning that several attractors or attractor and chaos coexist), basins of attraction and attractor crisis. This suggests that birth pulse provides a natural period or cyclicity that makes the dynamical behaviors more complex. Moreover, we show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulses, the population can sustain much higher harvesting effort if the mature fish is removed as early in the season as possible.

Severe fever with thrombocytopenia syndrome (SFTS) is an acute tick-borne disease caused by SFTS virus (SFTSV). In this paper, we use difference equations to establish a discrete tick-borne disease model with systemic and transovarial transmission. Using the method of the next generation of matrix, we get the basic reproduction number $R_0$ to determine whether SFTS will die out. Furthermore, we analyze the existence and stability of equilibrium points by $R_0$. In addition, the transcritical bifurcation property at the disease-free equilibrium point is discussed by deriving a equation describing the flow on the center manifold. Finally, we perform numerical simulations to verify the theoretical results.

Continuum models for granular flow generally give rise to systems of nonlinear partial differential equations that are linearly ill-posed. In this paper we introduce discreteness into an elastoplasticity model for granular flow by approximating spatial derivatives with finite differences. The resulting ordinary differential equations have bounded solutions for all time, a consequence of both discreteness and nonlinearity.

We study how the large-time behavior of solutions in this model depends on an elastic shear modulus ℰ. For large and moderate values of ℰ, the model has stable steady-state solutions with uniform shearing except for one shear band; almost all solutions tend to one of these as t→∞. However, when ℰ becomes sufficiently small, the single-shear-band solutions lose stability through a Hopf bifurcation. The value of ℰ at the bifurcation point is proportional to the ratio of the mesh size to the macroscopic length scale. These conclusions are established analytically through a careful estimation of the eigenvalues. In numerical simulations we find that: (i) after stability is lost, time-periodic solutions appear, containing both elastic and plastic waves, and (ii) the bifurcation diagram representing these solutions exhibits bi-stability.

Presented herein is a matrix method for buckling analysis of general frames based on the Hencky bar-chain model comprising of rigid segments connected by hinges with elastic rotational springs. Unlike the conventional matrix method of structural analysis based on the Euler–Bernoulli beam theory, the Hencky bar-chain model (HBM) matrix method allows one to readily handle the localized changes in end restraint conditions or localized structural changes (such as local damage or local stiffening) by simply tweaking the spring stiffnesses. The developed HBM matrix method was applied to solve some illustrative example problems to demonstrate its versatility in solving the buckling problem of beams and frames with various boundary conditions and local changes. It is hoped that this easy-to-code HBM matrix method will be useful to engineers in solving frame buckling problems.

Methods of comparing discrete and continuous cable models of single neurons and dynamical phenomena observed in neurobiology can be described with infinite-coupled systems of semilinear parabolic differential-functional equations of the reaction-diffusion-convection type or infinite systems of ordinary integro-differential equations. It is known that numerous problems in computational neuroscience use finite systems of equations based on the so-called compartmental model. It seems a natural idea to extend the results obtained in the theory of finite systems onto infinite systems. However, this requires stringent assumptions to be adopted to achieve compatibility. In most instances the dynamics of infinite systems behave differently to their finite-dimensional projections. The truncation method applied to infinite systems of equations and presented herein yields a truncated system consisting of the first N equations of the infinite system in N unknown functions. A solution of infinite system is defined as the limit when N → ∞ of the sequence of approximations {z_N}_N=1,2,…, where are defined as solutions of suitable finite truncated systems with corresponding initial-boundary conditions. Geometrically, it may be described as the projection of an infinite system of differential equations considered in a function abstract space of infinite dimension (such as Banach or Hilbert space) onto its finite-dimensional subspaces.

Recently, the description of immune response by discrete models has emerged to play an important role to study the problems in the area of human immunodeficiency virus type 1 (HIV-1) infection, leading to AIDS. As infection of target immune cells by HIV-1 mainly takes place in the lymphoid tissue, cellular automata (CA) models thus represent a significant step in understanding when the infected population is dispersed. Motivated by these, the studies of the dynamics of HIV-1 infection using CA in memory have been presented to recognize how CA have been developed for HIV-1 dynamics, which issues have been studied already and which issues still are objectives in future studies.

A nonclassical multiscale "morphological approach" devoted to highly-filled particulate composites was proposed by Nadot et al.¹ It is extended by developing complementary ingredients in order to describe the evolution of damage by decohesion of grain/matrix interfaces. A defect nucleation criterion allows the evolution of the number of defects to be followed, and a closure criterion monitors the respective proportions of open and closed defects. The nucleation criterion is associated with a critical value of the current distance between two points located on both sides of the interface considered, whereas the closure criterion is based on the normal component of the displacement jump across the interface. The "morphological approach" is tested at macroscopic and local scales for a random particulate microstructure under loading paths involving tension and compression. The corresponding results are a specific contribution of this work. At global scale, the numerical results presented show the effects of damage induced anisotropy and recovery of mechanical properties by defects closure. Thanks to explicit schematization of the microstructure, the discrete model provides, at microscale, the characteristics of damage (chronology of events, position and morphology of defects) as well as its effects on local strains.

A discrete time periodic n-species Lotka–Volterra type competitive model with delays is investigated. By using Gaines and Mawhin's continuation theorem based on the coincidence degree theory, a new sufficient condition on the existence of positive periodic solutions of the model is established.

Epidemic dynamics in networks have attracted a great deal of attention from researchers of many fields. In this paper, we mainly study the global behaviors of discrete-time epidemic model in heterogenous networks. By theoretical analysis, we show that the model can be characterized by the basic reproduction number R₀. When R₀ is smaller than unit, the disease-free equilibrium is globally stable, while R₀ is larger than unit, the unique positive equilibrium is globally attractive.

In this paper, we formulate a discrete two-stage model with two types of birth mechanisms, continuous and seasonal. We divide tick population into two subgroups, i.e. immature and mature. We also consider diapause as an important adaptive process for ticks in respond to climate change in both stages. We derive a formula for the inherent net reproductive numbers and explore their stability analysis. When breeding is continuous, there exists a unique globally asymptotically stable positive fixed point provided that the inherent net reproductive number is larger than one; the extinction fixed point is globally asymptotically stable if the inherent net reproductive number is less than one. When breeding is seasonal with 2-periodic birth function, there exists a unique globally asymptotically stable periodic solution provided that the inherent net reproductive number is larger than one, while the population goes to extinction if this value is less than one. For the cases with multi-periodic birth function, we use numerical simulation to compare their complex behaviors.

Phytoplanktons are drifting plants in an aquatic system. They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis. Zooplanktons are drifting animals found inside the aquatic bodies. For stable aquatic ecosystem, the growth of both Zooplankton and Phytoplankton should be in steady state but in previous eras, there has been a universal explosion in destructive Plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So, in this paper, a discretized two-dimensional Phytoplankton–Zooplankton model is investigated. The results for the existence and uniqueness, and conditions for local stability with topological classifications of the equilibrium solutions are determined. It is also exhibited that at trivial and semitrivial equilibrium solutions, discrete model does not undergo flip bifurcation, but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution under certain conditions. Further, state feedback method is deployed to control the chaos in the under consideration system. The extensive numerical simulations are provided to demonstrate theoretical results.

In this paper, a set of non-standard discrete models were constructed for the solution of non-homogenous second-order ordinary differential equation. We applied the method of non-local approximation and renormalization of the discretization functions to some problems and the result shows that the schemes behave qualitatively like the original equation.

A superintegrable finite model of the oscillator in two-dimensions is presented. It is defined on a uniform lattice of triangular shape. The wavefunctions are expressed in terms of bivariate Krawtchouk polynomials. The constants of motion form an SU(2) symmetry algebra.